pc5271AY2526wiki - User contributions [en]

Inductive Sensors of Ultra-high Sensitivity Based on Nonlinear Exceptional Point

2026-04-23T06:32:25Z

Siyu: /* Result display: steady oscillation */

== Introduction ==
This project is a implementation of the experiment reported in ''Observation of Nonlinear Exceptional Points with a Complete Basis in Dynamics'', adapted to our available laboratory equipment.<ref name="Bai2024">K. Bai, T.-R. Liu, L. Fang, J.-Z. Li, C. Lin, D. Wan, and M. Xiao, Observation of nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''132''', 073802 (2024).</ref> The circuit contains two coupled resonators. One resonator has gain, meaning that it can supply energy to the oscillation, while the other has loss, meaning that it dissipates energy. A system without energy conservation is called a '''non-Hermitian''' system. Such systems can show unusual degeneracies called '''exceptional points''' (EPs), where both the eigenfrequencies and the eigenmodes merge.

Exceptional points are interesting for sensing because the measured frequency can respond very strongly to a small perturbation. In an inductive sensor, the perturbation is a change in inductance. If a small change in inductance produces a large change in oscillation frequency, the circuit can in principle detect small physical changes. However, ordinary linear EPs also suffer from strong eigenmode nonorthogonality and a defective basis, which can increase noise sensitivity and complicate practical sensing.<ref name="Bai2024" />

The system studied in this project belongs to the recently developed class of '''nonlinear exceptional points''' (NEPs). The physical platform is a pair of coupled circuit resonators. Resonator <math>A</math> is made active by a saturable negative-resistance branch, while resonator <math>B</math> is intentionally lossy. Because the gain in resonator <math>A</math> depends on the oscillation amplitude, the steady-state problem is nonlinear and admits multiple self-consistent branches. At the singular operating point reported in Ref. [1], one stable steady branch and two auxiliary steady branches merge into a third-order nonlinear exceptional point, denoted <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM">Supplemental Material for K. Bai, T.-R. Liu, L. Fang, J.-Z. Li, C. Lin, D. Wan, and M. Xiao, Observation of nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''132''', 073802 (2024).</ref>

The experimentally important consequence is that the oscillation eigenfrequency obeys a cubic-root perturbation law near the singularity,

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3},
</math>

where <math>\delta L_A = L_A-L_{A,\mathrm{NEP}}</math> is the inductive detuning from the exceptional operating point. This response is steeper than the ordinary linear dependence of a conventional resonator and is therefore useful for ultra-sensitive inductive sensing.<ref name="Bai2024" /><ref name="Bai2024SM" />

A second and equally important point is that the cited work does not interpret the singularity as a conventional defective linear EP. Instead, the effective steady-state Hamiltonian remains non-defective in dynamics, and the associated Petermann factor stays finite at the operating point. In other words, the experiment realizes enhanced spectral responsivity without inheriting the strongest basis-collapse pathology of a standard linear EP.<ref name="Bai2024" /><ref name="Bai2024SM" />

In this project, our goal is to reproduce the experimental logic of Ref. [1] and its supplementary material under course-lab conditions. Unlike the PCB implementation in the paper, our circuit is assembled on a breadboard and the active branch uses a '''TL071''' operational amplifier. Our main observable is the steady oscillation frequency <math>f</math> as a function of the chosen inductance <math>L_A</math>. We tune <math>L_A</math> in two ways. For the main <math>f-L_A</math> sweep, we join small inductors in series so that the discrete inductance step is about <math>2~\mu\mathrm{H}</math>. In addition, we use a homebrew compressible coil to demonstrate the sensing mechanism directly. Compressing the coil changes its length and turn density while keeping the cross-sectional area approximately unchanged. We therefore use the <math>f-L_A</math> behavior in Fig. S11(a) of the supplementary material as the main experimental benchmark. Reproducing this trend is sufficient for demonstrating that our setup captures the intended nonlinear voltage-activated, self-stabilized oscillating response, although it is not a full reproduction of every diagnostic reported in Ref. [1].<ref name="Bai2024" /><ref name="Bai2024SM" />

== Team Members ==
Wang Peikun — E1538091@u.nus.edu 
Zhu Ziyang — E1583446@u.nus.edu 
Yuan Si-Yu — siyu_yuan@u.nus.edu 
Li Xunyu — xunyu@u.nus.edu

== Background Knowledge ==

=== Ordinary LC and RLC resonators ===
The basic building block of the experiment is an electrical resonator. An ideal LC resonator contains an inductor <math>L</math> and a capacitor <math>C</math>. Energy repeatedly moves between the magnetic field of the inductor and the electric field of the capacitor. The natural angular frequency is

<math display="block">
\omega_0=\frac{1}{\sqrt{LC}},
</math>

or, in ordinary frequency units,

<math display="block">
f_0=\frac{1}{2\pi\sqrt{LC}}.
</math>

This formula is important for the whole project because it explains why changing an inductance changes the measured frequency. A larger <math>L</math> gives a lower resonance frequency, while a smaller <math>L</math> gives a higher resonance frequency.

Real circuits are not ideal. Wires, inductors, capacitors, breadboard contacts, and measuring probes all dissipate energy. This loss is often represented by a resistor <math>R</math>, giving an RLC resonator. In a passive RLC circuit, the oscillation amplitude decays with time because energy is continuously lost as heat. If a circuit is to maintain a steady activated oscillation, the loss must be compensated by a gain element.

=== Gain, loss, and negative resistance ===
In electronics, gain can be represented as an effective negative resistance. A normal positive resistance removes energy from the resonator. A negative resistance does the opposite: it supplies energy to the resonator. In this experiment, resonator <math>A</math> contains an active op-amp branch that behaves like a negative resistance at small amplitude, while resonator <math>B</math> contains ordinary positive loss.

The gain cannot remain negative and constant forever. If it did, any activated oscillation would be amplified without bound until the op-amp clipped or the circuit failed. The important nonlinear ingredient is '''saturable gain''': the active branch gives strong gain at small amplitude but weaker gain at larger amplitude. The stable oscillation amplitude is selected when the supplied gain balances the total loss. This nonlinear steady state is why the experiment is a nonlinear dynamical problem rather than only a passive resonance measurement.

=== What "non-Hermitian" means in this circuit ===
In ordinary conservative physics, a Hermitian Hamiltonian describes a closed system whose total energy is conserved. Its eigenfrequencies are real, and its eigenmodes form a complete orthogonal basis. This is the familiar setting of many quantum-mechanics courses.

A circuit with gain and loss is an open system. Energy can enter through the op-amp branch and leave through resistive loss. The effective dynamical matrix is therefore generally '''non-Hermitian'''. In a non-Hermitian system, eigenfrequencies can be complex:

<math display="block">
\omega=\omega_\mathrm{r}+i\omega_\mathrm{i}.
</math>

The real part <math>\omega_\mathrm{r}</math> gives the oscillation frequency, while the imaginary part <math>\omega_\mathrm{i}</math> describes amplification or decay. A positive imaginary part means the mode is amplified, a negative imaginary part means it decays, and a stable steady oscillation corresponds to zero imaginary part, or equivalently zero net amplification. This is the sense in which the circuit is self-stabilized after voltage activation.

=== Exceptional points ===
In a usual Hermitian system, two modes can have the same eigenfrequency while still remaining two independent modes. This is an ordinary degeneracy. A non-Hermitian system can show a stronger kind of degeneracy called an '''exceptional point''' (EP). At an EP, not only the eigenvalues but also the eigenvectors coalesce. In simple language, two modes collapse into one mode.

This collapse makes EPs interesting for sensing. Near a second-order linear EP, a small perturbation can produce a square-root frequency splitting:

<math display="block">
\Delta f \propto \sqrt{\epsilon},
</math>

where <math>\epsilon</math> is the perturbation. This response is steeper than an ordinary linear response when <math>\epsilon</math> is very small. However, a conventional linear EP also has a serious drawback: the eigenbasis becomes defective, meaning that the system no longer has enough independent eigenvectors. This is associated with strong modal nonorthogonality and enhanced noise sensitivity. The Petermann factor is one common way to quantify this nonorthogonality.

=== Why the nonlinear exceptional point is different ===
The reference paper studies a nonlinear exceptional point. The word "nonlinear" is essential. The circuit does not have a fixed gain coefficient; instead, the gain depends on the oscillation amplitude. The steady state must therefore be solved self-consistently: the frequency, voltage amplitudes, and saturated gain must all agree with one another.

In the theory of Ref. [1], this nonlinear steady-state problem can reduce to a cubic equation for the oscillation frequency. A cubic equation can have a triple root. At the special operating point, one stable physical branch and two auxiliary branches merge, giving a third-order nonlinear exceptional point, written as <math>\mathrm{NEP}_3</math>. The stable physical branch is the one with real oscillation frequency, i.e. without an imaginary part. The auxiliary branches are needed to define the higher-order coalescence mathematically, but they are not physically stable operating states of the circuit. The corresponding frequency response has the form

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3}.
</math>

This cubic-root response is the key sensing signature. It is steeper than both an ordinary linear response and the square-root response of a second-order EP. The important conceptual point of Ref. [1] is that this enhanced response is obtained while the effective steady-state Hamiltonian can still retain a complete basis in dynamics.<ref name="Bai2024" /><ref name="Bai2024SM" />

=== Sensing mechanism ===
The circuit can be understood as a frequency-generating sensor. It converts an inductance change into a frequency change.

The first step is activation. A voltage is supplied to the resonators to excite the circuit into an oscillating state. The active branch in resonator <math>A</math> then supplies energy through an effective negative resistance and amplifies the selected oscillation mode.

The second step is self-stabilization. The gain is saturable, which means it becomes weaker when the voltage amplitude becomes larger. The activated oscillation therefore is not amplified forever. The circuit settles onto the physical steady branch where the imaginary part of the eigenfrequency is zero. At that point, the gain supplied by resonator <math>A</math> balances the loss in the circuit, especially the loss in resonator <math>B</math>.

The third step is frequency selection. Because resonators <math>A</math> and <math>B</math> are coupled, the final oscillation frequency is not simply the resonance frequency of either isolated resonator. It is selected by the whole coupled circuit: the inductors, capacitors, coupling elements, loss, and saturated gain all contribute.

The fourth step is sensing. The inductance <math>L_A</math> controls the bare frequency of resonator <math>A</math>. When <math>L_A</math> changes, the self-consistent frequency selected by the nonlinear circuit also changes. Therefore, tracking the spectral peak <math>f</math> tells us how the effective inductance has changed.

The fifth step is the nonlinear exceptional-point response. Near the nonlinear exceptional point discussed in Ref. [1], the physical steady branch and the two auxiliary branches coalesce in the nonlinear steady-state equation. The frequency shift of the stable branch can then scale as the cube root of the inductance perturbation. This means that a small inductance change can produce a comparatively large frequency change. Our breadboard experiment focuses on verifying the stable <math>f-L_A</math> branch corresponding to Fig. S11(a) of SM-4, which is the experimentally accessible part of this sensing mechanism.

=== Same mechanism in other platforms ===
Several references are included to show that this is a platform-independent sensing mechanism, not only a feature of our breadboard circuit. The common chain is

<math display="block">
\mathrm{physical\ perturbation}
\longrightarrow
\mathrm{parameter\ change}
\longrightarrow
\mathrm{eigenfrequency\ shift\ of\ a\ resonator/cavity\ with\ gain\ and\ loss}
\longrightarrow
\mathrm{higher\ order\ steady\ state\ response}
\longrightarrow
\mathrm{sensing}.
</math>

In our circuit, the physical perturbation is a change in the inductance <math>L_A</math>, and the measured output is the oscillation frequency <math>f</math>. In coupled optical, mechanical, electrical, or hybrid resonators, the perturbed parameter may instead be a cavity detuning, mechanical frequency, phase shift, magnetic-field-dependent transition, or another effective resonator parameter. The shared point is that gain, loss, coupling, and nonlinearity convert the parameter change into a higher-order steady-state spectral response.

The earlier theoretical work on nonlinear exceptional points showed that nonlinear coupled resonators can host higher-order NEPs while retaining a complete dynamical basis.<ref name="Bai2023">K. Bai, J.-Z. Li, T.-R. Liu, L. Fang, D. Wan, and M. Xiao, Nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''130''', 266901 (2023).</ref> The 2024 experiment implemented the same mechanism in coupled electrical resonators and measured the stable nonlinear frequency branch.<ref name="Bai2024" /><ref name="Bai2024SM" /> Silicon micromechanical resonators provide another realization: there, the measured output can be a phase difference, and the nonlinear exceptional-point response can enhance phase-shift sensing.<ref name="Xie2026">Y.-J. Xie, S.-Y. Zhang, M.-N. Zhang, R. Wang, L.-F. Wang, and Q.-A. Huang, Nonlinear exceptional points boost phase-shift sensing in silicon micromechanical resonators, ''Phys. Rev. Applied'' '''25''', 034039 (2026).</ref> Hybrid quantum systems with nonlinear bistability show a related route, where nonlinear transition dynamics can enhance sensing signal-to-noise ratio.<ref name="Wang2026">H. Wang ''et al.'', Exceptional sensitivity near the bistable transition point of a hybrid quantum system, ''Nature Physics'' '''22''', 577-583 (2026).</ref>

These examples support the same mechanism rather than unrelated applications. At the same time, enhanced response alone is not enough to prove better sensing. Noise can shift or weaken the apparent nonlinear exceptional point and can limit the signal-to-noise improvement.<ref name="Zheng2025">X. Zheng and Y. D. Chong, Noise constraints for nonlinear exceptional point sensing, ''Phys. Rev. Lett.'' '''134''', 133801 (2025).</ref> This is why our project makes a careful claim: we demonstrate the frequency-response mechanism on a breadboard, but we do not claim a complete metrological noise advantage.

== Report objectives and scope ==
The purpose of this work is to study the circuit principle behind '''nonlinear-exceptional-point-enhanced inductive sensing''' in a coupled electronic resonator platform. The scope is intentionally narrower than the full experiment reported in Ref. [1]. We are not attempting to redesign the active branch, derive all nonlinear bifurcation theory from first principles, reconstruct the auxiliary steady-state branches, or characterize every parasitic element. Instead, we adopt the framework of Ref. [1] and SM-4 and ask whether the breadboard circuit can reproduce the experimentally accessible frequency response.

The main target of our experiment is the <math>f-L_A</math> curve corresponding to Fig. S11(a) of SM-4. In the reference experiment, the steady-state frequency, voltage ratio <math>|V_A/V_B|</math>, and relative phase <math>\theta_A-\theta_B</math> are all measured as functions of <math>L_A</math>.<ref name="Bai2024SM" /> In our implementation, the frequency-versus-inductance curve is the primary observable. We tune <math>L_A</math> by connecting small inductors in series with approximately <math>2~\mu\mathrm{H}</math> resolution. We also use a homebrew coil in a separate demonstration where mechanical compression or insertion of an iron rod changes the effective inductance. The experimental goal is therefore to verify that the TL071 breadboard circuit settles into a stable oscillating state and that the measured frequency changes with <math>L_A</math> in the same qualitative manner as Fig. S11(a).

From the sensing perspective, the most relevant figure of merit is the local frequency responsivity. If

<math display="block">
|f-f_{\mathrm{NEP}}| = A |\delta L_A|^{1/3},
</math>

then differentiation gives

<math display="block">
S_f \equiv \left|\frac{df}{dL_A}\right| \propto |\delta L_A|^{-2/3}.
</math>

Thus, as the operating point approaches <math>L_{A,\mathrm{NEP}}</math>, the local spectral responsivity increases rapidly. The main practical constraint is that the system must remain on the intended stable branch while measurement noise and slow parameter drift are kept below the scale of the induced frequency shift.

== Setups ==
[[File:Experimental_setup.png|750px|thumb|center|Experimental setup adapted from Ref. [1].]]

=== Circuit topology and component implementation ===
The reference platform consists of two coupled RLC resonators. Both resonators contain the same nominal shunt capacitance <math>C_0</math>, while the inductances <math>L_A</math> and <math>L_B</math> set their bare resonance frequencies. The two resonators are coupled through a parallel resistor-capacitor channel characterized by <math>R_c</math> and <math>C_c</math>. Resonator <math>B</math> contains an explicit positive resistance <math>R_B</math> and therefore plays the role of the lossy resonator. Resonator <math>A</math>, by contrast, contains an active branch whose effective resistance is negative at small amplitude and becomes less negative as the oscillation amplitude increases, thereby providing saturable gain.<ref name="Bai2024" /><ref name="Bai2024SM" />

Our implementation follows this topology but is built on a breadboard rather than a custom PCB. The active branch is implemented with an op-amp negative-resistance circuit using a '''TL071''' operational amplifier. This is an important practical difference from the supplementary experiment, where the active components explicitly listed are a TI '''LM7171''' operational amplifier and an '''Onsemi BAV99L''' diode.<ref name="Bai2024SM" /> The TL071 substitution and the breadboard layout introduce different bandwidth limits, wiring parasitics, and contact resistances, so the numerical operating point is not expected to match the PCB experiment. For this reason, our comparison emphasizes the observable trend of <math>f</math> versus <math>L_A</math>.

The supplementary material reports the following representative component values for the PCB experiment:

<math display="block">
C_0 = 18.5~\mathrm{nF}, \qquad
C_c = 3.9~\mathrm{nF}, \qquad
R_c = 760~\Omega,
</math>

<math display="block">
L_B = 197.7~\mu\mathrm{H}, \qquad
R_B = 1314~\Omega.
</math>

A representative operating point shown in the main figure uses

<math display="block">
L_A = 245.4~\mu\mathrm{H}.
</math>

Our breadboard circuit follows the same functional topology, but its component values and parasitic elements are not identical to the PCB values above. The horizontal axis of our main <math>f-L_A</math> plot is therefore the series-combined <math>L_A</math> selected from our available small inductors, while the measured frequency <math>f</math> is the robust experimental readout.

=== Inductance tuning methods ===
We use two complementary methods to change the inductance seen by resonator <math>A</math>.

The first method is used for the main <math>f-L_A</math> sweep. Small inductors are connected in series so that the total inductance changes in discrete steps of approximately <math>2~\mu\mathrm{H}</math>. If the inductors are sufficiently separated so that their mutual magnetic coupling is small, the total series inductance is approximately the sum of the individual inductances,

<math display="block">
L_A \approx L_{\mathrm{base}}+\sum_i L_i .
</math>

This method is simple and reproducible. Adding one small inductor increases <math>L_A</math>; removing it decreases <math>L_A</math>. The circuit frequency is then measured for each chosen value. This produces a discrete version of the <math>f-L_A</math> curve in Fig. S11(a) of SM-4. The step size is not as fine as a commercial variable inductor, but <math>2~\mu\mathrm{H}</math> spacing is small enough to observe the frequency trend over the accessible range.

The second method is a homebrew compressible variable inductor. This coil is useful because it makes the sensing mechanism visible: a mechanical displacement changes the coil geometry, which changes the inductance, which then changes the oscillation frequency. The basic idea can be understood from the approximate solenoid formula

<math display="block">
L \approx \mu_\mathrm{eff}\frac{N^2 A}{\ell},
</math>

where <math>N</math> is the number of turns, <math>A</math> is the cross-sectional area, <math>\ell</math> is the coil length, and <math>\mu_\mathrm{eff}</math> describes the effective magnetic permeability of the core and surrounding region.<ref name="Chen2014">S.-C. Chen, D.-K. Le, and V.-S. Nguyen, Inductive displacement sensors with a notch filter for an active magnetic bearing system, ''Sensors'' '''14''', 12640-12657 (2014).</ref> During compression, the number of turns <math>N</math> is unchanged and the cross-sectional area <math>A</math> remains approximately unchanged, but the length <math>\ell</math> decreases. Equivalently, the turn density

<math display="block">
n=\frac{N}{\ell}
</math>

increases. The magnetic field produced by a solenoid is proportional to turn density. Therefore, compressing the coil increases the magnetic flux linkage through the turns and increases the inductance. Stretching the coil has the opposite effect.

This mechanical tuning is not identical to the precision tunable inductor used in the reference experiment. A real compressed coil is not an ideal long solenoid: the spacing between turns is finite, the end effects change as the coil length changes, nearby wires and the breadboard add parasitic capacitance, and the leads contribute additional series inductance and resistance. For this reason, the homebrew inductor is best treated as an experimentally characterized tuning element rather than an ideal component whose value is known exactly from geometry. In the present project, it provides a direct mechanical demonstration of how displacement can perturb <math>L_A</math>.

Experimentally, we measured the effective inductance of this homebrew variable inductor at different compression states. The compression process changes the coil length and wire density while leaving the coil area approximately fixed. Therefore, the tuning mechanism is physically the same as changing <math>\ell</math> and <math>n=N/\ell</math> in the solenoid model, rather than changing the number of turns or the coil diameter.

=== Kirchhoff model of the nonlinear circuit ===
A convenient state-space description uses the node voltages <math>V_A(t)</math> and <math>V_B(t)</math> together with the inductor currents <math>I_A(t)</math> and <math>I_B(t)</math>. A node voltage is the voltage measured at one resonator node relative to ground. Kirchhoff's current law says that the algebraic sum of all currents flowing away from a node must be zero. This gives a direct way to translate the circuit diagram into differential equations.

Denoting the amplitude-dependent conductance of the active branch by <math>G_A(|V_A|)</math>, where <math>G_A<0</math> in the small-signal regime, the nodal equations can be written as

<math display="block">
\begin{aligned}
C_0 \frac{dV_A}{dt} + I_A + G_A(|V_A|)\,V_A + \frac{V_A-V_B}{R_c}
+ C_c\frac{d(V_A-V_B)}{dt} &= 0,\\
C_0 \frac{dV_B}{dt} + I_B + \frac{V_B}{R_B} + \frac{V_B-V_A}{R_c}
+ C_c\frac{d(V_B-V_A)}{dt} &= 0,\\
L_A \frac{dI_A}{dt} &= V_A,\\
L_B \frac{dI_B}{dt} &= V_B.
\end{aligned}
</math>

Each term has a simple circuit meaning. The terms <math>C_0 dV_A/dt</math> and <math>C_0 dV_B/dt</math> are capacitor currents. The terms <math>I_A</math> and <math>I_B</math> are currents through the inductors. The term <math>G_A(|V_A|)V_A</math> is the active branch current; because <math>G_A</math> can be negative, this branch can inject energy instead of dissipating it. The term <math>V_B/R_B</math> is the ordinary loss current through the resistor in resonator <math>B</math>. The terms involving <math>R_c</math> and <math>C_c</math> describe current flowing between resonators <math>A</math> and <math>B</math>, so they are the electrical coupling between the two resonators.

These equations show explicitly how the experiment differs from a linear EP circuit. If <math>G_A</math> were a constant negative conductance, the model would reduce to a linear non-Hermitian dimer. Here, however, <math>G_A</math> depends on the voltage amplitude, so the steady state must be determined self-consistently together with the oscillation amplitude itself.

=== Harmonic-balance formulation of the steady state ===
After the transient has died away, the oscilloscope waveform is approximately periodic. The simplest approximation is therefore to keep only the dominant sinusoidal component. This is called a harmonic-balance approximation. For a stable activated oscillating state, one writes

<math display="block">
V_j(t)=\mathrm{Re}\!\left[\tilde V_j e^{i\omega t}\right],
\qquad j=A,B.
</math>

Substituting this ansatz into the circuit equations yields a nonlinear algebraic problem of the form

<math display="block">
\mathbf{Y}(\omega,|\tilde V_A|)
\begin{pmatrix}
\tilde V_A\\
\tilde V_B
\end{pmatrix}
=
\begin{pmatrix}
Y_A(\omega,|\tilde V_A|)+Y_c(\omega) & -Y_c(\omega)\\
-Y_c(\omega) & Y_B(\omega)+Y_c(\omega)
\end{pmatrix}
\begin{pmatrix}
\tilde V_A\\
\tilde V_B
\end{pmatrix}
=0,
</math>

with

<math display="block">
Y_A(\omega,|\tilde V_A|)=G_A(|\tilde V_A|)+i\omega C_0+\frac{1}{i\omega L_A},
</math>

<math display="block">
Y_B(\omega)=\frac{1}{R_B}+i\omega C_0+\frac{1}{i\omega L_B},
\qquad
Y_c(\omega)=\frac{1}{R_c}+i\omega C_c.
</math>

A nontrivial solution requires

<math display="block">
\det \mathbf{Y}(\omega,|\tilde V_A|)=0.
</math>

This determinant condition is the circuit analogue of an eigenfrequency condition. In a passive LC resonator, the resonance condition gives a single frequency <math>\omega_0=1/\sqrt{LC}</math>. In the present two-resonator active circuit, the same idea becomes a matrix equation because the two resonators are coupled. The amplitudes <math>\tilde V_A</math> and <math>\tilde V_B</math> are not arbitrary: a nonzero oscillation exists only when the admittance matrix has a zero eigenvalue.

Because the active admittance depends on amplitude, this condition is nonlinear and supports multiple branches. The supplementary material shows that the steady-state frequency can be reduced to a '''real-coefficient cubic polynomial''' in <math>\omega</math>. The existence of up to three real steady-state branches is therefore not accidental; it is the algebraic origin of the experimentally relevant <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM" />

=== Effective nonlinear-Hamiltonian picture ===
The same physics is often rewritten in coupled-mode form. This notation is common in non-Hermitian physics because it makes the circuit look like two coupled modes. The variables <math>\psi_A</math> and <math>\psi_B</math> represent the complex oscillation amplitudes in resonators <math>A</math> and <math>B</math>. The model is

<math display="block">
i\frac{d}{dt}
\begin{pmatrix}
\psi_A\\
\psi_B
\end{pmatrix}
=
\begin{pmatrix}
\omega_A+i\,g(|\psi_A|) & \kappa\\
\kappa & \omega_B-i\ell
\end{pmatrix}
\begin{pmatrix}
\psi_A\\
\psi_B
\end{pmatrix},
</math>

where <math>\omega_A</math> and <math>\omega_B</math> are the effective modal frequencies, <math>\kappa</math> is the coupling strength, <math>\ell</math> is the loss of resonator <math>B</math>, and <math>g(|\psi_A|)</math> is the saturable gain of resonator <math>A</math>. The steady-state problem then becomes

<math display="block">
H_s(\Psi)\Psi = \Omega \Psi,
</math>

with a real oscillation eigenfrequency <math>\Omega=2\pi f</math> for the stable oscillating branch. The word "real" is important: it means the imaginary part has vanished, so the mode is neither being amplified nor decaying in the steady state. In the cited experiment, this physical stable branch and two auxiliary steady branches coalesce at the singular point, producing a third-order nonlinear exceptional point rather than a conventional second-order linear EP. The auxiliary branches help form the cubic-root singularity but are not themselves stable physical outputs of the circuit.<ref name="Bai2024" /><ref name="Bai2024SM" />

== Measurements ==

=== Acquisition protocol and steady-state selection ===
The measurement is performed in the time domain. In the reference experiment, an arbitrary waveform generator first applies an external start-up signal of 1 V at 70 kHz in order to bring the system into the basin of attraction of the stable oscillating state.<ref name="Bai2024" /> In our breadboard measurement, an external voltage is likewise required to activate the resonators. The measured frequency is extracted from the activated steady waveform recorded on the oscilloscope.

The physical meaning of this procedure is important. The oscillation is not assumed to appear without activation; a supplied voltage is needed to bring the resonators into the relevant dynamical state. The measured oscillation frequency is then interpreted as the frequency selected by the nonlinear steady-state condition.

=== Extraction of the <math>f-L_A</math> observable ===
Ref. [1] does not treat the oscilloscope traces merely as qualitative evidence of oscillation. The supplementary explicitly states that the full reference experiment reports the steady-state frequency, the voltage ratio <math>|V_A/V_B|</math>, and the relative phase <math>\theta_A-\theta_B</math>, all extracted from the dominant peaks of the Fourier spectra of <math>V_A</math> and <math>V_B</math>.<ref name="Bai2024SM" />

Our available observable is the steady-state frequency. For each chosen value of <math>L_A</math>, the resonators are voltage-activated, the circuit reaches a stable oscillating state, and the dominant spectral peak is recorded. If we denote the Fourier transforms by

<math display="block">
\tilde V_A(f)=\mathcal{F}\{V_A(t)\},
\qquad
\tilde V_B(f)=\mathcal{F}\{V_B(t)\},
</math>

then the dominant oscillation frequency is obtained from

<math display="block">
f_0=\arg\max_f |\tilde V_A(f)| \approx \arg\max_f |\tilde V_B(f)|.
</math>

The experimental data set is therefore

<math display="block">
\{L_A,\ f_0(L_A)\},
</math>

which is compared with the frequency branch plotted in Fig. S11(a) of SM-4. In the full reference measurement, the amplitude ratio

<math display="block">
\rho = \frac{|\tilde V_A(f_0)|}{|\tilde V_B(f_0)|}
</math>

and relative phase

<math display="block">
\Delta\phi = \arg \tilde V_A(f_0)-\arg \tilde V_B(f_0)
</math>

provide additional state verification. Those quantities are useful but are not the main basis of our breadboard validation.

In practice, the same frequency can also be estimated directly from the time-domain period <math>T</math> using <math>f_0=1/T</math> when the waveform is clean. Fourier analysis is preferred because it uses a longer section of waveform and identifies the dominant oscillation peak even when there is small amplitude noise or waveform distortion.

=== Practical inductance sweep ===
In SM-4, the inductive tuning is carried out by varying <math>L_A</math>. The supplementary notes that the tunable inductor exhibits less than about 0.4% inductance variation over the frequency range of interest, which is sufficiently small for controlled sweeping near the singular point. To further suppress voltage-dependent drift during the sweep, the reference experiment slightly adjusts the active-branch resistance so as to maintain approximately <math>V_B \approx 2~\mathrm{V}</math> throughout the measurement series.<ref name="Bai2024SM" />

Our sweep uses controlled series combinations of small inductors, giving an approximately <math>2~\mu\mathrm{H}</math> step in <math>L_A</math>. The homebrew compressible coil is then used to show that a mechanical change of the sensing element also changes the inductance. The conclusion drawn from our data is therefore not a precision determination of <math>L_{A,\mathrm{NEP}}</math>, but a validation that the breadboard circuit produces the same kind of frequency response as the stable branch in Fig. S11(a).

=== Long-window spectral analysis and noise inspection ===
The supplementary also discusses long-time measurements over a window of <math>0<t<5~\mathrm{s}</math>. The motivation is that the noise around the stable oscillation is too small to be judged reliably from raw scope traces alone, so it is more informative to inspect the Fourier spectra and the extracted spectral center frequency and linewidth.<ref name="Bai2024SM" />

This is a useful methodological point for a sensing-oriented reproduction. In an NEP-based sensor, one ultimately cares about how precisely the frequency peak can be resolved and tracked. A long-window Fourier analysis is therefore closer to the sensing task than a purely visual time-domain inspection.

== Results ==

=== Result display: steady oscillation ===
[[File:PC5271inductive1.png|750px|thumb|center|(a) Breadboard implementation, and (b)representative steady oscillation waveform.]]

This figure shows (a) our circuit implemented on a breadboard and (b) a representative stabilised voltage waveform after the resonators are activated by the supplied voltage. The displayed waveform is the evidence used to identify a stable oscillation frequency.

=== Result display: <math>f-L_A</math> sweep ===
[[File:PC5271inductive3.jpeg|750px|thumb|center|Measured breadboard <math>f-L_A</math> curve.]]

This figure shows the measured oscillation frequency <math>f</math> as a function of <math>L_A</math>. The values of <math>L_A</math> in the main sweep are produced by connecting small inductors in series with approximately <math>2~\mu\mathrm{H}</math> step size.

=== Result display: iron-rod demonstration ===
[[File:PC5271inductive2.png|750px|thumb|center|Displacement measurement by (a) compressing the solenoid and (b) inserting iron rod into the homebrew coil.]]

This figure exhibits the resultant changes in the measured oscillation frequency of homebrew coil with (a) a compression and (b) inserted iron rod. The experimental variable is the coil length and rod position, and the measured output is the activated steady oscillation frequency. As show in the analysis above, the inductance shows an inversely proportional dependence on the coil length and proportion dependence on insertion length, with the constant residual being the baseline inductance in the circuit irrelevant to the coil.

== Evaluation ==

=== Stable activated oscillation ===
The first required experimental signature is that, after the resonators are activated by the supplied voltage, the circuit reaches a stable oscillating waveform instead of decaying immediately or being amplified without bound. A stable waveform means that the TL071 active branch supplies enough small-signal negative resistance to compensate loss, while gain saturation prevents unbounded amplification.

Physically, this behavior confirms the role of the nonlinear active branch. At small amplitude, the effective negative resistance in resonator <math>A</math> overcomes dissipation. At larger amplitude, the gain becomes weaker, so the oscillation amplitude is clamped. The system self-stabilizes because the physical steady branch is the one with zero imaginary part of the eigenfrequency. The auxiliary branches near the nonlinear exceptional point are not stable physical outputs; they are mathematical branches of the nonlinear steady-state equation. The experiment is therefore not a passive ring-down measurement; it is a nonlinear steady-state selection process.

=== Evaluation of the <math>f-L_A</math> trend ===
The main experimental comparison is the measured <math>f-L_A</math> trend against Fig. S11(a) of SM-4. The reference experiment reports the stable frequency branch of the nonlinear circuit as <math>L_A</math> is tuned.<ref name="Bai2024SM" /> This branch is the physical one because its eigenfrequency is real in the steady state. In our breadboard experiment, the same observable is measured using the series-inductor sweep.

It is useful to separate two statements. First, any LC resonator changes frequency when <math>L</math> changes, because <math>f_0=1/(2\pi\sqrt{LC})</math>. This alone would not demonstrate the nonlinear exceptional-point mechanism. Second, in the coupled active-lossy circuit, the frequency is not just the bare LC resonance frequency of one isolated inductor and capacitor. It is the frequency selected by the nonlinear steady-state balance of gain, loss, and coupling. The reason Fig. S11(a) matters is that it shows this selected stable branch as <math>L_A</math> is swept.

Our breadboard circuit uses different component values and a different op-amp from the PCB experiment, so exact numerical overlap is not expected. The meaningful evaluation is whether the measured frequency branch follows the same qualitative <math>f-L_A</math> behavior as the stable branch in SM-4. Observing that behavior is enough to show that the constructed circuit realizes the intended nonlinear voltage-activated oscillating mechanism.

=== Evaluation of the displacement demonstration ===
In the iron-rod demonstration, the homebrew coil becomes a displacement-dependent inductive element. As the rod enters the coil, the effective magnetic permeability inside the coil increases. In the solenoid expression

<math display="block">
L(x) \approx \mu_\mathrm{eff}(x)\frac{N^2 A}{\ell},
</math>

the insertion depth <math>x</math> changes <math>\mu_\mathrm{eff}</math>. The coil geometry <math>N</math>, <math>A</math>, and <math>\ell</math> are approximately fixed during this demonstration, so the dominant effect is the change in magnetic core material inside the coil. Inductive displacement sensors commonly use this principle: the position of a ferromagnetic object changes a coil inductance, and the electronics convert that inductance change into an electrical signal.<ref name="Chen2014" />

In our circuit, the electrical signal is the activated steady oscillation frequency rather than a bridge voltage. The sensing chain is

<math display="block">
\mathrm{rod\ displacement}\ x
\longrightarrow
\mathrm{effective\ inductance}\ L_A(x)
\longrightarrow
\mathrm{oscillation\ frequency}\ f(x).
</math>

This demonstration shows how the inductive readout can be turned into a displacement-sensing readout after calibration of the relation between rod position and frequency.

=== Cubic-root perturbation law and sensing meaning ===
Near the singular operating point, the eigenfrequency shift follows the hallmark cubic-root law,

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3}.
</math>

This law is the main sensing-related result of the reference experiment.<ref name="Bai2024" /><ref name="Bai2024SM" /> Compared with the linear response of an ordinary isolated resonator, the cubic-root dependence can produce a larger frequency displacement for the same small perturbation of inductance when the device is biased sufficiently close to <math>L_{A,\mathrm{NEP}}</math>.

A useful way to interpret the result is through the local derivative,

<math display="block">
\left|\frac{df}{dL_A}\right| \propto |\delta L_A|^{-2/3}.
</math>

This increasing local responsivity is why the circuit is attractive for sensing. Our experiment supports this sensing logic by showing that the breadboard circuit has a measurable stable frequency branch as <math>L_A</math> is changed. A full quantitative exponent extraction would require identifying <math>L_{A,\mathrm{NEP}}</math> and <math>f_{\mathrm{NEP}}</math>, then plotting <math>|f-f_{\mathrm{NEP}}|</math> against <math>|L_A-L_{A,\mathrm{NEP}}|</math> on logarithmic axes.

=== Basis completeness and noise robustness ===
The supplementary material evaluates the Petermann factor as a diagnostic of modal nonorthogonality and basis completeness. For a non-Hermitian eigenproblem, one may write

<math display="block">
\mathrm{PF}_n =
\frac{\langle \Phi_n^{L}|\Phi_n^{L}\rangle \langle \Phi_n^{R}|\Phi_n^{R}\rangle}
{\left|\langle \Phi_n^{L}|\Phi_n^{R}\rangle\right|^2},
</math>

where <math>|\Phi_n^{R}\rangle</math> and <math>|\Phi_n^{L}\rangle</math> are the right and left eigenvectors of the effective steady-state Hamiltonian. At a conventional linear EP, this quantity is associated with diverging excess-noise sensitivity. In Ref. [1], the cited analysis finds that the effective steady-state Hamiltonian remains non-defective and the Petermann factor is finite at the <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM" />

This point explains the significance of the reference experiment. The system does not merely realize a large mathematical responsivity; it realizes that responsivity in a way that avoids the most severe basis-collapse problem of a linear EP. Our breadboard experiment does not directly test basis completeness, but the result provides the theoretical reason why reproducing the stable <math>f-L_A</math> branch is physically meaningful.

=== Overall interpretation ===
In this platform, the measurand is encoded as a perturbation of <math>L_A</math>. Any physical mechanism that modifies the effective inductance of resonator <math>A</math>, for example displacement, magnetic coupling, or the presence of a nearby magnetic target, moves the system away from the operating point and shifts the oscillation frequency. The primary readout in our implementation is therefore spectral: we track <math>f</math> as <math>L_A</math> is changed.

Taken together, Ref. [1] establishes the complete sensing logic: a nonlinear active-lossy circuit is tuned to an <math>\mathrm{NEP}_3</math>; after voltage activation, the system settles onto a stable oscillating branch; the dominant spectral peak is tracked as the perturbation readout; and the enhanced cubic-root response is obtained without the defective-basis pathology of a conventional linear EP. Our breadboard result demonstrates the experimentally accessible core of this logic by reproducing the stable frequency response corresponding to Fig. S11(a) of SM-4.

== References ==
<references />

Inductive Sensors of Ultra-high Sensitivity Based on Nonlinear Exceptional Point

2026-04-23T06:31:25Z

Siyu: /* Result display: iron-rod demonstration */

== Introduction ==
This project is a implementation of the experiment reported in ''Observation of Nonlinear Exceptional Points with a Complete Basis in Dynamics'', adapted to our available laboratory equipment.<ref name="Bai2024">K. Bai, T.-R. Liu, L. Fang, J.-Z. Li, C. Lin, D. Wan, and M. Xiao, Observation of nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''132''', 073802 (2024).</ref> The circuit contains two coupled resonators. One resonator has gain, meaning that it can supply energy to the oscillation, while the other has loss, meaning that it dissipates energy. A system without energy conservation is called a '''non-Hermitian''' system. Such systems can show unusual degeneracies called '''exceptional points''' (EPs), where both the eigenfrequencies and the eigenmodes merge.

Exceptional points are interesting for sensing because the measured frequency can respond very strongly to a small perturbation. In an inductive sensor, the perturbation is a change in inductance. If a small change in inductance produces a large change in oscillation frequency, the circuit can in principle detect small physical changes. However, ordinary linear EPs also suffer from strong eigenmode nonorthogonality and a defective basis, which can increase noise sensitivity and complicate practical sensing.<ref name="Bai2024" />

The system studied in this project belongs to the recently developed class of '''nonlinear exceptional points''' (NEPs). The physical platform is a pair of coupled circuit resonators. Resonator <math>A</math> is made active by a saturable negative-resistance branch, while resonator <math>B</math> is intentionally lossy. Because the gain in resonator <math>A</math> depends on the oscillation amplitude, the steady-state problem is nonlinear and admits multiple self-consistent branches. At the singular operating point reported in Ref. [1], one stable steady branch and two auxiliary steady branches merge into a third-order nonlinear exceptional point, denoted <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM">Supplemental Material for K. Bai, T.-R. Liu, L. Fang, J.-Z. Li, C. Lin, D. Wan, and M. Xiao, Observation of nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''132''', 073802 (2024).</ref>

The experimentally important consequence is that the oscillation eigenfrequency obeys a cubic-root perturbation law near the singularity,

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3},
</math>

where <math>\delta L_A = L_A-L_{A,\mathrm{NEP}}</math> is the inductive detuning from the exceptional operating point. This response is steeper than the ordinary linear dependence of a conventional resonator and is therefore useful for ultra-sensitive inductive sensing.<ref name="Bai2024" /><ref name="Bai2024SM" />

A second and equally important point is that the cited work does not interpret the singularity as a conventional defective linear EP. Instead, the effective steady-state Hamiltonian remains non-defective in dynamics, and the associated Petermann factor stays finite at the operating point. In other words, the experiment realizes enhanced spectral responsivity without inheriting the strongest basis-collapse pathology of a standard linear EP.<ref name="Bai2024" /><ref name="Bai2024SM" />

In this project, our goal is to reproduce the experimental logic of Ref. [1] and its supplementary material under course-lab conditions. Unlike the PCB implementation in the paper, our circuit is assembled on a breadboard and the active branch uses a '''TL071''' operational amplifier. Our main observable is the steady oscillation frequency <math>f</math> as a function of the chosen inductance <math>L_A</math>. We tune <math>L_A</math> in two ways. For the main <math>f-L_A</math> sweep, we join small inductors in series so that the discrete inductance step is about <math>2~\mu\mathrm{H}</math>. In addition, we use a homebrew compressible coil to demonstrate the sensing mechanism directly. Compressing the coil changes its length and turn density while keeping the cross-sectional area approximately unchanged. We therefore use the <math>f-L_A</math> behavior in Fig. S11(a) of the supplementary material as the main experimental benchmark. Reproducing this trend is sufficient for demonstrating that our setup captures the intended nonlinear voltage-activated, self-stabilized oscillating response, although it is not a full reproduction of every diagnostic reported in Ref. [1].<ref name="Bai2024" /><ref name="Bai2024SM" />

== Team Members ==
Wang Peikun — E1538091@u.nus.edu 
Zhu Ziyang — E1583446@u.nus.edu 
Yuan Si-Yu — siyu_yuan@u.nus.edu 
Li Xunyu — xunyu@u.nus.edu

== Background Knowledge ==

=== Ordinary LC and RLC resonators ===
The basic building block of the experiment is an electrical resonator. An ideal LC resonator contains an inductor <math>L</math> and a capacitor <math>C</math>. Energy repeatedly moves between the magnetic field of the inductor and the electric field of the capacitor. The natural angular frequency is

<math display="block">
\omega_0=\frac{1}{\sqrt{LC}},
</math>

or, in ordinary frequency units,

<math display="block">
f_0=\frac{1}{2\pi\sqrt{LC}}.
</math>

This formula is important for the whole project because it explains why changing an inductance changes the measured frequency. A larger <math>L</math> gives a lower resonance frequency, while a smaller <math>L</math> gives a higher resonance frequency.

Real circuits are not ideal. Wires, inductors, capacitors, breadboard contacts, and measuring probes all dissipate energy. This loss is often represented by a resistor <math>R</math>, giving an RLC resonator. In a passive RLC circuit, the oscillation amplitude decays with time because energy is continuously lost as heat. If a circuit is to maintain a steady activated oscillation, the loss must be compensated by a gain element.

=== Gain, loss, and negative resistance ===
In electronics, gain can be represented as an effective negative resistance. A normal positive resistance removes energy from the resonator. A negative resistance does the opposite: it supplies energy to the resonator. In this experiment, resonator <math>A</math> contains an active op-amp branch that behaves like a negative resistance at small amplitude, while resonator <math>B</math> contains ordinary positive loss.

The gain cannot remain negative and constant forever. If it did, any activated oscillation would be amplified without bound until the op-amp clipped or the circuit failed. The important nonlinear ingredient is '''saturable gain''': the active branch gives strong gain at small amplitude but weaker gain at larger amplitude. The stable oscillation amplitude is selected when the supplied gain balances the total loss. This nonlinear steady state is why the experiment is a nonlinear dynamical problem rather than only a passive resonance measurement.

=== What "non-Hermitian" means in this circuit ===
In ordinary conservative physics, a Hermitian Hamiltonian describes a closed system whose total energy is conserved. Its eigenfrequencies are real, and its eigenmodes form a complete orthogonal basis. This is the familiar setting of many quantum-mechanics courses.

A circuit with gain and loss is an open system. Energy can enter through the op-amp branch and leave through resistive loss. The effective dynamical matrix is therefore generally '''non-Hermitian'''. In a non-Hermitian system, eigenfrequencies can be complex:

<math display="block">
\omega=\omega_\mathrm{r}+i\omega_\mathrm{i}.
</math>

The real part <math>\omega_\mathrm{r}</math> gives the oscillation frequency, while the imaginary part <math>\omega_\mathrm{i}</math> describes amplification or decay. A positive imaginary part means the mode is amplified, a negative imaginary part means it decays, and a stable steady oscillation corresponds to zero imaginary part, or equivalently zero net amplification. This is the sense in which the circuit is self-stabilized after voltage activation.

=== Exceptional points ===
In a usual Hermitian system, two modes can have the same eigenfrequency while still remaining two independent modes. This is an ordinary degeneracy. A non-Hermitian system can show a stronger kind of degeneracy called an '''exceptional point''' (EP). At an EP, not only the eigenvalues but also the eigenvectors coalesce. In simple language, two modes collapse into one mode.

This collapse makes EPs interesting for sensing. Near a second-order linear EP, a small perturbation can produce a square-root frequency splitting:

<math display="block">
\Delta f \propto \sqrt{\epsilon},
</math>

where <math>\epsilon</math> is the perturbation. This response is steeper than an ordinary linear response when <math>\epsilon</math> is very small. However, a conventional linear EP also has a serious drawback: the eigenbasis becomes defective, meaning that the system no longer has enough independent eigenvectors. This is associated with strong modal nonorthogonality and enhanced noise sensitivity. The Petermann factor is one common way to quantify this nonorthogonality.

=== Why the nonlinear exceptional point is different ===
The reference paper studies a nonlinear exceptional point. The word "nonlinear" is essential. The circuit does not have a fixed gain coefficient; instead, the gain depends on the oscillation amplitude. The steady state must therefore be solved self-consistently: the frequency, voltage amplitudes, and saturated gain must all agree with one another.

In the theory of Ref. [1], this nonlinear steady-state problem can reduce to a cubic equation for the oscillation frequency. A cubic equation can have a triple root. At the special operating point, one stable physical branch and two auxiliary branches merge, giving a third-order nonlinear exceptional point, written as <math>\mathrm{NEP}_3</math>. The stable physical branch is the one with real oscillation frequency, i.e. without an imaginary part. The auxiliary branches are needed to define the higher-order coalescence mathematically, but they are not physically stable operating states of the circuit. The corresponding frequency response has the form

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3}.
</math>

This cubic-root response is the key sensing signature. It is steeper than both an ordinary linear response and the square-root response of a second-order EP. The important conceptual point of Ref. [1] is that this enhanced response is obtained while the effective steady-state Hamiltonian can still retain a complete basis in dynamics.<ref name="Bai2024" /><ref name="Bai2024SM" />

=== Sensing mechanism ===
The circuit can be understood as a frequency-generating sensor. It converts an inductance change into a frequency change.

The first step is activation. A voltage is supplied to the resonators to excite the circuit into an oscillating state. The active branch in resonator <math>A</math> then supplies energy through an effective negative resistance and amplifies the selected oscillation mode.

The second step is self-stabilization. The gain is saturable, which means it becomes weaker when the voltage amplitude becomes larger. The activated oscillation therefore is not amplified forever. The circuit settles onto the physical steady branch where the imaginary part of the eigenfrequency is zero. At that point, the gain supplied by resonator <math>A</math> balances the loss in the circuit, especially the loss in resonator <math>B</math>.

The third step is frequency selection. Because resonators <math>A</math> and <math>B</math> are coupled, the final oscillation frequency is not simply the resonance frequency of either isolated resonator. It is selected by the whole coupled circuit: the inductors, capacitors, coupling elements, loss, and saturated gain all contribute.

The fourth step is sensing. The inductance <math>L_A</math> controls the bare frequency of resonator <math>A</math>. When <math>L_A</math> changes, the self-consistent frequency selected by the nonlinear circuit also changes. Therefore, tracking the spectral peak <math>f</math> tells us how the effective inductance has changed.

The fifth step is the nonlinear exceptional-point response. Near the nonlinear exceptional point discussed in Ref. [1], the physical steady branch and the two auxiliary branches coalesce in the nonlinear steady-state equation. The frequency shift of the stable branch can then scale as the cube root of the inductance perturbation. This means that a small inductance change can produce a comparatively large frequency change. Our breadboard experiment focuses on verifying the stable <math>f-L_A</math> branch corresponding to Fig. S11(a) of SM-4, which is the experimentally accessible part of this sensing mechanism.

=== Same mechanism in other platforms ===
Several references are included to show that this is a platform-independent sensing mechanism, not only a feature of our breadboard circuit. The common chain is

<math display="block">
\mathrm{physical\ perturbation}
\longrightarrow
\mathrm{parameter\ change}
\longrightarrow
\mathrm{eigenfrequency\ shift\ of\ a\ resonator/cavity\ with\ gain\ and\ loss}
\longrightarrow
\mathrm{higher\ order\ steady\ state\ response}
\longrightarrow
\mathrm{sensing}.
</math>

In our circuit, the physical perturbation is a change in the inductance <math>L_A</math>, and the measured output is the oscillation frequency <math>f</math>. In coupled optical, mechanical, electrical, or hybrid resonators, the perturbed parameter may instead be a cavity detuning, mechanical frequency, phase shift, magnetic-field-dependent transition, or another effective resonator parameter. The shared point is that gain, loss, coupling, and nonlinearity convert the parameter change into a higher-order steady-state spectral response.

The earlier theoretical work on nonlinear exceptional points showed that nonlinear coupled resonators can host higher-order NEPs while retaining a complete dynamical basis.<ref name="Bai2023">K. Bai, J.-Z. Li, T.-R. Liu, L. Fang, D. Wan, and M. Xiao, Nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''130''', 266901 (2023).</ref> The 2024 experiment implemented the same mechanism in coupled electrical resonators and measured the stable nonlinear frequency branch.<ref name="Bai2024" /><ref name="Bai2024SM" /> Silicon micromechanical resonators provide another realization: there, the measured output can be a phase difference, and the nonlinear exceptional-point response can enhance phase-shift sensing.<ref name="Xie2026">Y.-J. Xie, S.-Y. Zhang, M.-N. Zhang, R. Wang, L.-F. Wang, and Q.-A. Huang, Nonlinear exceptional points boost phase-shift sensing in silicon micromechanical resonators, ''Phys. Rev. Applied'' '''25''', 034039 (2026).</ref> Hybrid quantum systems with nonlinear bistability show a related route, where nonlinear transition dynamics can enhance sensing signal-to-noise ratio.<ref name="Wang2026">H. Wang ''et al.'', Exceptional sensitivity near the bistable transition point of a hybrid quantum system, ''Nature Physics'' '''22''', 577-583 (2026).</ref>

These examples support the same mechanism rather than unrelated applications. At the same time, enhanced response alone is not enough to prove better sensing. Noise can shift or weaken the apparent nonlinear exceptional point and can limit the signal-to-noise improvement.<ref name="Zheng2025">X. Zheng and Y. D. Chong, Noise constraints for nonlinear exceptional point sensing, ''Phys. Rev. Lett.'' '''134''', 133801 (2025).</ref> This is why our project makes a careful claim: we demonstrate the frequency-response mechanism on a breadboard, but we do not claim a complete metrological noise advantage.

== Report objectives and scope ==
The purpose of this work is to study the circuit principle behind '''nonlinear-exceptional-point-enhanced inductive sensing''' in a coupled electronic resonator platform. The scope is intentionally narrower than the full experiment reported in Ref. [1]. We are not attempting to redesign the active branch, derive all nonlinear bifurcation theory from first principles, reconstruct the auxiliary steady-state branches, or characterize every parasitic element. Instead, we adopt the framework of Ref. [1] and SM-4 and ask whether the breadboard circuit can reproduce the experimentally accessible frequency response.

The main target of our experiment is the <math>f-L_A</math> curve corresponding to Fig. S11(a) of SM-4. In the reference experiment, the steady-state frequency, voltage ratio <math>|V_A/V_B|</math>, and relative phase <math>\theta_A-\theta_B</math> are all measured as functions of <math>L_A</math>.<ref name="Bai2024SM" /> In our implementation, the frequency-versus-inductance curve is the primary observable. We tune <math>L_A</math> by connecting small inductors in series with approximately <math>2~\mu\mathrm{H}</math> resolution. We also use a homebrew coil in a separate demonstration where mechanical compression or insertion of an iron rod changes the effective inductance. The experimental goal is therefore to verify that the TL071 breadboard circuit settles into a stable oscillating state and that the measured frequency changes with <math>L_A</math> in the same qualitative manner as Fig. S11(a).

From the sensing perspective, the most relevant figure of merit is the local frequency responsivity. If

<math display="block">
|f-f_{\mathrm{NEP}}| = A |\delta L_A|^{1/3},
</math>

then differentiation gives

<math display="block">
S_f \equiv \left|\frac{df}{dL_A}\right| \propto |\delta L_A|^{-2/3}.
</math>

Thus, as the operating point approaches <math>L_{A,\mathrm{NEP}}</math>, the local spectral responsivity increases rapidly. The main practical constraint is that the system must remain on the intended stable branch while measurement noise and slow parameter drift are kept below the scale of the induced frequency shift.

== Setups ==
[[File:Experimental_setup.png|750px|thumb|center|Experimental setup adapted from Ref. [1].]]

=== Circuit topology and component implementation ===
The reference platform consists of two coupled RLC resonators. Both resonators contain the same nominal shunt capacitance <math>C_0</math>, while the inductances <math>L_A</math> and <math>L_B</math> set their bare resonance frequencies. The two resonators are coupled through a parallel resistor-capacitor channel characterized by <math>R_c</math> and <math>C_c</math>. Resonator <math>B</math> contains an explicit positive resistance <math>R_B</math> and therefore plays the role of the lossy resonator. Resonator <math>A</math>, by contrast, contains an active branch whose effective resistance is negative at small amplitude and becomes less negative as the oscillation amplitude increases, thereby providing saturable gain.<ref name="Bai2024" /><ref name="Bai2024SM" />

Our implementation follows this topology but is built on a breadboard rather than a custom PCB. The active branch is implemented with an op-amp negative-resistance circuit using a '''TL071''' operational amplifier. This is an important practical difference from the supplementary experiment, where the active components explicitly listed are a TI '''LM7171''' operational amplifier and an '''Onsemi BAV99L''' diode.<ref name="Bai2024SM" /> The TL071 substitution and the breadboard layout introduce different bandwidth limits, wiring parasitics, and contact resistances, so the numerical operating point is not expected to match the PCB experiment. For this reason, our comparison emphasizes the observable trend of <math>f</math> versus <math>L_A</math>.

The supplementary material reports the following representative component values for the PCB experiment:

<math display="block">
C_0 = 18.5~\mathrm{nF}, \qquad
C_c = 3.9~\mathrm{nF}, \qquad
R_c = 760~\Omega,
</math>

<math display="block">
L_B = 197.7~\mu\mathrm{H}, \qquad
R_B = 1314~\Omega.
</math>

A representative operating point shown in the main figure uses

<math display="block">
L_A = 245.4~\mu\mathrm{H}.
</math>

Our breadboard circuit follows the same functional topology, but its component values and parasitic elements are not identical to the PCB values above. The horizontal axis of our main <math>f-L_A</math> plot is therefore the series-combined <math>L_A</math> selected from our available small inductors, while the measured frequency <math>f</math> is the robust experimental readout.

=== Inductance tuning methods ===
We use two complementary methods to change the inductance seen by resonator <math>A</math>.

The first method is used for the main <math>f-L_A</math> sweep. Small inductors are connected in series so that the total inductance changes in discrete steps of approximately <math>2~\mu\mathrm{H}</math>. If the inductors are sufficiently separated so that their mutual magnetic coupling is small, the total series inductance is approximately the sum of the individual inductances,

<math display="block">
L_A \approx L_{\mathrm{base}}+\sum_i L_i .
</math>

This method is simple and reproducible. Adding one small inductor increases <math>L_A</math>; removing it decreases <math>L_A</math>. The circuit frequency is then measured for each chosen value. This produces a discrete version of the <math>f-L_A</math> curve in Fig. S11(a) of SM-4. The step size is not as fine as a commercial variable inductor, but <math>2~\mu\mathrm{H}</math> spacing is small enough to observe the frequency trend over the accessible range.

The second method is a homebrew compressible variable inductor. This coil is useful because it makes the sensing mechanism visible: a mechanical displacement changes the coil geometry, which changes the inductance, which then changes the oscillation frequency. The basic idea can be understood from the approximate solenoid formula

<math display="block">
L \approx \mu_\mathrm{eff}\frac{N^2 A}{\ell},
</math>

where <math>N</math> is the number of turns, <math>A</math> is the cross-sectional area, <math>\ell</math> is the coil length, and <math>\mu_\mathrm{eff}</math> describes the effective magnetic permeability of the core and surrounding region.<ref name="Chen2014">S.-C. Chen, D.-K. Le, and V.-S. Nguyen, Inductive displacement sensors with a notch filter for an active magnetic bearing system, ''Sensors'' '''14''', 12640-12657 (2014).</ref> During compression, the number of turns <math>N</math> is unchanged and the cross-sectional area <math>A</math> remains approximately unchanged, but the length <math>\ell</math> decreases. Equivalently, the turn density

<math display="block">
n=\frac{N}{\ell}
</math>

increases. The magnetic field produced by a solenoid is proportional to turn density. Therefore, compressing the coil increases the magnetic flux linkage through the turns and increases the inductance. Stretching the coil has the opposite effect.

This mechanical tuning is not identical to the precision tunable inductor used in the reference experiment. A real compressed coil is not an ideal long solenoid: the spacing between turns is finite, the end effects change as the coil length changes, nearby wires and the breadboard add parasitic capacitance, and the leads contribute additional series inductance and resistance. For this reason, the homebrew inductor is best treated as an experimentally characterized tuning element rather than an ideal component whose value is known exactly from geometry. In the present project, it provides a direct mechanical demonstration of how displacement can perturb <math>L_A</math>.

Experimentally, we measured the effective inductance of this homebrew variable inductor at different compression states. The compression process changes the coil length and wire density while leaving the coil area approximately fixed. Therefore, the tuning mechanism is physically the same as changing <math>\ell</math> and <math>n=N/\ell</math> in the solenoid model, rather than changing the number of turns or the coil diameter.

=== Kirchhoff model of the nonlinear circuit ===
A convenient state-space description uses the node voltages <math>V_A(t)</math> and <math>V_B(t)</math> together with the inductor currents <math>I_A(t)</math> and <math>I_B(t)</math>. A node voltage is the voltage measured at one resonator node relative to ground. Kirchhoff's current law says that the algebraic sum of all currents flowing away from a node must be zero. This gives a direct way to translate the circuit diagram into differential equations.

Denoting the amplitude-dependent conductance of the active branch by <math>G_A(|V_A|)</math>, where <math>G_A<0</math> in the small-signal regime, the nodal equations can be written as

<math display="block">
\begin{aligned}
C_0 \frac{dV_A}{dt} + I_A + G_A(|V_A|)\,V_A + \frac{V_A-V_B}{R_c}
+ C_c\frac{d(V_A-V_B)}{dt} &= 0,\\
C_0 \frac{dV_B}{dt} + I_B + \frac{V_B}{R_B} + \frac{V_B-V_A}{R_c}
+ C_c\frac{d(V_B-V_A)}{dt} &= 0,\\
L_A \frac{dI_A}{dt} &= V_A,\\
L_B \frac{dI_B}{dt} &= V_B.
\end{aligned}
</math>

Each term has a simple circuit meaning. The terms <math>C_0 dV_A/dt</math> and <math>C_0 dV_B/dt</math> are capacitor currents. The terms <math>I_A</math> and <math>I_B</math> are currents through the inductors. The term <math>G_A(|V_A|)V_A</math> is the active branch current; because <math>G_A</math> can be negative, this branch can inject energy instead of dissipating it. The term <math>V_B/R_B</math> is the ordinary loss current through the resistor in resonator <math>B</math>. The terms involving <math>R_c</math> and <math>C_c</math> describe current flowing between resonators <math>A</math> and <math>B</math>, so they are the electrical coupling between the two resonators.

These equations show explicitly how the experiment differs from a linear EP circuit. If <math>G_A</math> were a constant negative conductance, the model would reduce to a linear non-Hermitian dimer. Here, however, <math>G_A</math> depends on the voltage amplitude, so the steady state must be determined self-consistently together with the oscillation amplitude itself.

=== Harmonic-balance formulation of the steady state ===
After the transient has died away, the oscilloscope waveform is approximately periodic. The simplest approximation is therefore to keep only the dominant sinusoidal component. This is called a harmonic-balance approximation. For a stable activated oscillating state, one writes

<math display="block">
V_j(t)=\mathrm{Re}\!\left[\tilde V_j e^{i\omega t}\right],
\qquad j=A,B.
</math>

Substituting this ansatz into the circuit equations yields a nonlinear algebraic problem of the form

<math display="block">
\mathbf{Y}(\omega,|\tilde V_A|)
\begin{pmatrix}
\tilde V_A\\
\tilde V_B
\end{pmatrix}
=
\begin{pmatrix}
Y_A(\omega,|\tilde V_A|)+Y_c(\omega) & -Y_c(\omega)\\
-Y_c(\omega) & Y_B(\omega)+Y_c(\omega)
\end{pmatrix}
\begin{pmatrix}
\tilde V_A\\
\tilde V_B
\end{pmatrix}
=0,
</math>

with

<math display="block">
Y_A(\omega,|\tilde V_A|)=G_A(|\tilde V_A|)+i\omega C_0+\frac{1}{i\omega L_A},
</math>

<math display="block">
Y_B(\omega)=\frac{1}{R_B}+i\omega C_0+\frac{1}{i\omega L_B},
\qquad
Y_c(\omega)=\frac{1}{R_c}+i\omega C_c.
</math>

A nontrivial solution requires

<math display="block">
\det \mathbf{Y}(\omega,|\tilde V_A|)=0.
</math>

This determinant condition is the circuit analogue of an eigenfrequency condition. In a passive LC resonator, the resonance condition gives a single frequency <math>\omega_0=1/\sqrt{LC}</math>. In the present two-resonator active circuit, the same idea becomes a matrix equation because the two resonators are coupled. The amplitudes <math>\tilde V_A</math> and <math>\tilde V_B</math> are not arbitrary: a nonzero oscillation exists only when the admittance matrix has a zero eigenvalue.

Because the active admittance depends on amplitude, this condition is nonlinear and supports multiple branches. The supplementary material shows that the steady-state frequency can be reduced to a '''real-coefficient cubic polynomial''' in <math>\omega</math>. The existence of up to three real steady-state branches is therefore not accidental; it is the algebraic origin of the experimentally relevant <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM" />

=== Effective nonlinear-Hamiltonian picture ===
The same physics is often rewritten in coupled-mode form. This notation is common in non-Hermitian physics because it makes the circuit look like two coupled modes. The variables <math>\psi_A</math> and <math>\psi_B</math> represent the complex oscillation amplitudes in resonators <math>A</math> and <math>B</math>. The model is

<math display="block">
i\frac{d}{dt}
\begin{pmatrix}
\psi_A\\
\psi_B
\end{pmatrix}
=
\begin{pmatrix}
\omega_A+i\,g(|\psi_A|) & \kappa\\
\kappa & \omega_B-i\ell
\end{pmatrix}
\begin{pmatrix}
\psi_A\\
\psi_B
\end{pmatrix},
</math>

where <math>\omega_A</math> and <math>\omega_B</math> are the effective modal frequencies, <math>\kappa</math> is the coupling strength, <math>\ell</math> is the loss of resonator <math>B</math>, and <math>g(|\psi_A|)</math> is the saturable gain of resonator <math>A</math>. The steady-state problem then becomes

<math display="block">
H_s(\Psi)\Psi = \Omega \Psi,
</math>

with a real oscillation eigenfrequency <math>\Omega=2\pi f</math> for the stable oscillating branch. The word "real" is important: it means the imaginary part has vanished, so the mode is neither being amplified nor decaying in the steady state. In the cited experiment, this physical stable branch and two auxiliary steady branches coalesce at the singular point, producing a third-order nonlinear exceptional point rather than a conventional second-order linear EP. The auxiliary branches help form the cubic-root singularity but are not themselves stable physical outputs of the circuit.<ref name="Bai2024" /><ref name="Bai2024SM" />

== Measurements ==

=== Acquisition protocol and steady-state selection ===
The measurement is performed in the time domain. In the reference experiment, an arbitrary waveform generator first applies an external start-up signal of 1 V at 70 kHz in order to bring the system into the basin of attraction of the stable oscillating state.<ref name="Bai2024" /> In our breadboard measurement, an external voltage is likewise required to activate the resonators. The measured frequency is extracted from the activated steady waveform recorded on the oscilloscope.

The physical meaning of this procedure is important. The oscillation is not assumed to appear without activation; a supplied voltage is needed to bring the resonators into the relevant dynamical state. The measured oscillation frequency is then interpreted as the frequency selected by the nonlinear steady-state condition.

=== Extraction of the <math>f-L_A</math> observable ===
Ref. [1] does not treat the oscilloscope traces merely as qualitative evidence of oscillation. The supplementary explicitly states that the full reference experiment reports the steady-state frequency, the voltage ratio <math>|V_A/V_B|</math>, and the relative phase <math>\theta_A-\theta_B</math>, all extracted from the dominant peaks of the Fourier spectra of <math>V_A</math> and <math>V_B</math>.<ref name="Bai2024SM" />

Our available observable is the steady-state frequency. For each chosen value of <math>L_A</math>, the resonators are voltage-activated, the circuit reaches a stable oscillating state, and the dominant spectral peak is recorded. If we denote the Fourier transforms by

<math display="block">
\tilde V_A(f)=\mathcal{F}\{V_A(t)\},
\qquad
\tilde V_B(f)=\mathcal{F}\{V_B(t)\},
</math>

then the dominant oscillation frequency is obtained from

<math display="block">
f_0=\arg\max_f |\tilde V_A(f)| \approx \arg\max_f |\tilde V_B(f)|.
</math>

The experimental data set is therefore

<math display="block">
\{L_A,\ f_0(L_A)\},
</math>

which is compared with the frequency branch plotted in Fig. S11(a) of SM-4. In the full reference measurement, the amplitude ratio

<math display="block">
\rho = \frac{|\tilde V_A(f_0)|}{|\tilde V_B(f_0)|}
</math>

and relative phase

<math display="block">
\Delta\phi = \arg \tilde V_A(f_0)-\arg \tilde V_B(f_0)
</math>

provide additional state verification. Those quantities are useful but are not the main basis of our breadboard validation.

In practice, the same frequency can also be estimated directly from the time-domain period <math>T</math> using <math>f_0=1/T</math> when the waveform is clean. Fourier analysis is preferred because it uses a longer section of waveform and identifies the dominant oscillation peak even when there is small amplitude noise or waveform distortion.

=== Practical inductance sweep ===
In SM-4, the inductive tuning is carried out by varying <math>L_A</math>. The supplementary notes that the tunable inductor exhibits less than about 0.4% inductance variation over the frequency range of interest, which is sufficiently small for controlled sweeping near the singular point. To further suppress voltage-dependent drift during the sweep, the reference experiment slightly adjusts the active-branch resistance so as to maintain approximately <math>V_B \approx 2~\mathrm{V}</math> throughout the measurement series.<ref name="Bai2024SM" />

Our sweep uses controlled series combinations of small inductors, giving an approximately <math>2~\mu\mathrm{H}</math> step in <math>L_A</math>. The homebrew compressible coil is then used to show that a mechanical change of the sensing element also changes the inductance. The conclusion drawn from our data is therefore not a precision determination of <math>L_{A,\mathrm{NEP}}</math>, but a validation that the breadboard circuit produces the same kind of frequency response as the stable branch in Fig. S11(a).

=== Long-window spectral analysis and noise inspection ===
The supplementary also discusses long-time measurements over a window of <math>0<t<5~\mathrm{s}</math>. The motivation is that the noise around the stable oscillation is too small to be judged reliably from raw scope traces alone, so it is more informative to inspect the Fourier spectra and the extracted spectral center frequency and linewidth.<ref name="Bai2024SM" />

This is a useful methodological point for a sensing-oriented reproduction. In an NEP-based sensor, one ultimately cares about how precisely the frequency peak can be resolved and tracked. A long-window Fourier analysis is therefore closer to the sensing task than a purely visual time-domain inspection.

== Results ==

=== Result display: steady oscillation ===
[[File:PC5271inductive1.png|750px|thumb|center|Breadboard implementation and representative steady oscillation waveform.]]

This figure shows (a) our circuit implemented on a breadboard and (b) a representative stabilised voltage waveform after the resonators are activated by the supplied voltage. The displayed waveform is the evidence used to identify a stable oscillation frequency.

=== Result display: <math>f-L_A</math> sweep ===
[[File:PC5271inductive3.jpeg|750px|thumb|center|Measured breadboard <math>f-L_A</math> curve.]]

This figure shows the measured oscillation frequency <math>f</math> as a function of <math>L_A</math>. The values of <math>L_A</math> in the main sweep are produced by connecting small inductors in series with approximately <math>2~\mu\mathrm{H}</math> step size.

=== Result display: iron-rod demonstration ===
[[File:PC5271inductive2.png|750px|thumb|center|Displacement measurement by (a) compressing the solenoid and (b) inserting iron rod into the homebrew coil.]]

This figure exhibits the resultant changes in the measured oscillation frequency of homebrew coil with (a) a compression and (b) inserted iron rod. The experimental variable is the coil length and rod position, and the measured output is the activated steady oscillation frequency. As show in the analysis above, the inductance shows an inversely proportional dependence on the coil length and proportion dependence on insertion length, with the constant residual being the baseline inductance in the circuit irrelevant to the coil.

== Evaluation ==

=== Stable activated oscillation ===
The first required experimental signature is that, after the resonators are activated by the supplied voltage, the circuit reaches a stable oscillating waveform instead of decaying immediately or being amplified without bound. A stable waveform means that the TL071 active branch supplies enough small-signal negative resistance to compensate loss, while gain saturation prevents unbounded amplification.

Physically, this behavior confirms the role of the nonlinear active branch. At small amplitude, the effective negative resistance in resonator <math>A</math> overcomes dissipation. At larger amplitude, the gain becomes weaker, so the oscillation amplitude is clamped. The system self-stabilizes because the physical steady branch is the one with zero imaginary part of the eigenfrequency. The auxiliary branches near the nonlinear exceptional point are not stable physical outputs; they are mathematical branches of the nonlinear steady-state equation. The experiment is therefore not a passive ring-down measurement; it is a nonlinear steady-state selection process.

=== Evaluation of the <math>f-L_A</math> trend ===
The main experimental comparison is the measured <math>f-L_A</math> trend against Fig. S11(a) of SM-4. The reference experiment reports the stable frequency branch of the nonlinear circuit as <math>L_A</math> is tuned.<ref name="Bai2024SM" /> This branch is the physical one because its eigenfrequency is real in the steady state. In our breadboard experiment, the same observable is measured using the series-inductor sweep.

It is useful to separate two statements. First, any LC resonator changes frequency when <math>L</math> changes, because <math>f_0=1/(2\pi\sqrt{LC})</math>. This alone would not demonstrate the nonlinear exceptional-point mechanism. Second, in the coupled active-lossy circuit, the frequency is not just the bare LC resonance frequency of one isolated inductor and capacitor. It is the frequency selected by the nonlinear steady-state balance of gain, loss, and coupling. The reason Fig. S11(a) matters is that it shows this selected stable branch as <math>L_A</math> is swept.

Our breadboard circuit uses different component values and a different op-amp from the PCB experiment, so exact numerical overlap is not expected. The meaningful evaluation is whether the measured frequency branch follows the same qualitative <math>f-L_A</math> behavior as the stable branch in SM-4. Observing that behavior is enough to show that the constructed circuit realizes the intended nonlinear voltage-activated oscillating mechanism.

=== Evaluation of the displacement demonstration ===
In the iron-rod demonstration, the homebrew coil becomes a displacement-dependent inductive element. As the rod enters the coil, the effective magnetic permeability inside the coil increases. In the solenoid expression

<math display="block">
L(x) \approx \mu_\mathrm{eff}(x)\frac{N^2 A}{\ell},
</math>

the insertion depth <math>x</math> changes <math>\mu_\mathrm{eff}</math>. The coil geometry <math>N</math>, <math>A</math>, and <math>\ell</math> are approximately fixed during this demonstration, so the dominant effect is the change in magnetic core material inside the coil. Inductive displacement sensors commonly use this principle: the position of a ferromagnetic object changes a coil inductance, and the electronics convert that inductance change into an electrical signal.<ref name="Chen2014" />

In our circuit, the electrical signal is the activated steady oscillation frequency rather than a bridge voltage. The sensing chain is

<math display="block">
\mathrm{rod\ displacement}\ x
\longrightarrow
\mathrm{effective\ inductance}\ L_A(x)
\longrightarrow
\mathrm{oscillation\ frequency}\ f(x).
</math>

This demonstration shows how the inductive readout can be turned into a displacement-sensing readout after calibration of the relation between rod position and frequency.

=== Cubic-root perturbation law and sensing meaning ===
Near the singular operating point, the eigenfrequency shift follows the hallmark cubic-root law,

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3}.
</math>

This law is the main sensing-related result of the reference experiment.<ref name="Bai2024" /><ref name="Bai2024SM" /> Compared with the linear response of an ordinary isolated resonator, the cubic-root dependence can produce a larger frequency displacement for the same small perturbation of inductance when the device is biased sufficiently close to <math>L_{A,\mathrm{NEP}}</math>.

A useful way to interpret the result is through the local derivative,

<math display="block">
\left|\frac{df}{dL_A}\right| \propto |\delta L_A|^{-2/3}.
</math>

This increasing local responsivity is why the circuit is attractive for sensing. Our experiment supports this sensing logic by showing that the breadboard circuit has a measurable stable frequency branch as <math>L_A</math> is changed. A full quantitative exponent extraction would require identifying <math>L_{A,\mathrm{NEP}}</math> and <math>f_{\mathrm{NEP}}</math>, then plotting <math>|f-f_{\mathrm{NEP}}|</math> against <math>|L_A-L_{A,\mathrm{NEP}}|</math> on logarithmic axes.

=== Basis completeness and noise robustness ===
The supplementary material evaluates the Petermann factor as a diagnostic of modal nonorthogonality and basis completeness. For a non-Hermitian eigenproblem, one may write

<math display="block">
\mathrm{PF}_n =
\frac{\langle \Phi_n^{L}|\Phi_n^{L}\rangle \langle \Phi_n^{R}|\Phi_n^{R}\rangle}
{\left|\langle \Phi_n^{L}|\Phi_n^{R}\rangle\right|^2},
</math>

where <math>|\Phi_n^{R}\rangle</math> and <math>|\Phi_n^{L}\rangle</math> are the right and left eigenvectors of the effective steady-state Hamiltonian. At a conventional linear EP, this quantity is associated with diverging excess-noise sensitivity. In Ref. [1], the cited analysis finds that the effective steady-state Hamiltonian remains non-defective and the Petermann factor is finite at the <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM" />

This point explains the significance of the reference experiment. The system does not merely realize a large mathematical responsivity; it realizes that responsivity in a way that avoids the most severe basis-collapse problem of a linear EP. Our breadboard experiment does not directly test basis completeness, but the result provides the theoretical reason why reproducing the stable <math>f-L_A</math> branch is physically meaningful.

=== Overall interpretation ===
In this platform, the measurand is encoded as a perturbation of <math>L_A</math>. Any physical mechanism that modifies the effective inductance of resonator <math>A</math>, for example displacement, magnetic coupling, or the presence of a nearby magnetic target, moves the system away from the operating point and shifts the oscillation frequency. The primary readout in our implementation is therefore spectral: we track <math>f</math> as <math>L_A</math> is changed.

Taken together, Ref. [1] establishes the complete sensing logic: a nonlinear active-lossy circuit is tuned to an <math>\mathrm{NEP}_3</math>; after voltage activation, the system settles onto a stable oscillating branch; the dominant spectral peak is tracked as the perturbation readout; and the enhanced cubic-root response is obtained without the defective-basis pathology of a conventional linear EP. Our breadboard result demonstrates the experimentally accessible core of this logic by reproducing the stable frequency response corresponding to Fig. S11(a) of SM-4.

== References ==
<references />

Inductive Sensors of Ultra-high Sensitivity Based on Nonlinear Exceptional Point

2026-04-23T06:30:00Z

Siyu: /* Result display: iron-rod demonstration */

== Introduction ==
This project is a implementation of the experiment reported in ''Observation of Nonlinear Exceptional Points with a Complete Basis in Dynamics'', adapted to our available laboratory equipment.<ref name="Bai2024">K. Bai, T.-R. Liu, L. Fang, J.-Z. Li, C. Lin, D. Wan, and M. Xiao, Observation of nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''132''', 073802 (2024).</ref> The circuit contains two coupled resonators. One resonator has gain, meaning that it can supply energy to the oscillation, while the other has loss, meaning that it dissipates energy. A system without energy conservation is called a '''non-Hermitian''' system. Such systems can show unusual degeneracies called '''exceptional points''' (EPs), where both the eigenfrequencies and the eigenmodes merge.

Exceptional points are interesting for sensing because the measured frequency can respond very strongly to a small perturbation. In an inductive sensor, the perturbation is a change in inductance. If a small change in inductance produces a large change in oscillation frequency, the circuit can in principle detect small physical changes. However, ordinary linear EPs also suffer from strong eigenmode nonorthogonality and a defective basis, which can increase noise sensitivity and complicate practical sensing.<ref name="Bai2024" />

The system studied in this project belongs to the recently developed class of '''nonlinear exceptional points''' (NEPs). The physical platform is a pair of coupled circuit resonators. Resonator <math>A</math> is made active by a saturable negative-resistance branch, while resonator <math>B</math> is intentionally lossy. Because the gain in resonator <math>A</math> depends on the oscillation amplitude, the steady-state problem is nonlinear and admits multiple self-consistent branches. At the singular operating point reported in Ref. [1], one stable steady branch and two auxiliary steady branches merge into a third-order nonlinear exceptional point, denoted <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM">Supplemental Material for K. Bai, T.-R. Liu, L. Fang, J.-Z. Li, C. Lin, D. Wan, and M. Xiao, Observation of nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''132''', 073802 (2024).</ref>

The experimentally important consequence is that the oscillation eigenfrequency obeys a cubic-root perturbation law near the singularity,

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3},
</math>

where <math>\delta L_A = L_A-L_{A,\mathrm{NEP}}</math> is the inductive detuning from the exceptional operating point. This response is steeper than the ordinary linear dependence of a conventional resonator and is therefore useful for ultra-sensitive inductive sensing.<ref name="Bai2024" /><ref name="Bai2024SM" />

A second and equally important point is that the cited work does not interpret the singularity as a conventional defective linear EP. Instead, the effective steady-state Hamiltonian remains non-defective in dynamics, and the associated Petermann factor stays finite at the operating point. In other words, the experiment realizes enhanced spectral responsivity without inheriting the strongest basis-collapse pathology of a standard linear EP.<ref name="Bai2024" /><ref name="Bai2024SM" />

In this project, our goal is to reproduce the experimental logic of Ref. [1] and its supplementary material under course-lab conditions. Unlike the PCB implementation in the paper, our circuit is assembled on a breadboard and the active branch uses a '''TL071''' operational amplifier. Our main observable is the steady oscillation frequency <math>f</math> as a function of the chosen inductance <math>L_A</math>. We tune <math>L_A</math> in two ways. For the main <math>f-L_A</math> sweep, we join small inductors in series so that the discrete inductance step is about <math>2~\mu\mathrm{H}</math>. In addition, we use a homebrew compressible coil to demonstrate the sensing mechanism directly. Compressing the coil changes its length and turn density while keeping the cross-sectional area approximately unchanged. We therefore use the <math>f-L_A</math> behavior in Fig. S11(a) of the supplementary material as the main experimental benchmark. Reproducing this trend is sufficient for demonstrating that our setup captures the intended nonlinear voltage-activated, self-stabilized oscillating response, although it is not a full reproduction of every diagnostic reported in Ref. [1].<ref name="Bai2024" /><ref name="Bai2024SM" />

== Team Members ==
Wang Peikun — E1538091@u.nus.edu 
Zhu Ziyang — E1583446@u.nus.edu 
Yuan Si-Yu — siyu_yuan@u.nus.edu 
Li Xunyu — xunyu@u.nus.edu

== Background Knowledge ==

=== Ordinary LC and RLC resonators ===
The basic building block of the experiment is an electrical resonator. An ideal LC resonator contains an inductor <math>L</math> and a capacitor <math>C</math>. Energy repeatedly moves between the magnetic field of the inductor and the electric field of the capacitor. The natural angular frequency is

<math display="block">
\omega_0=\frac{1}{\sqrt{LC}},
</math>

or, in ordinary frequency units,

<math display="block">
f_0=\frac{1}{2\pi\sqrt{LC}}.
</math>

This formula is important for the whole project because it explains why changing an inductance changes the measured frequency. A larger <math>L</math> gives a lower resonance frequency, while a smaller <math>L</math> gives a higher resonance frequency.

Real circuits are not ideal. Wires, inductors, capacitors, breadboard contacts, and measuring probes all dissipate energy. This loss is often represented by a resistor <math>R</math>, giving an RLC resonator. In a passive RLC circuit, the oscillation amplitude decays with time because energy is continuously lost as heat. If a circuit is to maintain a steady activated oscillation, the loss must be compensated by a gain element.

=== Gain, loss, and negative resistance ===
In electronics, gain can be represented as an effective negative resistance. A normal positive resistance removes energy from the resonator. A negative resistance does the opposite: it supplies energy to the resonator. In this experiment, resonator <math>A</math> contains an active op-amp branch that behaves like a negative resistance at small amplitude, while resonator <math>B</math> contains ordinary positive loss.

The gain cannot remain negative and constant forever. If it did, any activated oscillation would be amplified without bound until the op-amp clipped or the circuit failed. The important nonlinear ingredient is '''saturable gain''': the active branch gives strong gain at small amplitude but weaker gain at larger amplitude. The stable oscillation amplitude is selected when the supplied gain balances the total loss. This nonlinear steady state is why the experiment is a nonlinear dynamical problem rather than only a passive resonance measurement.

=== What "non-Hermitian" means in this circuit ===
In ordinary conservative physics, a Hermitian Hamiltonian describes a closed system whose total energy is conserved. Its eigenfrequencies are real, and its eigenmodes form a complete orthogonal basis. This is the familiar setting of many quantum-mechanics courses.

A circuit with gain and loss is an open system. Energy can enter through the op-amp branch and leave through resistive loss. The effective dynamical matrix is therefore generally '''non-Hermitian'''. In a non-Hermitian system, eigenfrequencies can be complex:

<math display="block">
\omega=\omega_\mathrm{r}+i\omega_\mathrm{i}.
</math>

The real part <math>\omega_\mathrm{r}</math> gives the oscillation frequency, while the imaginary part <math>\omega_\mathrm{i}</math> describes amplification or decay. A positive imaginary part means the mode is amplified, a negative imaginary part means it decays, and a stable steady oscillation corresponds to zero imaginary part, or equivalently zero net amplification. This is the sense in which the circuit is self-stabilized after voltage activation.

=== Exceptional points ===
In a usual Hermitian system, two modes can have the same eigenfrequency while still remaining two independent modes. This is an ordinary degeneracy. A non-Hermitian system can show a stronger kind of degeneracy called an '''exceptional point''' (EP). At an EP, not only the eigenvalues but also the eigenvectors coalesce. In simple language, two modes collapse into one mode.

This collapse makes EPs interesting for sensing. Near a second-order linear EP, a small perturbation can produce a square-root frequency splitting:

<math display="block">
\Delta f \propto \sqrt{\epsilon},
</math>

where <math>\epsilon</math> is the perturbation. This response is steeper than an ordinary linear response when <math>\epsilon</math> is very small. However, a conventional linear EP also has a serious drawback: the eigenbasis becomes defective, meaning that the system no longer has enough independent eigenvectors. This is associated with strong modal nonorthogonality and enhanced noise sensitivity. The Petermann factor is one common way to quantify this nonorthogonality.

=== Why the nonlinear exceptional point is different ===
The reference paper studies a nonlinear exceptional point. The word "nonlinear" is essential. The circuit does not have a fixed gain coefficient; instead, the gain depends on the oscillation amplitude. The steady state must therefore be solved self-consistently: the frequency, voltage amplitudes, and saturated gain must all agree with one another.

In the theory of Ref. [1], this nonlinear steady-state problem can reduce to a cubic equation for the oscillation frequency. A cubic equation can have a triple root. At the special operating point, one stable physical branch and two auxiliary branches merge, giving a third-order nonlinear exceptional point, written as <math>\mathrm{NEP}_3</math>. The stable physical branch is the one with real oscillation frequency, i.e. without an imaginary part. The auxiliary branches are needed to define the higher-order coalescence mathematically, but they are not physically stable operating states of the circuit. The corresponding frequency response has the form

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3}.
</math>

This cubic-root response is the key sensing signature. It is steeper than both an ordinary linear response and the square-root response of a second-order EP. The important conceptual point of Ref. [1] is that this enhanced response is obtained while the effective steady-state Hamiltonian can still retain a complete basis in dynamics.<ref name="Bai2024" /><ref name="Bai2024SM" />

=== Sensing mechanism ===
The circuit can be understood as a frequency-generating sensor. It converts an inductance change into a frequency change.

The first step is activation. A voltage is supplied to the resonators to excite the circuit into an oscillating state. The active branch in resonator <math>A</math> then supplies energy through an effective negative resistance and amplifies the selected oscillation mode.

The second step is self-stabilization. The gain is saturable, which means it becomes weaker when the voltage amplitude becomes larger. The activated oscillation therefore is not amplified forever. The circuit settles onto the physical steady branch where the imaginary part of the eigenfrequency is zero. At that point, the gain supplied by resonator <math>A</math> balances the loss in the circuit, especially the loss in resonator <math>B</math>.

The third step is frequency selection. Because resonators <math>A</math> and <math>B</math> are coupled, the final oscillation frequency is not simply the resonance frequency of either isolated resonator. It is selected by the whole coupled circuit: the inductors, capacitors, coupling elements, loss, and saturated gain all contribute.

The fourth step is sensing. The inductance <math>L_A</math> controls the bare frequency of resonator <math>A</math>. When <math>L_A</math> changes, the self-consistent frequency selected by the nonlinear circuit also changes. Therefore, tracking the spectral peak <math>f</math> tells us how the effective inductance has changed.

The fifth step is the nonlinear exceptional-point response. Near the nonlinear exceptional point discussed in Ref. [1], the physical steady branch and the two auxiliary branches coalesce in the nonlinear steady-state equation. The frequency shift of the stable branch can then scale as the cube root of the inductance perturbation. This means that a small inductance change can produce a comparatively large frequency change. Our breadboard experiment focuses on verifying the stable <math>f-L_A</math> branch corresponding to Fig. S11(a) of SM-4, which is the experimentally accessible part of this sensing mechanism.

=== Same mechanism in other platforms ===
Several references are included to show that this is a platform-independent sensing mechanism, not only a feature of our breadboard circuit. The common chain is

<math display="block">
\mathrm{physical\ perturbation}
\longrightarrow
\mathrm{parameter\ change}
\longrightarrow
\mathrm{eigenfrequency\ shift\ of\ a\ resonator/cavity\ with\ gain\ and\ loss}
\longrightarrow
\mathrm{higher\ order\ steady\ state\ response}
\longrightarrow
\mathrm{sensing}.
</math>

In our circuit, the physical perturbation is a change in the inductance <math>L_A</math>, and the measured output is the oscillation frequency <math>f</math>. In coupled optical, mechanical, electrical, or hybrid resonators, the perturbed parameter may instead be a cavity detuning, mechanical frequency, phase shift, magnetic-field-dependent transition, or another effective resonator parameter. The shared point is that gain, loss, coupling, and nonlinearity convert the parameter change into a higher-order steady-state spectral response.

The earlier theoretical work on nonlinear exceptional points showed that nonlinear coupled resonators can host higher-order NEPs while retaining a complete dynamical basis.<ref name="Bai2023">K. Bai, J.-Z. Li, T.-R. Liu, L. Fang, D. Wan, and M. Xiao, Nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''130''', 266901 (2023).</ref> The 2024 experiment implemented the same mechanism in coupled electrical resonators and measured the stable nonlinear frequency branch.<ref name="Bai2024" /><ref name="Bai2024SM" /> Silicon micromechanical resonators provide another realization: there, the measured output can be a phase difference, and the nonlinear exceptional-point response can enhance phase-shift sensing.<ref name="Xie2026">Y.-J. Xie, S.-Y. Zhang, M.-N. Zhang, R. Wang, L.-F. Wang, and Q.-A. Huang, Nonlinear exceptional points boost phase-shift sensing in silicon micromechanical resonators, ''Phys. Rev. Applied'' '''25''', 034039 (2026).</ref> Hybrid quantum systems with nonlinear bistability show a related route, where nonlinear transition dynamics can enhance sensing signal-to-noise ratio.<ref name="Wang2026">H. Wang ''et al.'', Exceptional sensitivity near the bistable transition point of a hybrid quantum system, ''Nature Physics'' '''22''', 577-583 (2026).</ref>

These examples support the same mechanism rather than unrelated applications. At the same time, enhanced response alone is not enough to prove better sensing. Noise can shift or weaken the apparent nonlinear exceptional point and can limit the signal-to-noise improvement.<ref name="Zheng2025">X. Zheng and Y. D. Chong, Noise constraints for nonlinear exceptional point sensing, ''Phys. Rev. Lett.'' '''134''', 133801 (2025).</ref> This is why our project makes a careful claim: we demonstrate the frequency-response mechanism on a breadboard, but we do not claim a complete metrological noise advantage.

== Report objectives and scope ==
The purpose of this work is to study the circuit principle behind '''nonlinear-exceptional-point-enhanced inductive sensing''' in a coupled electronic resonator platform. The scope is intentionally narrower than the full experiment reported in Ref. [1]. We are not attempting to redesign the active branch, derive all nonlinear bifurcation theory from first principles, reconstruct the auxiliary steady-state branches, or characterize every parasitic element. Instead, we adopt the framework of Ref. [1] and SM-4 and ask whether the breadboard circuit can reproduce the experimentally accessible frequency response.

The main target of our experiment is the <math>f-L_A</math> curve corresponding to Fig. S11(a) of SM-4. In the reference experiment, the steady-state frequency, voltage ratio <math>|V_A/V_B|</math>, and relative phase <math>\theta_A-\theta_B</math> are all measured as functions of <math>L_A</math>.<ref name="Bai2024SM" /> In our implementation, the frequency-versus-inductance curve is the primary observable. We tune <math>L_A</math> by connecting small inductors in series with approximately <math>2~\mu\mathrm{H}</math> resolution. We also use a homebrew coil in a separate demonstration where mechanical compression or insertion of an iron rod changes the effective inductance. The experimental goal is therefore to verify that the TL071 breadboard circuit settles into a stable oscillating state and that the measured frequency changes with <math>L_A</math> in the same qualitative manner as Fig. S11(a).

From the sensing perspective, the most relevant figure of merit is the local frequency responsivity. If

<math display="block">
|f-f_{\mathrm{NEP}}| = A |\delta L_A|^{1/3},
</math>

then differentiation gives

<math display="block">
S_f \equiv \left|\frac{df}{dL_A}\right| \propto |\delta L_A|^{-2/3}.
</math>

Thus, as the operating point approaches <math>L_{A,\mathrm{NEP}}</math>, the local spectral responsivity increases rapidly. The main practical constraint is that the system must remain on the intended stable branch while measurement noise and slow parameter drift are kept below the scale of the induced frequency shift.

== Setups ==
[[File:Experimental_setup.png|750px|thumb|center|Experimental setup adapted from Ref. [1].]]

=== Circuit topology and component implementation ===
The reference platform consists of two coupled RLC resonators. Both resonators contain the same nominal shunt capacitance <math>C_0</math>, while the inductances <math>L_A</math> and <math>L_B</math> set their bare resonance frequencies. The two resonators are coupled through a parallel resistor-capacitor channel characterized by <math>R_c</math> and <math>C_c</math>. Resonator <math>B</math> contains an explicit positive resistance <math>R_B</math> and therefore plays the role of the lossy resonator. Resonator <math>A</math>, by contrast, contains an active branch whose effective resistance is negative at small amplitude and becomes less negative as the oscillation amplitude increases, thereby providing saturable gain.<ref name="Bai2024" /><ref name="Bai2024SM" />

Our implementation follows this topology but is built on a breadboard rather than a custom PCB. The active branch is implemented with an op-amp negative-resistance circuit using a '''TL071''' operational amplifier. This is an important practical difference from the supplementary experiment, where the active components explicitly listed are a TI '''LM7171''' operational amplifier and an '''Onsemi BAV99L''' diode.<ref name="Bai2024SM" /> The TL071 substitution and the breadboard layout introduce different bandwidth limits, wiring parasitics, and contact resistances, so the numerical operating point is not expected to match the PCB experiment. For this reason, our comparison emphasizes the observable trend of <math>f</math> versus <math>L_A</math>.

The supplementary material reports the following representative component values for the PCB experiment:

<math display="block">
C_0 = 18.5~\mathrm{nF}, \qquad
C_c = 3.9~\mathrm{nF}, \qquad
R_c = 760~\Omega,
</math>

<math display="block">
L_B = 197.7~\mu\mathrm{H}, \qquad
R_B = 1314~\Omega.
</math>

A representative operating point shown in the main figure uses

<math display="block">
L_A = 245.4~\mu\mathrm{H}.
</math>

Our breadboard circuit follows the same functional topology, but its component values and parasitic elements are not identical to the PCB values above. The horizontal axis of our main <math>f-L_A</math> plot is therefore the series-combined <math>L_A</math> selected from our available small inductors, while the measured frequency <math>f</math> is the robust experimental readout.

=== Inductance tuning methods ===
We use two complementary methods to change the inductance seen by resonator <math>A</math>.

The first method is used for the main <math>f-L_A</math> sweep. Small inductors are connected in series so that the total inductance changes in discrete steps of approximately <math>2~\mu\mathrm{H}</math>. If the inductors are sufficiently separated so that their mutual magnetic coupling is small, the total series inductance is approximately the sum of the individual inductances,

<math display="block">
L_A \approx L_{\mathrm{base}}+\sum_i L_i .
</math>

This method is simple and reproducible. Adding one small inductor increases <math>L_A</math>; removing it decreases <math>L_A</math>. The circuit frequency is then measured for each chosen value. This produces a discrete version of the <math>f-L_A</math> curve in Fig. S11(a) of SM-4. The step size is not as fine as a commercial variable inductor, but <math>2~\mu\mathrm{H}</math> spacing is small enough to observe the frequency trend over the accessible range.

The second method is a homebrew compressible variable inductor. This coil is useful because it makes the sensing mechanism visible: a mechanical displacement changes the coil geometry, which changes the inductance, which then changes the oscillation frequency. The basic idea can be understood from the approximate solenoid formula

<math display="block">
L \approx \mu_\mathrm{eff}\frac{N^2 A}{\ell},
</math>

where <math>N</math> is the number of turns, <math>A</math> is the cross-sectional area, <math>\ell</math> is the coil length, and <math>\mu_\mathrm{eff}</math> describes the effective magnetic permeability of the core and surrounding region.<ref name="Chen2014">S.-C. Chen, D.-K. Le, and V.-S. Nguyen, Inductive displacement sensors with a notch filter for an active magnetic bearing system, ''Sensors'' '''14''', 12640-12657 (2014).</ref> During compression, the number of turns <math>N</math> is unchanged and the cross-sectional area <math>A</math> remains approximately unchanged, but the length <math>\ell</math> decreases. Equivalently, the turn density

<math display="block">
n=\frac{N}{\ell}
</math>

increases. The magnetic field produced by a solenoid is proportional to turn density. Therefore, compressing the coil increases the magnetic flux linkage through the turns and increases the inductance. Stretching the coil has the opposite effect.

This mechanical tuning is not identical to the precision tunable inductor used in the reference experiment. A real compressed coil is not an ideal long solenoid: the spacing between turns is finite, the end effects change as the coil length changes, nearby wires and the breadboard add parasitic capacitance, and the leads contribute additional series inductance and resistance. For this reason, the homebrew inductor is best treated as an experimentally characterized tuning element rather than an ideal component whose value is known exactly from geometry. In the present project, it provides a direct mechanical demonstration of how displacement can perturb <math>L_A</math>.

Experimentally, we measured the effective inductance of this homebrew variable inductor at different compression states. The compression process changes the coil length and wire density while leaving the coil area approximately fixed. Therefore, the tuning mechanism is physically the same as changing <math>\ell</math> and <math>n=N/\ell</math> in the solenoid model, rather than changing the number of turns or the coil diameter.

=== Kirchhoff model of the nonlinear circuit ===
A convenient state-space description uses the node voltages <math>V_A(t)</math> and <math>V_B(t)</math> together with the inductor currents <math>I_A(t)</math> and <math>I_B(t)</math>. A node voltage is the voltage measured at one resonator node relative to ground. Kirchhoff's current law says that the algebraic sum of all currents flowing away from a node must be zero. This gives a direct way to translate the circuit diagram into differential equations.

Denoting the amplitude-dependent conductance of the active branch by <math>G_A(|V_A|)</math>, where <math>G_A<0</math> in the small-signal regime, the nodal equations can be written as

<math display="block">
\begin{aligned}
C_0 \frac{dV_A}{dt} + I_A + G_A(|V_A|)\,V_A + \frac{V_A-V_B}{R_c}
+ C_c\frac{d(V_A-V_B)}{dt} &= 0,\\
C_0 \frac{dV_B}{dt} + I_B + \frac{V_B}{R_B} + \frac{V_B-V_A}{R_c}
+ C_c\frac{d(V_B-V_A)}{dt} &= 0,\\
L_A \frac{dI_A}{dt} &= V_A,\\
L_B \frac{dI_B}{dt} &= V_B.
\end{aligned}
</math>

Each term has a simple circuit meaning. The terms <math>C_0 dV_A/dt</math> and <math>C_0 dV_B/dt</math> are capacitor currents. The terms <math>I_A</math> and <math>I_B</math> are currents through the inductors. The term <math>G_A(|V_A|)V_A</math> is the active branch current; because <math>G_A</math> can be negative, this branch can inject energy instead of dissipating it. The term <math>V_B/R_B</math> is the ordinary loss current through the resistor in resonator <math>B</math>. The terms involving <math>R_c</math> and <math>C_c</math> describe current flowing between resonators <math>A</math> and <math>B</math>, so they are the electrical coupling between the two resonators.

These equations show explicitly how the experiment differs from a linear EP circuit. If <math>G_A</math> were a constant negative conductance, the model would reduce to a linear non-Hermitian dimer. Here, however, <math>G_A</math> depends on the voltage amplitude, so the steady state must be determined self-consistently together with the oscillation amplitude itself.

=== Harmonic-balance formulation of the steady state ===
After the transient has died away, the oscilloscope waveform is approximately periodic. The simplest approximation is therefore to keep only the dominant sinusoidal component. This is called a harmonic-balance approximation. For a stable activated oscillating state, one writes

<math display="block">
V_j(t)=\mathrm{Re}\!\left[\tilde V_j e^{i\omega t}\right],
\qquad j=A,B.
</math>

Substituting this ansatz into the circuit equations yields a nonlinear algebraic problem of the form

<math display="block">
\mathbf{Y}(\omega,|\tilde V_A|)
\begin{pmatrix}
\tilde V_A\\
\tilde V_B
\end{pmatrix}
=
\begin{pmatrix}
Y_A(\omega,|\tilde V_A|)+Y_c(\omega) & -Y_c(\omega)\\
-Y_c(\omega) & Y_B(\omega)+Y_c(\omega)
\end{pmatrix}
\begin{pmatrix}
\tilde V_A\\
\tilde V_B
\end{pmatrix}
=0,
</math>

with

<math display="block">
Y_A(\omega,|\tilde V_A|)=G_A(|\tilde V_A|)+i\omega C_0+\frac{1}{i\omega L_A},
</math>

<math display="block">
Y_B(\omega)=\frac{1}{R_B}+i\omega C_0+\frac{1}{i\omega L_B},
\qquad
Y_c(\omega)=\frac{1}{R_c}+i\omega C_c.
</math>

A nontrivial solution requires

<math display="block">
\det \mathbf{Y}(\omega,|\tilde V_A|)=0.
</math>

This determinant condition is the circuit analogue of an eigenfrequency condition. In a passive LC resonator, the resonance condition gives a single frequency <math>\omega_0=1/\sqrt{LC}</math>. In the present two-resonator active circuit, the same idea becomes a matrix equation because the two resonators are coupled. The amplitudes <math>\tilde V_A</math> and <math>\tilde V_B</math> are not arbitrary: a nonzero oscillation exists only when the admittance matrix has a zero eigenvalue.

Because the active admittance depends on amplitude, this condition is nonlinear and supports multiple branches. The supplementary material shows that the steady-state frequency can be reduced to a '''real-coefficient cubic polynomial''' in <math>\omega</math>. The existence of up to three real steady-state branches is therefore not accidental; it is the algebraic origin of the experimentally relevant <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM" />

=== Effective nonlinear-Hamiltonian picture ===
The same physics is often rewritten in coupled-mode form. This notation is common in non-Hermitian physics because it makes the circuit look like two coupled modes. The variables <math>\psi_A</math> and <math>\psi_B</math> represent the complex oscillation amplitudes in resonators <math>A</math> and <math>B</math>. The model is

<math display="block">
i\frac{d}{dt}
\begin{pmatrix}
\psi_A\\
\psi_B
\end{pmatrix}
=
\begin{pmatrix}
\omega_A+i\,g(|\psi_A|) & \kappa\\
\kappa & \omega_B-i\ell
\end{pmatrix}
\begin{pmatrix}
\psi_A\\
\psi_B
\end{pmatrix},
</math>

where <math>\omega_A</math> and <math>\omega_B</math> are the effective modal frequencies, <math>\kappa</math> is the coupling strength, <math>\ell</math> is the loss of resonator <math>B</math>, and <math>g(|\psi_A|)</math> is the saturable gain of resonator <math>A</math>. The steady-state problem then becomes

<math display="block">
H_s(\Psi)\Psi = \Omega \Psi,
</math>

with a real oscillation eigenfrequency <math>\Omega=2\pi f</math> for the stable oscillating branch. The word "real" is important: it means the imaginary part has vanished, so the mode is neither being amplified nor decaying in the steady state. In the cited experiment, this physical stable branch and two auxiliary steady branches coalesce at the singular point, producing a third-order nonlinear exceptional point rather than a conventional second-order linear EP. The auxiliary branches help form the cubic-root singularity but are not themselves stable physical outputs of the circuit.<ref name="Bai2024" /><ref name="Bai2024SM" />

== Measurements ==

=== Acquisition protocol and steady-state selection ===
The measurement is performed in the time domain. In the reference experiment, an arbitrary waveform generator first applies an external start-up signal of 1 V at 70 kHz in order to bring the system into the basin of attraction of the stable oscillating state.<ref name="Bai2024" /> In our breadboard measurement, an external voltage is likewise required to activate the resonators. The measured frequency is extracted from the activated steady waveform recorded on the oscilloscope.

The physical meaning of this procedure is important. The oscillation is not assumed to appear without activation; a supplied voltage is needed to bring the resonators into the relevant dynamical state. The measured oscillation frequency is then interpreted as the frequency selected by the nonlinear steady-state condition.

=== Extraction of the <math>f-L_A</math> observable ===
Ref. [1] does not treat the oscilloscope traces merely as qualitative evidence of oscillation. The supplementary explicitly states that the full reference experiment reports the steady-state frequency, the voltage ratio <math>|V_A/V_B|</math>, and the relative phase <math>\theta_A-\theta_B</math>, all extracted from the dominant peaks of the Fourier spectra of <math>V_A</math> and <math>V_B</math>.<ref name="Bai2024SM" />

Our available observable is the steady-state frequency. For each chosen value of <math>L_A</math>, the resonators are voltage-activated, the circuit reaches a stable oscillating state, and the dominant spectral peak is recorded. If we denote the Fourier transforms by

<math display="block">
\tilde V_A(f)=\mathcal{F}\{V_A(t)\},
\qquad
\tilde V_B(f)=\mathcal{F}\{V_B(t)\},
</math>

then the dominant oscillation frequency is obtained from

<math display="block">
f_0=\arg\max_f |\tilde V_A(f)| \approx \arg\max_f |\tilde V_B(f)|.
</math>

The experimental data set is therefore

<math display="block">
\{L_A,\ f_0(L_A)\},
</math>

which is compared with the frequency branch plotted in Fig. S11(a) of SM-4. In the full reference measurement, the amplitude ratio

<math display="block">
\rho = \frac{|\tilde V_A(f_0)|}{|\tilde V_B(f_0)|}
</math>

and relative phase

<math display="block">
\Delta\phi = \arg \tilde V_A(f_0)-\arg \tilde V_B(f_0)
</math>

provide additional state verification. Those quantities are useful but are not the main basis of our breadboard validation.

In practice, the same frequency can also be estimated directly from the time-domain period <math>T</math> using <math>f_0=1/T</math> when the waveform is clean. Fourier analysis is preferred because it uses a longer section of waveform and identifies the dominant oscillation peak even when there is small amplitude noise or waveform distortion.

=== Practical inductance sweep ===
In SM-4, the inductive tuning is carried out by varying <math>L_A</math>. The supplementary notes that the tunable inductor exhibits less than about 0.4% inductance variation over the frequency range of interest, which is sufficiently small for controlled sweeping near the singular point. To further suppress voltage-dependent drift during the sweep, the reference experiment slightly adjusts the active-branch resistance so as to maintain approximately <math>V_B \approx 2~\mathrm{V}</math> throughout the measurement series.<ref name="Bai2024SM" />

Our sweep uses controlled series combinations of small inductors, giving an approximately <math>2~\mu\mathrm{H}</math> step in <math>L_A</math>. The homebrew compressible coil is then used to show that a mechanical change of the sensing element also changes the inductance. The conclusion drawn from our data is therefore not a precision determination of <math>L_{A,\mathrm{NEP}}</math>, but a validation that the breadboard circuit produces the same kind of frequency response as the stable branch in Fig. S11(a).

=== Long-window spectral analysis and noise inspection ===
The supplementary also discusses long-time measurements over a window of <math>0<t<5~\mathrm{s}</math>. The motivation is that the noise around the stable oscillation is too small to be judged reliably from raw scope traces alone, so it is more informative to inspect the Fourier spectra and the extracted spectral center frequency and linewidth.<ref name="Bai2024SM" />

This is a useful methodological point for a sensing-oriented reproduction. In an NEP-based sensor, one ultimately cares about how precisely the frequency peak can be resolved and tracked. A long-window Fourier analysis is therefore closer to the sensing task than a purely visual time-domain inspection.

== Results ==

=== Result display: steady oscillation ===
[[File:PC5271inductive1.png|750px|thumb|center|Breadboard implementation and representative steady oscillation waveform.]]

This figure shows (a) our circuit implemented on a breadboard and (b) a representative stabilised voltage waveform after the resonators are activated by the supplied voltage. The displayed waveform is the evidence used to identify a stable oscillation frequency.

=== Result display: <math>f-L_A</math> sweep ===
[[File:PC5271inductive3.jpeg|750px|thumb|center|Measured breadboard <math>f-L_A</math> curve.]]

This figure shows the measured oscillation frequency <math>f</math> as a function of <math>L_A</math>. The values of <math>L_A</math> in the main sweep are produced by connecting small inductors in series with approximately <math>2~\mu\mathrm{H}</math> step size.

=== Result display: iron-rod demonstration ===
[[File:PC5271inductive2.png|750px|thumb|center|Displacement measurement by compressing the solenoid and inserting iron rod into the homebrew coil.]]

This figure exhibits the resultant changes in the measured oscillation frequency of homebrew coil with (a) a compression and (b) inserted iron rod. The experimental variable is the coil length and rod position, and the measured output is the activated steady oscillation frequency. As show in the analysis above, the inductance shows an inversely proportional dependence on the coil length and proportion dependence on insertion length, with the constant residual being the baseline inductance in the circuit irrelevant to the coil.

== Evaluation ==

=== Stable activated oscillation ===
The first required experimental signature is that, after the resonators are activated by the supplied voltage, the circuit reaches a stable oscillating waveform instead of decaying immediately or being amplified without bound. A stable waveform means that the TL071 active branch supplies enough small-signal negative resistance to compensate loss, while gain saturation prevents unbounded amplification.

Physically, this behavior confirms the role of the nonlinear active branch. At small amplitude, the effective negative resistance in resonator <math>A</math> overcomes dissipation. At larger amplitude, the gain becomes weaker, so the oscillation amplitude is clamped. The system self-stabilizes because the physical steady branch is the one with zero imaginary part of the eigenfrequency. The auxiliary branches near the nonlinear exceptional point are not stable physical outputs; they are mathematical branches of the nonlinear steady-state equation. The experiment is therefore not a passive ring-down measurement; it is a nonlinear steady-state selection process.

=== Evaluation of the <math>f-L_A</math> trend ===
The main experimental comparison is the measured <math>f-L_A</math> trend against Fig. S11(a) of SM-4. The reference experiment reports the stable frequency branch of the nonlinear circuit as <math>L_A</math> is tuned.<ref name="Bai2024SM" /> This branch is the physical one because its eigenfrequency is real in the steady state. In our breadboard experiment, the same observable is measured using the series-inductor sweep.

It is useful to separate two statements. First, any LC resonator changes frequency when <math>L</math> changes, because <math>f_0=1/(2\pi\sqrt{LC})</math>. This alone would not demonstrate the nonlinear exceptional-point mechanism. Second, in the coupled active-lossy circuit, the frequency is not just the bare LC resonance frequency of one isolated inductor and capacitor. It is the frequency selected by the nonlinear steady-state balance of gain, loss, and coupling. The reason Fig. S11(a) matters is that it shows this selected stable branch as <math>L_A</math> is swept.

Our breadboard circuit uses different component values and a different op-amp from the PCB experiment, so exact numerical overlap is not expected. The meaningful evaluation is whether the measured frequency branch follows the same qualitative <math>f-L_A</math> behavior as the stable branch in SM-4. Observing that behavior is enough to show that the constructed circuit realizes the intended nonlinear voltage-activated oscillating mechanism.

=== Evaluation of the displacement demonstration ===
In the iron-rod demonstration, the homebrew coil becomes a displacement-dependent inductive element. As the rod enters the coil, the effective magnetic permeability inside the coil increases. In the solenoid expression

<math display="block">
L(x) \approx \mu_\mathrm{eff}(x)\frac{N^2 A}{\ell},
</math>

the insertion depth <math>x</math> changes <math>\mu_\mathrm{eff}</math>. The coil geometry <math>N</math>, <math>A</math>, and <math>\ell</math> are approximately fixed during this demonstration, so the dominant effect is the change in magnetic core material inside the coil. Inductive displacement sensors commonly use this principle: the position of a ferromagnetic object changes a coil inductance, and the electronics convert that inductance change into an electrical signal.<ref name="Chen2014" />

In our circuit, the electrical signal is the activated steady oscillation frequency rather than a bridge voltage. The sensing chain is

<math display="block">
\mathrm{rod\ displacement}\ x
\longrightarrow
\mathrm{effective\ inductance}\ L_A(x)
\longrightarrow
\mathrm{oscillation\ frequency}\ f(x).
</math>

This demonstration shows how the inductive readout can be turned into a displacement-sensing readout after calibration of the relation between rod position and frequency.

=== Cubic-root perturbation law and sensing meaning ===
Near the singular operating point, the eigenfrequency shift follows the hallmark cubic-root law,

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3}.
</math>

This law is the main sensing-related result of the reference experiment.<ref name="Bai2024" /><ref name="Bai2024SM" /> Compared with the linear response of an ordinary isolated resonator, the cubic-root dependence can produce a larger frequency displacement for the same small perturbation of inductance when the device is biased sufficiently close to <math>L_{A,\mathrm{NEP}}</math>.

A useful way to interpret the result is through the local derivative,

<math display="block">
\left|\frac{df}{dL_A}\right| \propto |\delta L_A|^{-2/3}.
</math>

This increasing local responsivity is why the circuit is attractive for sensing. Our experiment supports this sensing logic by showing that the breadboard circuit has a measurable stable frequency branch as <math>L_A</math> is changed. A full quantitative exponent extraction would require identifying <math>L_{A,\mathrm{NEP}}</math> and <math>f_{\mathrm{NEP}}</math>, then plotting <math>|f-f_{\mathrm{NEP}}|</math> against <math>|L_A-L_{A,\mathrm{NEP}}|</math> on logarithmic axes.

=== Basis completeness and noise robustness ===
The supplementary material evaluates the Petermann factor as a diagnostic of modal nonorthogonality and basis completeness. For a non-Hermitian eigenproblem, one may write

<math display="block">
\mathrm{PF}_n =
\frac{\langle \Phi_n^{L}|\Phi_n^{L}\rangle \langle \Phi_n^{R}|\Phi_n^{R}\rangle}
{\left|\langle \Phi_n^{L}|\Phi_n^{R}\rangle\right|^2},
</math>

where <math>|\Phi_n^{R}\rangle</math> and <math>|\Phi_n^{L}\rangle</math> are the right and left eigenvectors of the effective steady-state Hamiltonian. At a conventional linear EP, this quantity is associated with diverging excess-noise sensitivity. In Ref. [1], the cited analysis finds that the effective steady-state Hamiltonian remains non-defective and the Petermann factor is finite at the <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM" />

This point explains the significance of the reference experiment. The system does not merely realize a large mathematical responsivity; it realizes that responsivity in a way that avoids the most severe basis-collapse problem of a linear EP. Our breadboard experiment does not directly test basis completeness, but the result provides the theoretical reason why reproducing the stable <math>f-L_A</math> branch is physically meaningful.

=== Overall interpretation ===
In this platform, the measurand is encoded as a perturbation of <math>L_A</math>. Any physical mechanism that modifies the effective inductance of resonator <math>A</math>, for example displacement, magnetic coupling, or the presence of a nearby magnetic target, moves the system away from the operating point and shifts the oscillation frequency. The primary readout in our implementation is therefore spectral: we track <math>f</math> as <math>L_A</math> is changed.

Taken together, Ref. [1] establishes the complete sensing logic: a nonlinear active-lossy circuit is tuned to an <math>\mathrm{NEP}_3</math>; after voltage activation, the system settles onto a stable oscillating branch; the dominant spectral peak is tracked as the perturbation readout; and the enhanced cubic-root response is obtained without the defective-basis pathology of a conventional linear EP. Our breadboard result demonstrates the experimentally accessible core of this logic by reproducing the stable frequency response corresponding to Fig. S11(a) of SM-4.

== References ==
<references />

Inductive Sensors of Ultra-high Sensitivity Based on Nonlinear Exceptional Point

2026-04-23T06:29:19Z

Siyu: /* Result display: steady oscillation */

== Introduction ==
This project is a implementation of the experiment reported in ''Observation of Nonlinear Exceptional Points with a Complete Basis in Dynamics'', adapted to our available laboratory equipment.<ref name="Bai2024">K. Bai, T.-R. Liu, L. Fang, J.-Z. Li, C. Lin, D. Wan, and M. Xiao, Observation of nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''132''', 073802 (2024).</ref> The circuit contains two coupled resonators. One resonator has gain, meaning that it can supply energy to the oscillation, while the other has loss, meaning that it dissipates energy. A system without energy conservation is called a '''non-Hermitian''' system. Such systems can show unusual degeneracies called '''exceptional points''' (EPs), where both the eigenfrequencies and the eigenmodes merge.

Exceptional points are interesting for sensing because the measured frequency can respond very strongly to a small perturbation. In an inductive sensor, the perturbation is a change in inductance. If a small change in inductance produces a large change in oscillation frequency, the circuit can in principle detect small physical changes. However, ordinary linear EPs also suffer from strong eigenmode nonorthogonality and a defective basis, which can increase noise sensitivity and complicate practical sensing.<ref name="Bai2024" />

The system studied in this project belongs to the recently developed class of '''nonlinear exceptional points''' (NEPs). The physical platform is a pair of coupled circuit resonators. Resonator <math>A</math> is made active by a saturable negative-resistance branch, while resonator <math>B</math> is intentionally lossy. Because the gain in resonator <math>A</math> depends on the oscillation amplitude, the steady-state problem is nonlinear and admits multiple self-consistent branches. At the singular operating point reported in Ref. [1], one stable steady branch and two auxiliary steady branches merge into a third-order nonlinear exceptional point, denoted <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM">Supplemental Material for K. Bai, T.-R. Liu, L. Fang, J.-Z. Li, C. Lin, D. Wan, and M. Xiao, Observation of nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''132''', 073802 (2024).</ref>

The experimentally important consequence is that the oscillation eigenfrequency obeys a cubic-root perturbation law near the singularity,

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3},
</math>

where <math>\delta L_A = L_A-L_{A,\mathrm{NEP}}</math> is the inductive detuning from the exceptional operating point. This response is steeper than the ordinary linear dependence of a conventional resonator and is therefore useful for ultra-sensitive inductive sensing.<ref name="Bai2024" /><ref name="Bai2024SM" />

A second and equally important point is that the cited work does not interpret the singularity as a conventional defective linear EP. Instead, the effective steady-state Hamiltonian remains non-defective in dynamics, and the associated Petermann factor stays finite at the operating point. In other words, the experiment realizes enhanced spectral responsivity without inheriting the strongest basis-collapse pathology of a standard linear EP.<ref name="Bai2024" /><ref name="Bai2024SM" />

In this project, our goal is to reproduce the experimental logic of Ref. [1] and its supplementary material under course-lab conditions. Unlike the PCB implementation in the paper, our circuit is assembled on a breadboard and the active branch uses a '''TL071''' operational amplifier. Our main observable is the steady oscillation frequency <math>f</math> as a function of the chosen inductance <math>L_A</math>. We tune <math>L_A</math> in two ways. For the main <math>f-L_A</math> sweep, we join small inductors in series so that the discrete inductance step is about <math>2~\mu\mathrm{H}</math>. In addition, we use a homebrew compressible coil to demonstrate the sensing mechanism directly. Compressing the coil changes its length and turn density while keeping the cross-sectional area approximately unchanged. We therefore use the <math>f-L_A</math> behavior in Fig. S11(a) of the supplementary material as the main experimental benchmark. Reproducing this trend is sufficient for demonstrating that our setup captures the intended nonlinear voltage-activated, self-stabilized oscillating response, although it is not a full reproduction of every diagnostic reported in Ref. [1].<ref name="Bai2024" /><ref name="Bai2024SM" />

== Team Members ==
Wang Peikun — E1538091@u.nus.edu 
Zhu Ziyang — E1583446@u.nus.edu 
Yuan Si-Yu — siyu_yuan@u.nus.edu 
Li Xunyu — xunyu@u.nus.edu

== Background Knowledge ==

=== Ordinary LC and RLC resonators ===
The basic building block of the experiment is an electrical resonator. An ideal LC resonator contains an inductor <math>L</math> and a capacitor <math>C</math>. Energy repeatedly moves between the magnetic field of the inductor and the electric field of the capacitor. The natural angular frequency is

<math display="block">
\omega_0=\frac{1}{\sqrt{LC}},
</math>

or, in ordinary frequency units,

<math display="block">
f_0=\frac{1}{2\pi\sqrt{LC}}.
</math>

This formula is important for the whole project because it explains why changing an inductance changes the measured frequency. A larger <math>L</math> gives a lower resonance frequency, while a smaller <math>L</math> gives a higher resonance frequency.

Real circuits are not ideal. Wires, inductors, capacitors, breadboard contacts, and measuring probes all dissipate energy. This loss is often represented by a resistor <math>R</math>, giving an RLC resonator. In a passive RLC circuit, the oscillation amplitude decays with time because energy is continuously lost as heat. If a circuit is to maintain a steady activated oscillation, the loss must be compensated by a gain element.

=== Gain, loss, and negative resistance ===
In electronics, gain can be represented as an effective negative resistance. A normal positive resistance removes energy from the resonator. A negative resistance does the opposite: it supplies energy to the resonator. In this experiment, resonator <math>A</math> contains an active op-amp branch that behaves like a negative resistance at small amplitude, while resonator <math>B</math> contains ordinary positive loss.

The gain cannot remain negative and constant forever. If it did, any activated oscillation would be amplified without bound until the op-amp clipped or the circuit failed. The important nonlinear ingredient is '''saturable gain''': the active branch gives strong gain at small amplitude but weaker gain at larger amplitude. The stable oscillation amplitude is selected when the supplied gain balances the total loss. This nonlinear steady state is why the experiment is a nonlinear dynamical problem rather than only a passive resonance measurement.

=== What "non-Hermitian" means in this circuit ===
In ordinary conservative physics, a Hermitian Hamiltonian describes a closed system whose total energy is conserved. Its eigenfrequencies are real, and its eigenmodes form a complete orthogonal basis. This is the familiar setting of many quantum-mechanics courses.

A circuit with gain and loss is an open system. Energy can enter through the op-amp branch and leave through resistive loss. The effective dynamical matrix is therefore generally '''non-Hermitian'''. In a non-Hermitian system, eigenfrequencies can be complex:

<math display="block">
\omega=\omega_\mathrm{r}+i\omega_\mathrm{i}.
</math>

The real part <math>\omega_\mathrm{r}</math> gives the oscillation frequency, while the imaginary part <math>\omega_\mathrm{i}</math> describes amplification or decay. A positive imaginary part means the mode is amplified, a negative imaginary part means it decays, and a stable steady oscillation corresponds to zero imaginary part, or equivalently zero net amplification. This is the sense in which the circuit is self-stabilized after voltage activation.

=== Exceptional points ===
In a usual Hermitian system, two modes can have the same eigenfrequency while still remaining two independent modes. This is an ordinary degeneracy. A non-Hermitian system can show a stronger kind of degeneracy called an '''exceptional point''' (EP). At an EP, not only the eigenvalues but also the eigenvectors coalesce. In simple language, two modes collapse into one mode.

This collapse makes EPs interesting for sensing. Near a second-order linear EP, a small perturbation can produce a square-root frequency splitting:

<math display="block">
\Delta f \propto \sqrt{\epsilon},
</math>

where <math>\epsilon</math> is the perturbation. This response is steeper than an ordinary linear response when <math>\epsilon</math> is very small. However, a conventional linear EP also has a serious drawback: the eigenbasis becomes defective, meaning that the system no longer has enough independent eigenvectors. This is associated with strong modal nonorthogonality and enhanced noise sensitivity. The Petermann factor is one common way to quantify this nonorthogonality.

=== Why the nonlinear exceptional point is different ===
The reference paper studies a nonlinear exceptional point. The word "nonlinear" is essential. The circuit does not have a fixed gain coefficient; instead, the gain depends on the oscillation amplitude. The steady state must therefore be solved self-consistently: the frequency, voltage amplitudes, and saturated gain must all agree with one another.

In the theory of Ref. [1], this nonlinear steady-state problem can reduce to a cubic equation for the oscillation frequency. A cubic equation can have a triple root. At the special operating point, one stable physical branch and two auxiliary branches merge, giving a third-order nonlinear exceptional point, written as <math>\mathrm{NEP}_3</math>. The stable physical branch is the one with real oscillation frequency, i.e. without an imaginary part. The auxiliary branches are needed to define the higher-order coalescence mathematically, but they are not physically stable operating states of the circuit. The corresponding frequency response has the form

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3}.
</math>

This cubic-root response is the key sensing signature. It is steeper than both an ordinary linear response and the square-root response of a second-order EP. The important conceptual point of Ref. [1] is that this enhanced response is obtained while the effective steady-state Hamiltonian can still retain a complete basis in dynamics.<ref name="Bai2024" /><ref name="Bai2024SM" />

=== Sensing mechanism ===
The circuit can be understood as a frequency-generating sensor. It converts an inductance change into a frequency change.

The first step is activation. A voltage is supplied to the resonators to excite the circuit into an oscillating state. The active branch in resonator <math>A</math> then supplies energy through an effective negative resistance and amplifies the selected oscillation mode.

The second step is self-stabilization. The gain is saturable, which means it becomes weaker when the voltage amplitude becomes larger. The activated oscillation therefore is not amplified forever. The circuit settles onto the physical steady branch where the imaginary part of the eigenfrequency is zero. At that point, the gain supplied by resonator <math>A</math> balances the loss in the circuit, especially the loss in resonator <math>B</math>.

The third step is frequency selection. Because resonators <math>A</math> and <math>B</math> are coupled, the final oscillation frequency is not simply the resonance frequency of either isolated resonator. It is selected by the whole coupled circuit: the inductors, capacitors, coupling elements, loss, and saturated gain all contribute.

The fourth step is sensing. The inductance <math>L_A</math> controls the bare frequency of resonator <math>A</math>. When <math>L_A</math> changes, the self-consistent frequency selected by the nonlinear circuit also changes. Therefore, tracking the spectral peak <math>f</math> tells us how the effective inductance has changed.

The fifth step is the nonlinear exceptional-point response. Near the nonlinear exceptional point discussed in Ref. [1], the physical steady branch and the two auxiliary branches coalesce in the nonlinear steady-state equation. The frequency shift of the stable branch can then scale as the cube root of the inductance perturbation. This means that a small inductance change can produce a comparatively large frequency change. Our breadboard experiment focuses on verifying the stable <math>f-L_A</math> branch corresponding to Fig. S11(a) of SM-4, which is the experimentally accessible part of this sensing mechanism.

=== Same mechanism in other platforms ===
Several references are included to show that this is a platform-independent sensing mechanism, not only a feature of our breadboard circuit. The common chain is

<math display="block">
\mathrm{physical\ perturbation}
\longrightarrow
\mathrm{parameter\ change}
\longrightarrow
\mathrm{eigenfrequency\ shift\ of\ a\ resonator/cavity\ with\ gain\ and\ loss}
\longrightarrow
\mathrm{higher\ order\ steady\ state\ response}
\longrightarrow
\mathrm{sensing}.
</math>

In our circuit, the physical perturbation is a change in the inductance <math>L_A</math>, and the measured output is the oscillation frequency <math>f</math>. In coupled optical, mechanical, electrical, or hybrid resonators, the perturbed parameter may instead be a cavity detuning, mechanical frequency, phase shift, magnetic-field-dependent transition, or another effective resonator parameter. The shared point is that gain, loss, coupling, and nonlinearity convert the parameter change into a higher-order steady-state spectral response.

The earlier theoretical work on nonlinear exceptional points showed that nonlinear coupled resonators can host higher-order NEPs while retaining a complete dynamical basis.<ref name="Bai2023">K. Bai, J.-Z. Li, T.-R. Liu, L. Fang, D. Wan, and M. Xiao, Nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''130''', 266901 (2023).</ref> The 2024 experiment implemented the same mechanism in coupled electrical resonators and measured the stable nonlinear frequency branch.<ref name="Bai2024" /><ref name="Bai2024SM" /> Silicon micromechanical resonators provide another realization: there, the measured output can be a phase difference, and the nonlinear exceptional-point response can enhance phase-shift sensing.<ref name="Xie2026">Y.-J. Xie, S.-Y. Zhang, M.-N. Zhang, R. Wang, L.-F. Wang, and Q.-A. Huang, Nonlinear exceptional points boost phase-shift sensing in silicon micromechanical resonators, ''Phys. Rev. Applied'' '''25''', 034039 (2026).</ref> Hybrid quantum systems with nonlinear bistability show a related route, where nonlinear transition dynamics can enhance sensing signal-to-noise ratio.<ref name="Wang2026">H. Wang ''et al.'', Exceptional sensitivity near the bistable transition point of a hybrid quantum system, ''Nature Physics'' '''22''', 577-583 (2026).</ref>

These examples support the same mechanism rather than unrelated applications. At the same time, enhanced response alone is not enough to prove better sensing. Noise can shift or weaken the apparent nonlinear exceptional point and can limit the signal-to-noise improvement.<ref name="Zheng2025">X. Zheng and Y. D. Chong, Noise constraints for nonlinear exceptional point sensing, ''Phys. Rev. Lett.'' '''134''', 133801 (2025).</ref> This is why our project makes a careful claim: we demonstrate the frequency-response mechanism on a breadboard, but we do not claim a complete metrological noise advantage.

== Report objectives and scope ==
The purpose of this work is to study the circuit principle behind '''nonlinear-exceptional-point-enhanced inductive sensing''' in a coupled electronic resonator platform. The scope is intentionally narrower than the full experiment reported in Ref. [1]. We are not attempting to redesign the active branch, derive all nonlinear bifurcation theory from first principles, reconstruct the auxiliary steady-state branches, or characterize every parasitic element. Instead, we adopt the framework of Ref. [1] and SM-4 and ask whether the breadboard circuit can reproduce the experimentally accessible frequency response.

The main target of our experiment is the <math>f-L_A</math> curve corresponding to Fig. S11(a) of SM-4. In the reference experiment, the steady-state frequency, voltage ratio <math>|V_A/V_B|</math>, and relative phase <math>\theta_A-\theta_B</math> are all measured as functions of <math>L_A</math>.<ref name="Bai2024SM" /> In our implementation, the frequency-versus-inductance curve is the primary observable. We tune <math>L_A</math> by connecting small inductors in series with approximately <math>2~\mu\mathrm{H}</math> resolution. We also use a homebrew coil in a separate demonstration where mechanical compression or insertion of an iron rod changes the effective inductance. The experimental goal is therefore to verify that the TL071 breadboard circuit settles into a stable oscillating state and that the measured frequency changes with <math>L_A</math> in the same qualitative manner as Fig. S11(a).

From the sensing perspective, the most relevant figure of merit is the local frequency responsivity. If

<math display="block">
|f-f_{\mathrm{NEP}}| = A |\delta L_A|^{1/3},
</math>

then differentiation gives

<math display="block">
S_f \equiv \left|\frac{df}{dL_A}\right| \propto |\delta L_A|^{-2/3}.
</math>

Thus, as the operating point approaches <math>L_{A,\mathrm{NEP}}</math>, the local spectral responsivity increases rapidly. The main practical constraint is that the system must remain on the intended stable branch while measurement noise and slow parameter drift are kept below the scale of the induced frequency shift.

== Setups ==
[[File:Experimental_setup.png|750px|thumb|center|Experimental setup adapted from Ref. [1].]]

=== Circuit topology and component implementation ===
The reference platform consists of two coupled RLC resonators. Both resonators contain the same nominal shunt capacitance <math>C_0</math>, while the inductances <math>L_A</math> and <math>L_B</math> set their bare resonance frequencies. The two resonators are coupled through a parallel resistor-capacitor channel characterized by <math>R_c</math> and <math>C_c</math>. Resonator <math>B</math> contains an explicit positive resistance <math>R_B</math> and therefore plays the role of the lossy resonator. Resonator <math>A</math>, by contrast, contains an active branch whose effective resistance is negative at small amplitude and becomes less negative as the oscillation amplitude increases, thereby providing saturable gain.<ref name="Bai2024" /><ref name="Bai2024SM" />

Our implementation follows this topology but is built on a breadboard rather than a custom PCB. The active branch is implemented with an op-amp negative-resistance circuit using a '''TL071''' operational amplifier. This is an important practical difference from the supplementary experiment, where the active components explicitly listed are a TI '''LM7171''' operational amplifier and an '''Onsemi BAV99L''' diode.<ref name="Bai2024SM" /> The TL071 substitution and the breadboard layout introduce different bandwidth limits, wiring parasitics, and contact resistances, so the numerical operating point is not expected to match the PCB experiment. For this reason, our comparison emphasizes the observable trend of <math>f</math> versus <math>L_A</math>.

The supplementary material reports the following representative component values for the PCB experiment:

<math display="block">
C_0 = 18.5~\mathrm{nF}, \qquad
C_c = 3.9~\mathrm{nF}, \qquad
R_c = 760~\Omega,
</math>

<math display="block">
L_B = 197.7~\mu\mathrm{H}, \qquad
R_B = 1314~\Omega.
</math>

A representative operating point shown in the main figure uses

<math display="block">
L_A = 245.4~\mu\mathrm{H}.
</math>

Our breadboard circuit follows the same functional topology, but its component values and parasitic elements are not identical to the PCB values above. The horizontal axis of our main <math>f-L_A</math> plot is therefore the series-combined <math>L_A</math> selected from our available small inductors, while the measured frequency <math>f</math> is the robust experimental readout.

=== Inductance tuning methods ===
We use two complementary methods to change the inductance seen by resonator <math>A</math>.

The first method is used for the main <math>f-L_A</math> sweep. Small inductors are connected in series so that the total inductance changes in discrete steps of approximately <math>2~\mu\mathrm{H}</math>. If the inductors are sufficiently separated so that their mutual magnetic coupling is small, the total series inductance is approximately the sum of the individual inductances,

<math display="block">
L_A \approx L_{\mathrm{base}}+\sum_i L_i .
</math>

This method is simple and reproducible. Adding one small inductor increases <math>L_A</math>; removing it decreases <math>L_A</math>. The circuit frequency is then measured for each chosen value. This produces a discrete version of the <math>f-L_A</math> curve in Fig. S11(a) of SM-4. The step size is not as fine as a commercial variable inductor, but <math>2~\mu\mathrm{H}</math> spacing is small enough to observe the frequency trend over the accessible range.

The second method is a homebrew compressible variable inductor. This coil is useful because it makes the sensing mechanism visible: a mechanical displacement changes the coil geometry, which changes the inductance, which then changes the oscillation frequency. The basic idea can be understood from the approximate solenoid formula

<math display="block">
L \approx \mu_\mathrm{eff}\frac{N^2 A}{\ell},
</math>

where <math>N</math> is the number of turns, <math>A</math> is the cross-sectional area, <math>\ell</math> is the coil length, and <math>\mu_\mathrm{eff}</math> describes the effective magnetic permeability of the core and surrounding region.<ref name="Chen2014">S.-C. Chen, D.-K. Le, and V.-S. Nguyen, Inductive displacement sensors with a notch filter for an active magnetic bearing system, ''Sensors'' '''14''', 12640-12657 (2014).</ref> During compression, the number of turns <math>N</math> is unchanged and the cross-sectional area <math>A</math> remains approximately unchanged, but the length <math>\ell</math> decreases. Equivalently, the turn density

<math display="block">
n=\frac{N}{\ell}
</math>

increases. The magnetic field produced by a solenoid is proportional to turn density. Therefore, compressing the coil increases the magnetic flux linkage through the turns and increases the inductance. Stretching the coil has the opposite effect.

This mechanical tuning is not identical to the precision tunable inductor used in the reference experiment. A real compressed coil is not an ideal long solenoid: the spacing between turns is finite, the end effects change as the coil length changes, nearby wires and the breadboard add parasitic capacitance, and the leads contribute additional series inductance and resistance. For this reason, the homebrew inductor is best treated as an experimentally characterized tuning element rather than an ideal component whose value is known exactly from geometry. In the present project, it provides a direct mechanical demonstration of how displacement can perturb <math>L_A</math>.

Experimentally, we measured the effective inductance of this homebrew variable inductor at different compression states. The compression process changes the coil length and wire density while leaving the coil area approximately fixed. Therefore, the tuning mechanism is physically the same as changing <math>\ell</math> and <math>n=N/\ell</math> in the solenoid model, rather than changing the number of turns or the coil diameter.

=== Kirchhoff model of the nonlinear circuit ===
A convenient state-space description uses the node voltages <math>V_A(t)</math> and <math>V_B(t)</math> together with the inductor currents <math>I_A(t)</math> and <math>I_B(t)</math>. A node voltage is the voltage measured at one resonator node relative to ground. Kirchhoff's current law says that the algebraic sum of all currents flowing away from a node must be zero. This gives a direct way to translate the circuit diagram into differential equations.

Denoting the amplitude-dependent conductance of the active branch by <math>G_A(|V_A|)</math>, where <math>G_A<0</math> in the small-signal regime, the nodal equations can be written as

<math display="block">
\begin{aligned}
C_0 \frac{dV_A}{dt} + I_A + G_A(|V_A|)\,V_A + \frac{V_A-V_B}{R_c}
+ C_c\frac{d(V_A-V_B)}{dt} &= 0,\\
C_0 \frac{dV_B}{dt} + I_B + \frac{V_B}{R_B} + \frac{V_B-V_A}{R_c}
+ C_c\frac{d(V_B-V_A)}{dt} &= 0,\\
L_A \frac{dI_A}{dt} &= V_A,\\
L_B \frac{dI_B}{dt} &= V_B.
\end{aligned}
</math>

Each term has a simple circuit meaning. The terms <math>C_0 dV_A/dt</math> and <math>C_0 dV_B/dt</math> are capacitor currents. The terms <math>I_A</math> and <math>I_B</math> are currents through the inductors. The term <math>G_A(|V_A|)V_A</math> is the active branch current; because <math>G_A</math> can be negative, this branch can inject energy instead of dissipating it. The term <math>V_B/R_B</math> is the ordinary loss current through the resistor in resonator <math>B</math>. The terms involving <math>R_c</math> and <math>C_c</math> describe current flowing between resonators <math>A</math> and <math>B</math>, so they are the electrical coupling between the two resonators.

These equations show explicitly how the experiment differs from a linear EP circuit. If <math>G_A</math> were a constant negative conductance, the model would reduce to a linear non-Hermitian dimer. Here, however, <math>G_A</math> depends on the voltage amplitude, so the steady state must be determined self-consistently together with the oscillation amplitude itself.

=== Harmonic-balance formulation of the steady state ===
After the transient has died away, the oscilloscope waveform is approximately periodic. The simplest approximation is therefore to keep only the dominant sinusoidal component. This is called a harmonic-balance approximation. For a stable activated oscillating state, one writes

<math display="block">
V_j(t)=\mathrm{Re}\!\left[\tilde V_j e^{i\omega t}\right],
\qquad j=A,B.
</math>

Substituting this ansatz into the circuit equations yields a nonlinear algebraic problem of the form

<math display="block">
\mathbf{Y}(\omega,|\tilde V_A|)
\begin{pmatrix}
\tilde V_A\\
\tilde V_B
\end{pmatrix}
=
\begin{pmatrix}
Y_A(\omega,|\tilde V_A|)+Y_c(\omega) & -Y_c(\omega)\\
-Y_c(\omega) & Y_B(\omega)+Y_c(\omega)
\end{pmatrix}
\begin{pmatrix}
\tilde V_A\\
\tilde V_B
\end{pmatrix}
=0,
</math>

with

<math display="block">
Y_A(\omega,|\tilde V_A|)=G_A(|\tilde V_A|)+i\omega C_0+\frac{1}{i\omega L_A},
</math>

<math display="block">
Y_B(\omega)=\frac{1}{R_B}+i\omega C_0+\frac{1}{i\omega L_B},
\qquad
Y_c(\omega)=\frac{1}{R_c}+i\omega C_c.
</math>

A nontrivial solution requires

<math display="block">
\det \mathbf{Y}(\omega,|\tilde V_A|)=0.
</math>

This determinant condition is the circuit analogue of an eigenfrequency condition. In a passive LC resonator, the resonance condition gives a single frequency <math>\omega_0=1/\sqrt{LC}</math>. In the present two-resonator active circuit, the same idea becomes a matrix equation because the two resonators are coupled. The amplitudes <math>\tilde V_A</math> and <math>\tilde V_B</math> are not arbitrary: a nonzero oscillation exists only when the admittance matrix has a zero eigenvalue.

Because the active admittance depends on amplitude, this condition is nonlinear and supports multiple branches. The supplementary material shows that the steady-state frequency can be reduced to a '''real-coefficient cubic polynomial''' in <math>\omega</math>. The existence of up to three real steady-state branches is therefore not accidental; it is the algebraic origin of the experimentally relevant <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM" />

=== Effective nonlinear-Hamiltonian picture ===
The same physics is often rewritten in coupled-mode form. This notation is common in non-Hermitian physics because it makes the circuit look like two coupled modes. The variables <math>\psi_A</math> and <math>\psi_B</math> represent the complex oscillation amplitudes in resonators <math>A</math> and <math>B</math>. The model is

<math display="block">
i\frac{d}{dt}
\begin{pmatrix}
\psi_A\\
\psi_B
\end{pmatrix}
=
\begin{pmatrix}
\omega_A+i\,g(|\psi_A|) & \kappa\\
\kappa & \omega_B-i\ell
\end{pmatrix}
\begin{pmatrix}
\psi_A\\
\psi_B
\end{pmatrix},
</math>

where <math>\omega_A</math> and <math>\omega_B</math> are the effective modal frequencies, <math>\kappa</math> is the coupling strength, <math>\ell</math> is the loss of resonator <math>B</math>, and <math>g(|\psi_A|)</math> is the saturable gain of resonator <math>A</math>. The steady-state problem then becomes

<math display="block">
H_s(\Psi)\Psi = \Omega \Psi,
</math>

with a real oscillation eigenfrequency <math>\Omega=2\pi f</math> for the stable oscillating branch. The word "real" is important: it means the imaginary part has vanished, so the mode is neither being amplified nor decaying in the steady state. In the cited experiment, this physical stable branch and two auxiliary steady branches coalesce at the singular point, producing a third-order nonlinear exceptional point rather than a conventional second-order linear EP. The auxiliary branches help form the cubic-root singularity but are not themselves stable physical outputs of the circuit.<ref name="Bai2024" /><ref name="Bai2024SM" />

== Measurements ==

=== Acquisition protocol and steady-state selection ===
The measurement is performed in the time domain. In the reference experiment, an arbitrary waveform generator first applies an external start-up signal of 1 V at 70 kHz in order to bring the system into the basin of attraction of the stable oscillating state.<ref name="Bai2024" /> In our breadboard measurement, an external voltage is likewise required to activate the resonators. The measured frequency is extracted from the activated steady waveform recorded on the oscilloscope.

The physical meaning of this procedure is important. The oscillation is not assumed to appear without activation; a supplied voltage is needed to bring the resonators into the relevant dynamical state. The measured oscillation frequency is then interpreted as the frequency selected by the nonlinear steady-state condition.

=== Extraction of the <math>f-L_A</math> observable ===
Ref. [1] does not treat the oscilloscope traces merely as qualitative evidence of oscillation. The supplementary explicitly states that the full reference experiment reports the steady-state frequency, the voltage ratio <math>|V_A/V_B|</math>, and the relative phase <math>\theta_A-\theta_B</math>, all extracted from the dominant peaks of the Fourier spectra of <math>V_A</math> and <math>V_B</math>.<ref name="Bai2024SM" />

Our available observable is the steady-state frequency. For each chosen value of <math>L_A</math>, the resonators are voltage-activated, the circuit reaches a stable oscillating state, and the dominant spectral peak is recorded. If we denote the Fourier transforms by

<math display="block">
\tilde V_A(f)=\mathcal{F}\{V_A(t)\},
\qquad
\tilde V_B(f)=\mathcal{F}\{V_B(t)\},
</math>

then the dominant oscillation frequency is obtained from

<math display="block">
f_0=\arg\max_f |\tilde V_A(f)| \approx \arg\max_f |\tilde V_B(f)|.
</math>

The experimental data set is therefore

<math display="block">
\{L_A,\ f_0(L_A)\},
</math>

which is compared with the frequency branch plotted in Fig. S11(a) of SM-4. In the full reference measurement, the amplitude ratio

<math display="block">
\rho = \frac{|\tilde V_A(f_0)|}{|\tilde V_B(f_0)|}
</math>

and relative phase

<math display="block">
\Delta\phi = \arg \tilde V_A(f_0)-\arg \tilde V_B(f_0)
</math>

provide additional state verification. Those quantities are useful but are not the main basis of our breadboard validation.

In practice, the same frequency can also be estimated directly from the time-domain period <math>T</math> using <math>f_0=1/T</math> when the waveform is clean. Fourier analysis is preferred because it uses a longer section of waveform and identifies the dominant oscillation peak even when there is small amplitude noise or waveform distortion.

=== Practical inductance sweep ===
In SM-4, the inductive tuning is carried out by varying <math>L_A</math>. The supplementary notes that the tunable inductor exhibits less than about 0.4% inductance variation over the frequency range of interest, which is sufficiently small for controlled sweeping near the singular point. To further suppress voltage-dependent drift during the sweep, the reference experiment slightly adjusts the active-branch resistance so as to maintain approximately <math>V_B \approx 2~\mathrm{V}</math> throughout the measurement series.<ref name="Bai2024SM" />

Our sweep uses controlled series combinations of small inductors, giving an approximately <math>2~\mu\mathrm{H}</math> step in <math>L_A</math>. The homebrew compressible coil is then used to show that a mechanical change of the sensing element also changes the inductance. The conclusion drawn from our data is therefore not a precision determination of <math>L_{A,\mathrm{NEP}}</math>, but a validation that the breadboard circuit produces the same kind of frequency response as the stable branch in Fig. S11(a).

=== Long-window spectral analysis and noise inspection ===
The supplementary also discusses long-time measurements over a window of <math>0<t<5~\mathrm{s}</math>. The motivation is that the noise around the stable oscillation is too small to be judged reliably from raw scope traces alone, so it is more informative to inspect the Fourier spectra and the extracted spectral center frequency and linewidth.<ref name="Bai2024SM" />

This is a useful methodological point for a sensing-oriented reproduction. In an NEP-based sensor, one ultimately cares about how precisely the frequency peak can be resolved and tracked. A long-window Fourier analysis is therefore closer to the sensing task than a purely visual time-domain inspection.

== Results ==

=== Result display: steady oscillation ===
[[File:PC5271inductive1.png|750px|thumb|center|Breadboard implementation and representative steady oscillation waveform.]]

This figure shows (a) our circuit implemented on a breadboard and (b) a representative stabilised voltage waveform after the resonators are activated by the supplied voltage. The displayed waveform is the evidence used to identify a stable oscillation frequency.

=== Result display: <math>f-L_A</math> sweep ===
[[File:PC5271inductive3.jpeg|750px|thumb|center|Measured breadboard <math>f-L_A</math> curve.]]

This figure shows the measured oscillation frequency <math>f</math> as a function of <math>L_A</math>. The values of <math>L_A</math> in the main sweep are produced by connecting small inductors in series with approximately <math>2~\mu\mathrm{H}</math> step size.

=== Result display: iron-rod demonstration ===
[[File:PC5271inductive2.png|750px|thumb|center|Displacement measurement by compressing the solenoid and inserting iron rod into the homebrew coil.]]

This figure exhibits the resultant changes in the measured oscillation frequency of homebrew coil with a compression (a) and inserted iron rod (b). The experimental variable is the coil length and rod position, and the measured output is the activated steady oscillation frequency. As show in the analysis above, the inductance shows an inversely proportional dependence on the coil length and proportion dependence on insertion length, with the constant residual being the baseline inductance in the circuit irrelevant to the coil.

== Evaluation ==

=== Stable activated oscillation ===
The first required experimental signature is that, after the resonators are activated by the supplied voltage, the circuit reaches a stable oscillating waveform instead of decaying immediately or being amplified without bound. A stable waveform means that the TL071 active branch supplies enough small-signal negative resistance to compensate loss, while gain saturation prevents unbounded amplification.

Physically, this behavior confirms the role of the nonlinear active branch. At small amplitude, the effective negative resistance in resonator <math>A</math> overcomes dissipation. At larger amplitude, the gain becomes weaker, so the oscillation amplitude is clamped. The system self-stabilizes because the physical steady branch is the one with zero imaginary part of the eigenfrequency. The auxiliary branches near the nonlinear exceptional point are not stable physical outputs; they are mathematical branches of the nonlinear steady-state equation. The experiment is therefore not a passive ring-down measurement; it is a nonlinear steady-state selection process.

=== Evaluation of the <math>f-L_A</math> trend ===
The main experimental comparison is the measured <math>f-L_A</math> trend against Fig. S11(a) of SM-4. The reference experiment reports the stable frequency branch of the nonlinear circuit as <math>L_A</math> is tuned.<ref name="Bai2024SM" /> This branch is the physical one because its eigenfrequency is real in the steady state. In our breadboard experiment, the same observable is measured using the series-inductor sweep.

It is useful to separate two statements. First, any LC resonator changes frequency when <math>L</math> changes, because <math>f_0=1/(2\pi\sqrt{LC})</math>. This alone would not demonstrate the nonlinear exceptional-point mechanism. Second, in the coupled active-lossy circuit, the frequency is not just the bare LC resonance frequency of one isolated inductor and capacitor. It is the frequency selected by the nonlinear steady-state balance of gain, loss, and coupling. The reason Fig. S11(a) matters is that it shows this selected stable branch as <math>L_A</math> is swept.

Our breadboard circuit uses different component values and a different op-amp from the PCB experiment, so exact numerical overlap is not expected. The meaningful evaluation is whether the measured frequency branch follows the same qualitative <math>f-L_A</math> behavior as the stable branch in SM-4. Observing that behavior is enough to show that the constructed circuit realizes the intended nonlinear voltage-activated oscillating mechanism.

=== Evaluation of the displacement demonstration ===
In the iron-rod demonstration, the homebrew coil becomes a displacement-dependent inductive element. As the rod enters the coil, the effective magnetic permeability inside the coil increases. In the solenoid expression

<math display="block">
L(x) \approx \mu_\mathrm{eff}(x)\frac{N^2 A}{\ell},
</math>

the insertion depth <math>x</math> changes <math>\mu_\mathrm{eff}</math>. The coil geometry <math>N</math>, <math>A</math>, and <math>\ell</math> are approximately fixed during this demonstration, so the dominant effect is the change in magnetic core material inside the coil. Inductive displacement sensors commonly use this principle: the position of a ferromagnetic object changes a coil inductance, and the electronics convert that inductance change into an electrical signal.<ref name="Chen2014" />

In our circuit, the electrical signal is the activated steady oscillation frequency rather than a bridge voltage. The sensing chain is

<math display="block">
\mathrm{rod\ displacement}\ x
\longrightarrow
\mathrm{effective\ inductance}\ L_A(x)
\longrightarrow
\mathrm{oscillation\ frequency}\ f(x).
</math>

This demonstration shows how the inductive readout can be turned into a displacement-sensing readout after calibration of the relation between rod position and frequency.

=== Cubic-root perturbation law and sensing meaning ===
Near the singular operating point, the eigenfrequency shift follows the hallmark cubic-root law,

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3}.
</math>

This law is the main sensing-related result of the reference experiment.<ref name="Bai2024" /><ref name="Bai2024SM" /> Compared with the linear response of an ordinary isolated resonator, the cubic-root dependence can produce a larger frequency displacement for the same small perturbation of inductance when the device is biased sufficiently close to <math>L_{A,\mathrm{NEP}}</math>.

A useful way to interpret the result is through the local derivative,

<math display="block">
\left|\frac{df}{dL_A}\right| \propto |\delta L_A|^{-2/3}.
</math>

This increasing local responsivity is why the circuit is attractive for sensing. Our experiment supports this sensing logic by showing that the breadboard circuit has a measurable stable frequency branch as <math>L_A</math> is changed. A full quantitative exponent extraction would require identifying <math>L_{A,\mathrm{NEP}}</math> and <math>f_{\mathrm{NEP}}</math>, then plotting <math>|f-f_{\mathrm{NEP}}|</math> against <math>|L_A-L_{A,\mathrm{NEP}}|</math> on logarithmic axes.

=== Basis completeness and noise robustness ===
The supplementary material evaluates the Petermann factor as a diagnostic of modal nonorthogonality and basis completeness. For a non-Hermitian eigenproblem, one may write

<math display="block">
\mathrm{PF}_n =
\frac{\langle \Phi_n^{L}|\Phi_n^{L}\rangle \langle \Phi_n^{R}|\Phi_n^{R}\rangle}
{\left|\langle \Phi_n^{L}|\Phi_n^{R}\rangle\right|^2},
</math>

where <math>|\Phi_n^{R}\rangle</math> and <math>|\Phi_n^{L}\rangle</math> are the right and left eigenvectors of the effective steady-state Hamiltonian. At a conventional linear EP, this quantity is associated with diverging excess-noise sensitivity. In Ref. [1], the cited analysis finds that the effective steady-state Hamiltonian remains non-defective and the Petermann factor is finite at the <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM" />

This point explains the significance of the reference experiment. The system does not merely realize a large mathematical responsivity; it realizes that responsivity in a way that avoids the most severe basis-collapse problem of a linear EP. Our breadboard experiment does not directly test basis completeness, but the result provides the theoretical reason why reproducing the stable <math>f-L_A</math> branch is physically meaningful.

=== Overall interpretation ===
In this platform, the measurand is encoded as a perturbation of <math>L_A</math>. Any physical mechanism that modifies the effective inductance of resonator <math>A</math>, for example displacement, magnetic coupling, or the presence of a nearby magnetic target, moves the system away from the operating point and shifts the oscillation frequency. The primary readout in our implementation is therefore spectral: we track <math>f</math> as <math>L_A</math> is changed.

Taken together, Ref. [1] establishes the complete sensing logic: a nonlinear active-lossy circuit is tuned to an <math>\mathrm{NEP}_3</math>; after voltage activation, the system settles onto a stable oscillating branch; the dominant spectral peak is tracked as the perturbation readout; and the enhanced cubic-root response is obtained without the defective-basis pathology of a conventional linear EP. Our breadboard result demonstrates the experimentally accessible core of this logic by reproducing the stable frequency response corresponding to Fig. S11(a) of SM-4.

== References ==
<references />

Inductive Sensors of Ultra-high Sensitivity Based on Nonlinear Exceptional Point

2026-04-23T06:28:41Z

Siyu: /* Result display: iron-rod demonstration */

== Introduction ==
This project is a implementation of the experiment reported in ''Observation of Nonlinear Exceptional Points with a Complete Basis in Dynamics'', adapted to our available laboratory equipment.<ref name="Bai2024">K. Bai, T.-R. Liu, L. Fang, J.-Z. Li, C. Lin, D. Wan, and M. Xiao, Observation of nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''132''', 073802 (2024).</ref> The circuit contains two coupled resonators. One resonator has gain, meaning that it can supply energy to the oscillation, while the other has loss, meaning that it dissipates energy. A system without energy conservation is called a '''non-Hermitian''' system. Such systems can show unusual degeneracies called '''exceptional points''' (EPs), where both the eigenfrequencies and the eigenmodes merge.

Exceptional points are interesting for sensing because the measured frequency can respond very strongly to a small perturbation. In an inductive sensor, the perturbation is a change in inductance. If a small change in inductance produces a large change in oscillation frequency, the circuit can in principle detect small physical changes. However, ordinary linear EPs also suffer from strong eigenmode nonorthogonality and a defective basis, which can increase noise sensitivity and complicate practical sensing.<ref name="Bai2024" />

The system studied in this project belongs to the recently developed class of '''nonlinear exceptional points''' (NEPs). The physical platform is a pair of coupled circuit resonators. Resonator <math>A</math> is made active by a saturable negative-resistance branch, while resonator <math>B</math> is intentionally lossy. Because the gain in resonator <math>A</math> depends on the oscillation amplitude, the steady-state problem is nonlinear and admits multiple self-consistent branches. At the singular operating point reported in Ref. [1], one stable steady branch and two auxiliary steady branches merge into a third-order nonlinear exceptional point, denoted <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM">Supplemental Material for K. Bai, T.-R. Liu, L. Fang, J.-Z. Li, C. Lin, D. Wan, and M. Xiao, Observation of nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''132''', 073802 (2024).</ref>

The experimentally important consequence is that the oscillation eigenfrequency obeys a cubic-root perturbation law near the singularity,

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3},
</math>

where <math>\delta L_A = L_A-L_{A,\mathrm{NEP}}</math> is the inductive detuning from the exceptional operating point. This response is steeper than the ordinary linear dependence of a conventional resonator and is therefore useful for ultra-sensitive inductive sensing.<ref name="Bai2024" /><ref name="Bai2024SM" />

A second and equally important point is that the cited work does not interpret the singularity as a conventional defective linear EP. Instead, the effective steady-state Hamiltonian remains non-defective in dynamics, and the associated Petermann factor stays finite at the operating point. In other words, the experiment realizes enhanced spectral responsivity without inheriting the strongest basis-collapse pathology of a standard linear EP.<ref name="Bai2024" /><ref name="Bai2024SM" />

In this project, our goal is to reproduce the experimental logic of Ref. [1] and its supplementary material under course-lab conditions. Unlike the PCB implementation in the paper, our circuit is assembled on a breadboard and the active branch uses a '''TL071''' operational amplifier. Our main observable is the steady oscillation frequency <math>f</math> as a function of the chosen inductance <math>L_A</math>. We tune <math>L_A</math> in two ways. For the main <math>f-L_A</math> sweep, we join small inductors in series so that the discrete inductance step is about <math>2~\mu\mathrm{H}</math>. In addition, we use a homebrew compressible coil to demonstrate the sensing mechanism directly. Compressing the coil changes its length and turn density while keeping the cross-sectional area approximately unchanged. We therefore use the <math>f-L_A</math> behavior in Fig. S11(a) of the supplementary material as the main experimental benchmark. Reproducing this trend is sufficient for demonstrating that our setup captures the intended nonlinear voltage-activated, self-stabilized oscillating response, although it is not a full reproduction of every diagnostic reported in Ref. [1].<ref name="Bai2024" /><ref name="Bai2024SM" />

== Team Members ==
Wang Peikun — E1538091@u.nus.edu 
Zhu Ziyang — E1583446@u.nus.edu 
Yuan Si-Yu — siyu_yuan@u.nus.edu 
Li Xunyu — xunyu@u.nus.edu

== Background Knowledge ==

=== Ordinary LC and RLC resonators ===
The basic building block of the experiment is an electrical resonator. An ideal LC resonator contains an inductor <math>L</math> and a capacitor <math>C</math>. Energy repeatedly moves between the magnetic field of the inductor and the electric field of the capacitor. The natural angular frequency is

<math display="block">
\omega_0=\frac{1}{\sqrt{LC}},
</math>

or, in ordinary frequency units,

<math display="block">
f_0=\frac{1}{2\pi\sqrt{LC}}.
</math>

This formula is important for the whole project because it explains why changing an inductance changes the measured frequency. A larger <math>L</math> gives a lower resonance frequency, while a smaller <math>L</math> gives a higher resonance frequency.

Real circuits are not ideal. Wires, inductors, capacitors, breadboard contacts, and measuring probes all dissipate energy. This loss is often represented by a resistor <math>R</math>, giving an RLC resonator. In a passive RLC circuit, the oscillation amplitude decays with time because energy is continuously lost as heat. If a circuit is to maintain a steady activated oscillation, the loss must be compensated by a gain element.

=== Gain, loss, and negative resistance ===
In electronics, gain can be represented as an effective negative resistance. A normal positive resistance removes energy from the resonator. A negative resistance does the opposite: it supplies energy to the resonator. In this experiment, resonator <math>A</math> contains an active op-amp branch that behaves like a negative resistance at small amplitude, while resonator <math>B</math> contains ordinary positive loss.

The gain cannot remain negative and constant forever. If it did, any activated oscillation would be amplified without bound until the op-amp clipped or the circuit failed. The important nonlinear ingredient is '''saturable gain''': the active branch gives strong gain at small amplitude but weaker gain at larger amplitude. The stable oscillation amplitude is selected when the supplied gain balances the total loss. This nonlinear steady state is why the experiment is a nonlinear dynamical problem rather than only a passive resonance measurement.

=== What "non-Hermitian" means in this circuit ===
In ordinary conservative physics, a Hermitian Hamiltonian describes a closed system whose total energy is conserved. Its eigenfrequencies are real, and its eigenmodes form a complete orthogonal basis. This is the familiar setting of many quantum-mechanics courses.

A circuit with gain and loss is an open system. Energy can enter through the op-amp branch and leave through resistive loss. The effective dynamical matrix is therefore generally '''non-Hermitian'''. In a non-Hermitian system, eigenfrequencies can be complex:

<math display="block">
\omega=\omega_\mathrm{r}+i\omega_\mathrm{i}.
</math>

The real part <math>\omega_\mathrm{r}</math> gives the oscillation frequency, while the imaginary part <math>\omega_\mathrm{i}</math> describes amplification or decay. A positive imaginary part means the mode is amplified, a negative imaginary part means it decays, and a stable steady oscillation corresponds to zero imaginary part, or equivalently zero net amplification. This is the sense in which the circuit is self-stabilized after voltage activation.

=== Exceptional points ===
In a usual Hermitian system, two modes can have the same eigenfrequency while still remaining two independent modes. This is an ordinary degeneracy. A non-Hermitian system can show a stronger kind of degeneracy called an '''exceptional point''' (EP). At an EP, not only the eigenvalues but also the eigenvectors coalesce. In simple language, two modes collapse into one mode.

This collapse makes EPs interesting for sensing. Near a second-order linear EP, a small perturbation can produce a square-root frequency splitting:

<math display="block">
\Delta f \propto \sqrt{\epsilon},
</math>

where <math>\epsilon</math> is the perturbation. This response is steeper than an ordinary linear response when <math>\epsilon</math> is very small. However, a conventional linear EP also has a serious drawback: the eigenbasis becomes defective, meaning that the system no longer has enough independent eigenvectors. This is associated with strong modal nonorthogonality and enhanced noise sensitivity. The Petermann factor is one common way to quantify this nonorthogonality.

=== Why the nonlinear exceptional point is different ===
The reference paper studies a nonlinear exceptional point. The word "nonlinear" is essential. The circuit does not have a fixed gain coefficient; instead, the gain depends on the oscillation amplitude. The steady state must therefore be solved self-consistently: the frequency, voltage amplitudes, and saturated gain must all agree with one another.

In the theory of Ref. [1], this nonlinear steady-state problem can reduce to a cubic equation for the oscillation frequency. A cubic equation can have a triple root. At the special operating point, one stable physical branch and two auxiliary branches merge, giving a third-order nonlinear exceptional point, written as <math>\mathrm{NEP}_3</math>. The stable physical branch is the one with real oscillation frequency, i.e. without an imaginary part. The auxiliary branches are needed to define the higher-order coalescence mathematically, but they are not physically stable operating states of the circuit. The corresponding frequency response has the form

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3}.
</math>

This cubic-root response is the key sensing signature. It is steeper than both an ordinary linear response and the square-root response of a second-order EP. The important conceptual point of Ref. [1] is that this enhanced response is obtained while the effective steady-state Hamiltonian can still retain a complete basis in dynamics.<ref name="Bai2024" /><ref name="Bai2024SM" />

=== Sensing mechanism ===
The circuit can be understood as a frequency-generating sensor. It converts an inductance change into a frequency change.

The first step is activation. A voltage is supplied to the resonators to excite the circuit into an oscillating state. The active branch in resonator <math>A</math> then supplies energy through an effective negative resistance and amplifies the selected oscillation mode.

The second step is self-stabilization. The gain is saturable, which means it becomes weaker when the voltage amplitude becomes larger. The activated oscillation therefore is not amplified forever. The circuit settles onto the physical steady branch where the imaginary part of the eigenfrequency is zero. At that point, the gain supplied by resonator <math>A</math> balances the loss in the circuit, especially the loss in resonator <math>B</math>.

The third step is frequency selection. Because resonators <math>A</math> and <math>B</math> are coupled, the final oscillation frequency is not simply the resonance frequency of either isolated resonator. It is selected by the whole coupled circuit: the inductors, capacitors, coupling elements, loss, and saturated gain all contribute.

The fourth step is sensing. The inductance <math>L_A</math> controls the bare frequency of resonator <math>A</math>. When <math>L_A</math> changes, the self-consistent frequency selected by the nonlinear circuit also changes. Therefore, tracking the spectral peak <math>f</math> tells us how the effective inductance has changed.

The fifth step is the nonlinear exceptional-point response. Near the nonlinear exceptional point discussed in Ref. [1], the physical steady branch and the two auxiliary branches coalesce in the nonlinear steady-state equation. The frequency shift of the stable branch can then scale as the cube root of the inductance perturbation. This means that a small inductance change can produce a comparatively large frequency change. Our breadboard experiment focuses on verifying the stable <math>f-L_A</math> branch corresponding to Fig. S11(a) of SM-4, which is the experimentally accessible part of this sensing mechanism.

=== Same mechanism in other platforms ===
Several references are included to show that this is a platform-independent sensing mechanism, not only a feature of our breadboard circuit. The common chain is

<math display="block">
\mathrm{physical\ perturbation}
\longrightarrow
\mathrm{parameter\ change}
\longrightarrow
\mathrm{eigenfrequency\ shift\ of\ a\ resonator/cavity\ with\ gain\ and\ loss}
\longrightarrow
\mathrm{higher\ order\ steady\ state\ response}
\longrightarrow
\mathrm{sensing}.
</math>

In our circuit, the physical perturbation is a change in the inductance <math>L_A</math>, and the measured output is the oscillation frequency <math>f</math>. In coupled optical, mechanical, electrical, or hybrid resonators, the perturbed parameter may instead be a cavity detuning, mechanical frequency, phase shift, magnetic-field-dependent transition, or another effective resonator parameter. The shared point is that gain, loss, coupling, and nonlinearity convert the parameter change into a higher-order steady-state spectral response.

The earlier theoretical work on nonlinear exceptional points showed that nonlinear coupled resonators can host higher-order NEPs while retaining a complete dynamical basis.<ref name="Bai2023">K. Bai, J.-Z. Li, T.-R. Liu, L. Fang, D. Wan, and M. Xiao, Nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''130''', 266901 (2023).</ref> The 2024 experiment implemented the same mechanism in coupled electrical resonators and measured the stable nonlinear frequency branch.<ref name="Bai2024" /><ref name="Bai2024SM" /> Silicon micromechanical resonators provide another realization: there, the measured output can be a phase difference, and the nonlinear exceptional-point response can enhance phase-shift sensing.<ref name="Xie2026">Y.-J. Xie, S.-Y. Zhang, M.-N. Zhang, R. Wang, L.-F. Wang, and Q.-A. Huang, Nonlinear exceptional points boost phase-shift sensing in silicon micromechanical resonators, ''Phys. Rev. Applied'' '''25''', 034039 (2026).</ref> Hybrid quantum systems with nonlinear bistability show a related route, where nonlinear transition dynamics can enhance sensing signal-to-noise ratio.<ref name="Wang2026">H. Wang ''et al.'', Exceptional sensitivity near the bistable transition point of a hybrid quantum system, ''Nature Physics'' '''22''', 577-583 (2026).</ref>

These examples support the same mechanism rather than unrelated applications. At the same time, enhanced response alone is not enough to prove better sensing. Noise can shift or weaken the apparent nonlinear exceptional point and can limit the signal-to-noise improvement.<ref name="Zheng2025">X. Zheng and Y. D. Chong, Noise constraints for nonlinear exceptional point sensing, ''Phys. Rev. Lett.'' '''134''', 133801 (2025).</ref> This is why our project makes a careful claim: we demonstrate the frequency-response mechanism on a breadboard, but we do not claim a complete metrological noise advantage.

== Report objectives and scope ==
The purpose of this work is to study the circuit principle behind '''nonlinear-exceptional-point-enhanced inductive sensing''' in a coupled electronic resonator platform. The scope is intentionally narrower than the full experiment reported in Ref. [1]. We are not attempting to redesign the active branch, derive all nonlinear bifurcation theory from first principles, reconstruct the auxiliary steady-state branches, or characterize every parasitic element. Instead, we adopt the framework of Ref. [1] and SM-4 and ask whether the breadboard circuit can reproduce the experimentally accessible frequency response.

The main target of our experiment is the <math>f-L_A</math> curve corresponding to Fig. S11(a) of SM-4. In the reference experiment, the steady-state frequency, voltage ratio <math>|V_A/V_B|</math>, and relative phase <math>\theta_A-\theta_B</math> are all measured as functions of <math>L_A</math>.<ref name="Bai2024SM" /> In our implementation, the frequency-versus-inductance curve is the primary observable. We tune <math>L_A</math> by connecting small inductors in series with approximately <math>2~\mu\mathrm{H}</math> resolution. We also use a homebrew coil in a separate demonstration where mechanical compression or insertion of an iron rod changes the effective inductance. The experimental goal is therefore to verify that the TL071 breadboard circuit settles into a stable oscillating state and that the measured frequency changes with <math>L_A</math> in the same qualitative manner as Fig. S11(a).

From the sensing perspective, the most relevant figure of merit is the local frequency responsivity. If

<math display="block">
|f-f_{\mathrm{NEP}}| = A |\delta L_A|^{1/3},
</math>

then differentiation gives

<math display="block">
S_f \equiv \left|\frac{df}{dL_A}\right| \propto |\delta L_A|^{-2/3}.
</math>

Thus, as the operating point approaches <math>L_{A,\mathrm{NEP}}</math>, the local spectral responsivity increases rapidly. The main practical constraint is that the system must remain on the intended stable branch while measurement noise and slow parameter drift are kept below the scale of the induced frequency shift.

== Setups ==
[[File:Experimental_setup.png|750px|thumb|center|Experimental setup adapted from Ref. [1].]]

=== Circuit topology and component implementation ===
The reference platform consists of two coupled RLC resonators. Both resonators contain the same nominal shunt capacitance <math>C_0</math>, while the inductances <math>L_A</math> and <math>L_B</math> set their bare resonance frequencies. The two resonators are coupled through a parallel resistor-capacitor channel characterized by <math>R_c</math> and <math>C_c</math>. Resonator <math>B</math> contains an explicit positive resistance <math>R_B</math> and therefore plays the role of the lossy resonator. Resonator <math>A</math>, by contrast, contains an active branch whose effective resistance is negative at small amplitude and becomes less negative as the oscillation amplitude increases, thereby providing saturable gain.<ref name="Bai2024" /><ref name="Bai2024SM" />

Our implementation follows this topology but is built on a breadboard rather than a custom PCB. The active branch is implemented with an op-amp negative-resistance circuit using a '''TL071''' operational amplifier. This is an important practical difference from the supplementary experiment, where the active components explicitly listed are a TI '''LM7171''' operational amplifier and an '''Onsemi BAV99L''' diode.<ref name="Bai2024SM" /> The TL071 substitution and the breadboard layout introduce different bandwidth limits, wiring parasitics, and contact resistances, so the numerical operating point is not expected to match the PCB experiment. For this reason, our comparison emphasizes the observable trend of <math>f</math> versus <math>L_A</math>.

The supplementary material reports the following representative component values for the PCB experiment:

<math display="block">
C_0 = 18.5~\mathrm{nF}, \qquad
C_c = 3.9~\mathrm{nF}, \qquad
R_c = 760~\Omega,
</math>

<math display="block">
L_B = 197.7~\mu\mathrm{H}, \qquad
R_B = 1314~\Omega.
</math>

A representative operating point shown in the main figure uses

<math display="block">
L_A = 245.4~\mu\mathrm{H}.
</math>

Our breadboard circuit follows the same functional topology, but its component values and parasitic elements are not identical to the PCB values above. The horizontal axis of our main <math>f-L_A</math> plot is therefore the series-combined <math>L_A</math> selected from our available small inductors, while the measured frequency <math>f</math> is the robust experimental readout.

=== Inductance tuning methods ===
We use two complementary methods to change the inductance seen by resonator <math>A</math>.

The first method is used for the main <math>f-L_A</math> sweep. Small inductors are connected in series so that the total inductance changes in discrete steps of approximately <math>2~\mu\mathrm{H}</math>. If the inductors are sufficiently separated so that their mutual magnetic coupling is small, the total series inductance is approximately the sum of the individual inductances,

<math display="block">
L_A \approx L_{\mathrm{base}}+\sum_i L_i .
</math>

This method is simple and reproducible. Adding one small inductor increases <math>L_A</math>; removing it decreases <math>L_A</math>. The circuit frequency is then measured for each chosen value. This produces a discrete version of the <math>f-L_A</math> curve in Fig. S11(a) of SM-4. The step size is not as fine as a commercial variable inductor, but <math>2~\mu\mathrm{H}</math> spacing is small enough to observe the frequency trend over the accessible range.

The second method is a homebrew compressible variable inductor. This coil is useful because it makes the sensing mechanism visible: a mechanical displacement changes the coil geometry, which changes the inductance, which then changes the oscillation frequency. The basic idea can be understood from the approximate solenoid formula

<math display="block">
L \approx \mu_\mathrm{eff}\frac{N^2 A}{\ell},
</math>

where <math>N</math> is the number of turns, <math>A</math> is the cross-sectional area, <math>\ell</math> is the coil length, and <math>\mu_\mathrm{eff}</math> describes the effective magnetic permeability of the core and surrounding region.<ref name="Chen2014">S.-C. Chen, D.-K. Le, and V.-S. Nguyen, Inductive displacement sensors with a notch filter for an active magnetic bearing system, ''Sensors'' '''14''', 12640-12657 (2014).</ref> During compression, the number of turns <math>N</math> is unchanged and the cross-sectional area <math>A</math> remains approximately unchanged, but the length <math>\ell</math> decreases. Equivalently, the turn density

<math display="block">
n=\frac{N}{\ell}
</math>

increases. The magnetic field produced by a solenoid is proportional to turn density. Therefore, compressing the coil increases the magnetic flux linkage through the turns and increases the inductance. Stretching the coil has the opposite effect.

This mechanical tuning is not identical to the precision tunable inductor used in the reference experiment. A real compressed coil is not an ideal long solenoid: the spacing between turns is finite, the end effects change as the coil length changes, nearby wires and the breadboard add parasitic capacitance, and the leads contribute additional series inductance and resistance. For this reason, the homebrew inductor is best treated as an experimentally characterized tuning element rather than an ideal component whose value is known exactly from geometry. In the present project, it provides a direct mechanical demonstration of how displacement can perturb <math>L_A</math>.

Experimentally, we measured the effective inductance of this homebrew variable inductor at different compression states. The compression process changes the coil length and wire density while leaving the coil area approximately fixed. Therefore, the tuning mechanism is physically the same as changing <math>\ell</math> and <math>n=N/\ell</math> in the solenoid model, rather than changing the number of turns or the coil diameter.

=== Kirchhoff model of the nonlinear circuit ===
A convenient state-space description uses the node voltages <math>V_A(t)</math> and <math>V_B(t)</math> together with the inductor currents <math>I_A(t)</math> and <math>I_B(t)</math>. A node voltage is the voltage measured at one resonator node relative to ground. Kirchhoff's current law says that the algebraic sum of all currents flowing away from a node must be zero. This gives a direct way to translate the circuit diagram into differential equations.

Denoting the amplitude-dependent conductance of the active branch by <math>G_A(|V_A|)</math>, where <math>G_A<0</math> in the small-signal regime, the nodal equations can be written as

<math display="block">
\begin{aligned}
C_0 \frac{dV_A}{dt} + I_A + G_A(|V_A|)\,V_A + \frac{V_A-V_B}{R_c}
+ C_c\frac{d(V_A-V_B)}{dt} &= 0,\\
C_0 \frac{dV_B}{dt} + I_B + \frac{V_B}{R_B} + \frac{V_B-V_A}{R_c}
+ C_c\frac{d(V_B-V_A)}{dt} &= 0,\\
L_A \frac{dI_A}{dt} &= V_A,\\
L_B \frac{dI_B}{dt} &= V_B.
\end{aligned}
</math>

Each term has a simple circuit meaning. The terms <math>C_0 dV_A/dt</math> and <math>C_0 dV_B/dt</math> are capacitor currents. The terms <math>I_A</math> and <math>I_B</math> are currents through the inductors. The term <math>G_A(|V_A|)V_A</math> is the active branch current; because <math>G_A</math> can be negative, this branch can inject energy instead of dissipating it. The term <math>V_B/R_B</math> is the ordinary loss current through the resistor in resonator <math>B</math>. The terms involving <math>R_c</math> and <math>C_c</math> describe current flowing between resonators <math>A</math> and <math>B</math>, so they are the electrical coupling between the two resonators.

These equations show explicitly how the experiment differs from a linear EP circuit. If <math>G_A</math> were a constant negative conductance, the model would reduce to a linear non-Hermitian dimer. Here, however, <math>G_A</math> depends on the voltage amplitude, so the steady state must be determined self-consistently together with the oscillation amplitude itself.

=== Harmonic-balance formulation of the steady state ===
After the transient has died away, the oscilloscope waveform is approximately periodic. The simplest approximation is therefore to keep only the dominant sinusoidal component. This is called a harmonic-balance approximation. For a stable activated oscillating state, one writes

<math display="block">
V_j(t)=\mathrm{Re}\!\left[\tilde V_j e^{i\omega t}\right],
\qquad j=A,B.
</math>

Substituting this ansatz into the circuit equations yields a nonlinear algebraic problem of the form

<math display="block">
\mathbf{Y}(\omega,|\tilde V_A|)
\begin{pmatrix}
\tilde V_A\\
\tilde V_B
\end{pmatrix}
=
\begin{pmatrix}
Y_A(\omega,|\tilde V_A|)+Y_c(\omega) & -Y_c(\omega)\\
-Y_c(\omega) & Y_B(\omega)+Y_c(\omega)
\end{pmatrix}
\begin{pmatrix}
\tilde V_A\\
\tilde V_B
\end{pmatrix}
=0,
</math>

with

<math display="block">
Y_A(\omega,|\tilde V_A|)=G_A(|\tilde V_A|)+i\omega C_0+\frac{1}{i\omega L_A},
</math>

<math display="block">
Y_B(\omega)=\frac{1}{R_B}+i\omega C_0+\frac{1}{i\omega L_B},
\qquad
Y_c(\omega)=\frac{1}{R_c}+i\omega C_c.
</math>

A nontrivial solution requires

<math display="block">
\det \mathbf{Y}(\omega,|\tilde V_A|)=0.
</math>

This determinant condition is the circuit analogue of an eigenfrequency condition. In a passive LC resonator, the resonance condition gives a single frequency <math>\omega_0=1/\sqrt{LC}</math>. In the present two-resonator active circuit, the same idea becomes a matrix equation because the two resonators are coupled. The amplitudes <math>\tilde V_A</math> and <math>\tilde V_B</math> are not arbitrary: a nonzero oscillation exists only when the admittance matrix has a zero eigenvalue.

Because the active admittance depends on amplitude, this condition is nonlinear and supports multiple branches. The supplementary material shows that the steady-state frequency can be reduced to a '''real-coefficient cubic polynomial''' in <math>\omega</math>. The existence of up to three real steady-state branches is therefore not accidental; it is the algebraic origin of the experimentally relevant <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM" />

=== Effective nonlinear-Hamiltonian picture ===
The same physics is often rewritten in coupled-mode form. This notation is common in non-Hermitian physics because it makes the circuit look like two coupled modes. The variables <math>\psi_A</math> and <math>\psi_B</math> represent the complex oscillation amplitudes in resonators <math>A</math> and <math>B</math>. The model is

<math display="block">
i\frac{d}{dt}
\begin{pmatrix}
\psi_A\\
\psi_B
\end{pmatrix}
=
\begin{pmatrix}
\omega_A+i\,g(|\psi_A|) & \kappa\\
\kappa & \omega_B-i\ell
\end{pmatrix}
\begin{pmatrix}
\psi_A\\
\psi_B
\end{pmatrix},
</math>

where <math>\omega_A</math> and <math>\omega_B</math> are the effective modal frequencies, <math>\kappa</math> is the coupling strength, <math>\ell</math> is the loss of resonator <math>B</math>, and <math>g(|\psi_A|)</math> is the saturable gain of resonator <math>A</math>. The steady-state problem then becomes

<math display="block">
H_s(\Psi)\Psi = \Omega \Psi,
</math>

with a real oscillation eigenfrequency <math>\Omega=2\pi f</math> for the stable oscillating branch. The word "real" is important: it means the imaginary part has vanished, so the mode is neither being amplified nor decaying in the steady state. In the cited experiment, this physical stable branch and two auxiliary steady branches coalesce at the singular point, producing a third-order nonlinear exceptional point rather than a conventional second-order linear EP. The auxiliary branches help form the cubic-root singularity but are not themselves stable physical outputs of the circuit.<ref name="Bai2024" /><ref name="Bai2024SM" />

== Measurements ==

=== Acquisition protocol and steady-state selection ===
The measurement is performed in the time domain. In the reference experiment, an arbitrary waveform generator first applies an external start-up signal of 1 V at 70 kHz in order to bring the system into the basin of attraction of the stable oscillating state.<ref name="Bai2024" /> In our breadboard measurement, an external voltage is likewise required to activate the resonators. The measured frequency is extracted from the activated steady waveform recorded on the oscilloscope.

The physical meaning of this procedure is important. The oscillation is not assumed to appear without activation; a supplied voltage is needed to bring the resonators into the relevant dynamical state. The measured oscillation frequency is then interpreted as the frequency selected by the nonlinear steady-state condition.

=== Extraction of the <math>f-L_A</math> observable ===
Ref. [1] does not treat the oscilloscope traces merely as qualitative evidence of oscillation. The supplementary explicitly states that the full reference experiment reports the steady-state frequency, the voltage ratio <math>|V_A/V_B|</math>, and the relative phase <math>\theta_A-\theta_B</math>, all extracted from the dominant peaks of the Fourier spectra of <math>V_A</math> and <math>V_B</math>.<ref name="Bai2024SM" />

Our available observable is the steady-state frequency. For each chosen value of <math>L_A</math>, the resonators are voltage-activated, the circuit reaches a stable oscillating state, and the dominant spectral peak is recorded. If we denote the Fourier transforms by

<math display="block">
\tilde V_A(f)=\mathcal{F}\{V_A(t)\},
\qquad
\tilde V_B(f)=\mathcal{F}\{V_B(t)\},
</math>

then the dominant oscillation frequency is obtained from

<math display="block">
f_0=\arg\max_f |\tilde V_A(f)| \approx \arg\max_f |\tilde V_B(f)|.
</math>

The experimental data set is therefore

<math display="block">
\{L_A,\ f_0(L_A)\},
</math>

which is compared with the frequency branch plotted in Fig. S11(a) of SM-4. In the full reference measurement, the amplitude ratio

<math display="block">
\rho = \frac{|\tilde V_A(f_0)|}{|\tilde V_B(f_0)|}
</math>

and relative phase

<math display="block">
\Delta\phi = \arg \tilde V_A(f_0)-\arg \tilde V_B(f_0)
</math>

provide additional state verification. Those quantities are useful but are not the main basis of our breadboard validation.

In practice, the same frequency can also be estimated directly from the time-domain period <math>T</math> using <math>f_0=1/T</math> when the waveform is clean. Fourier analysis is preferred because it uses a longer section of waveform and identifies the dominant oscillation peak even when there is small amplitude noise or waveform distortion.

=== Practical inductance sweep ===
In SM-4, the inductive tuning is carried out by varying <math>L_A</math>. The supplementary notes that the tunable inductor exhibits less than about 0.4% inductance variation over the frequency range of interest, which is sufficiently small for controlled sweeping near the singular point. To further suppress voltage-dependent drift during the sweep, the reference experiment slightly adjusts the active-branch resistance so as to maintain approximately <math>V_B \approx 2~\mathrm{V}</math> throughout the measurement series.<ref name="Bai2024SM" />

Our sweep uses controlled series combinations of small inductors, giving an approximately <math>2~\mu\mathrm{H}</math> step in <math>L_A</math>. The homebrew compressible coil is then used to show that a mechanical change of the sensing element also changes the inductance. The conclusion drawn from our data is therefore not a precision determination of <math>L_{A,\mathrm{NEP}}</math>, but a validation that the breadboard circuit produces the same kind of frequency response as the stable branch in Fig. S11(a).

=== Long-window spectral analysis and noise inspection ===
The supplementary also discusses long-time measurements over a window of <math>0<t<5~\mathrm{s}</math>. The motivation is that the noise around the stable oscillation is too small to be judged reliably from raw scope traces alone, so it is more informative to inspect the Fourier spectra and the extracted spectral center frequency and linewidth.<ref name="Bai2024SM" />

This is a useful methodological point for a sensing-oriented reproduction. In an NEP-based sensor, one ultimately cares about how precisely the frequency peak can be resolved and tracked. A long-window Fourier analysis is therefore closer to the sensing task than a purely visual time-domain inspection.

== Results ==

=== Result display: steady oscillation ===
[[File:PC5271inductive1.png|750px|thumb|center|Breadboard implementation and representative steady oscillation waveform.]]

This figure shows our circuit implemented on a breadboard and a representative stabilised voltage waveform after the resonators are activated by the supplied voltage. The displayed waveform is the evidence used to identify a stable oscillation frequency.

=== Result display: <math>f-L_A</math> sweep ===
[[File:PC5271inductive3.jpeg|750px|thumb|center|Measured breadboard <math>f-L_A</math> curve.]]

This figure shows the measured oscillation frequency <math>f</math> as a function of <math>L_A</math>. The values of <math>L_A</math> in the main sweep are produced by connecting small inductors in series with approximately <math>2~\mu\mathrm{H}</math> step size.

=== Result display: iron-rod demonstration ===
[[File:PC5271inductive2.png|750px|thumb|center|Displacement measurement by compressing the solenoid and inserting iron rod into the homebrew coil.]]

This figure exhibits the resultant changes in the measured oscillation frequency of homebrew coil with a compression (a) and inserted iron rod (b). The experimental variable is the coil length and rod position, and the measured output is the activated steady oscillation frequency. As show in the analysis above, the inductance shows an inversely proportional dependence on the coil length and proportion dependence on insertion length, with the constant residual being the baseline inductance in the circuit irrelevant to the coil.

== Evaluation ==

=== Stable activated oscillation ===
The first required experimental signature is that, after the resonators are activated by the supplied voltage, the circuit reaches a stable oscillating waveform instead of decaying immediately or being amplified without bound. A stable waveform means that the TL071 active branch supplies enough small-signal negative resistance to compensate loss, while gain saturation prevents unbounded amplification.

Physically, this behavior confirms the role of the nonlinear active branch. At small amplitude, the effective negative resistance in resonator <math>A</math> overcomes dissipation. At larger amplitude, the gain becomes weaker, so the oscillation amplitude is clamped. The system self-stabilizes because the physical steady branch is the one with zero imaginary part of the eigenfrequency. The auxiliary branches near the nonlinear exceptional point are not stable physical outputs; they are mathematical branches of the nonlinear steady-state equation. The experiment is therefore not a passive ring-down measurement; it is a nonlinear steady-state selection process.

=== Evaluation of the <math>f-L_A</math> trend ===
The main experimental comparison is the measured <math>f-L_A</math> trend against Fig. S11(a) of SM-4. The reference experiment reports the stable frequency branch of the nonlinear circuit as <math>L_A</math> is tuned.<ref name="Bai2024SM" /> This branch is the physical one because its eigenfrequency is real in the steady state. In our breadboard experiment, the same observable is measured using the series-inductor sweep.

It is useful to separate two statements. First, any LC resonator changes frequency when <math>L</math> changes, because <math>f_0=1/(2\pi\sqrt{LC})</math>. This alone would not demonstrate the nonlinear exceptional-point mechanism. Second, in the coupled active-lossy circuit, the frequency is not just the bare LC resonance frequency of one isolated inductor and capacitor. It is the frequency selected by the nonlinear steady-state balance of gain, loss, and coupling. The reason Fig. S11(a) matters is that it shows this selected stable branch as <math>L_A</math> is swept.

Our breadboard circuit uses different component values and a different op-amp from the PCB experiment, so exact numerical overlap is not expected. The meaningful evaluation is whether the measured frequency branch follows the same qualitative <math>f-L_A</math> behavior as the stable branch in SM-4. Observing that behavior is enough to show that the constructed circuit realizes the intended nonlinear voltage-activated oscillating mechanism.

=== Evaluation of the displacement demonstration ===
In the iron-rod demonstration, the homebrew coil becomes a displacement-dependent inductive element. As the rod enters the coil, the effective magnetic permeability inside the coil increases. In the solenoid expression

<math display="block">
L(x) \approx \mu_\mathrm{eff}(x)\frac{N^2 A}{\ell},
</math>

the insertion depth <math>x</math> changes <math>\mu_\mathrm{eff}</math>. The coil geometry <math>N</math>, <math>A</math>, and <math>\ell</math> are approximately fixed during this demonstration, so the dominant effect is the change in magnetic core material inside the coil. Inductive displacement sensors commonly use this principle: the position of a ferromagnetic object changes a coil inductance, and the electronics convert that inductance change into an electrical signal.<ref name="Chen2014" />

In our circuit, the electrical signal is the activated steady oscillation frequency rather than a bridge voltage. The sensing chain is

<math display="block">
\mathrm{rod\ displacement}\ x
\longrightarrow
\mathrm{effective\ inductance}\ L_A(x)
\longrightarrow
\mathrm{oscillation\ frequency}\ f(x).
</math>

This demonstration shows how the inductive readout can be turned into a displacement-sensing readout after calibration of the relation between rod position and frequency.

=== Cubic-root perturbation law and sensing meaning ===
Near the singular operating point, the eigenfrequency shift follows the hallmark cubic-root law,

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3}.
</math>

This law is the main sensing-related result of the reference experiment.<ref name="Bai2024" /><ref name="Bai2024SM" /> Compared with the linear response of an ordinary isolated resonator, the cubic-root dependence can produce a larger frequency displacement for the same small perturbation of inductance when the device is biased sufficiently close to <math>L_{A,\mathrm{NEP}}</math>.

A useful way to interpret the result is through the local derivative,

<math display="block">
\left|\frac{df}{dL_A}\right| \propto |\delta L_A|^{-2/3}.
</math>

This increasing local responsivity is why the circuit is attractive for sensing. Our experiment supports this sensing logic by showing that the breadboard circuit has a measurable stable frequency branch as <math>L_A</math> is changed. A full quantitative exponent extraction would require identifying <math>L_{A,\mathrm{NEP}}</math> and <math>f_{\mathrm{NEP}}</math>, then plotting <math>|f-f_{\mathrm{NEP}}|</math> against <math>|L_A-L_{A,\mathrm{NEP}}|</math> on logarithmic axes.

=== Basis completeness and noise robustness ===
The supplementary material evaluates the Petermann factor as a diagnostic of modal nonorthogonality and basis completeness. For a non-Hermitian eigenproblem, one may write

<math display="block">
\mathrm{PF}_n =
\frac{\langle \Phi_n^{L}|\Phi_n^{L}\rangle \langle \Phi_n^{R}|\Phi_n^{R}\rangle}
{\left|\langle \Phi_n^{L}|\Phi_n^{R}\rangle\right|^2},
</math>

where <math>|\Phi_n^{R}\rangle</math> and <math>|\Phi_n^{L}\rangle</math> are the right and left eigenvectors of the effective steady-state Hamiltonian. At a conventional linear EP, this quantity is associated with diverging excess-noise sensitivity. In Ref. [1], the cited analysis finds that the effective steady-state Hamiltonian remains non-defective and the Petermann factor is finite at the <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM" />

This point explains the significance of the reference experiment. The system does not merely realize a large mathematical responsivity; it realizes that responsivity in a way that avoids the most severe basis-collapse problem of a linear EP. Our breadboard experiment does not directly test basis completeness, but the result provides the theoretical reason why reproducing the stable <math>f-L_A</math> branch is physically meaningful.

=== Overall interpretation ===
In this platform, the measurand is encoded as a perturbation of <math>L_A</math>. Any physical mechanism that modifies the effective inductance of resonator <math>A</math>, for example displacement, magnetic coupling, or the presence of a nearby magnetic target, moves the system away from the operating point and shifts the oscillation frequency. The primary readout in our implementation is therefore spectral: we track <math>f</math> as <math>L_A</math> is changed.

Taken together, Ref. [1] establishes the complete sensing logic: a nonlinear active-lossy circuit is tuned to an <math>\mathrm{NEP}_3</math>; after voltage activation, the system settles onto a stable oscillating branch; the dominant spectral peak is tracked as the perturbation readout; and the enhanced cubic-root response is obtained without the defective-basis pathology of a conventional linear EP. Our breadboard result demonstrates the experimentally accessible core of this logic by reproducing the stable frequency response corresponding to Fig. S11(a) of SM-4.

== References ==
<references />

Inductive Sensors of Ultra-high Sensitivity Based on Nonlinear Exceptional Point

2026-04-23T06:22:08Z

Siyu: /* Result display: f-L_A sweep */

== Introduction ==
This project is a implementation of the experiment reported in ''Observation of Nonlinear Exceptional Points with a Complete Basis in Dynamics'', adapted to our available laboratory equipment.<ref name="Bai2024">K. Bai, T.-R. Liu, L. Fang, J.-Z. Li, C. Lin, D. Wan, and M. Xiao, Observation of nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''132''', 073802 (2024).</ref> The circuit contains two coupled resonators. One resonator has gain, meaning that it can supply energy to the oscillation, while the other has loss, meaning that it dissipates energy. A system without energy conservation is called a '''non-Hermitian''' system. Such systems can show unusual degeneracies called '''exceptional points''' (EPs), where both the eigenfrequencies and the eigenmodes merge.

Exceptional points are interesting for sensing because the measured frequency can respond very strongly to a small perturbation. In an inductive sensor, the perturbation is a change in inductance. If a small change in inductance produces a large change in oscillation frequency, the circuit can in principle detect small physical changes. However, ordinary linear EPs also suffer from strong eigenmode nonorthogonality and a defective basis, which can increase noise sensitivity and complicate practical sensing.<ref name="Bai2024" />

The system studied in this project belongs to the recently developed class of '''nonlinear exceptional points''' (NEPs). The physical platform is a pair of coupled circuit resonators. Resonator <math>A</math> is made active by a saturable negative-resistance branch, while resonator <math>B</math> is intentionally lossy. Because the gain in resonator <math>A</math> depends on the oscillation amplitude, the steady-state problem is nonlinear and admits multiple self-consistent branches. At the singular operating point reported in Ref. [1], one stable steady branch and two auxiliary steady branches merge into a third-order nonlinear exceptional point, denoted <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM">Supplemental Material for K. Bai, T.-R. Liu, L. Fang, J.-Z. Li, C. Lin, D. Wan, and M. Xiao, Observation of nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''132''', 073802 (2024).</ref>

The experimentally important consequence is that the oscillation eigenfrequency obeys a cubic-root perturbation law near the singularity,

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3},
</math>

where <math>\delta L_A = L_A-L_{A,\mathrm{NEP}}</math> is the inductive detuning from the exceptional operating point. This response is steeper than the ordinary linear dependence of a conventional resonator and is therefore useful for ultra-sensitive inductive sensing.<ref name="Bai2024" /><ref name="Bai2024SM" />

A second and equally important point is that the cited work does not interpret the singularity as a conventional defective linear EP. Instead, the effective steady-state Hamiltonian remains non-defective in dynamics, and the associated Petermann factor stays finite at the operating point. In other words, the experiment realizes enhanced spectral responsivity without inheriting the strongest basis-collapse pathology of a standard linear EP.<ref name="Bai2024" /><ref name="Bai2024SM" />

In this project, our goal is to reproduce the experimental logic of Ref. [1] and its supplementary material under course-lab conditions. Unlike the PCB implementation in the paper, our circuit is assembled on a breadboard and the active branch uses a '''TL071''' operational amplifier. Our main observable is the steady oscillation frequency <math>f</math> as a function of the chosen inductance <math>L_A</math>. We tune <math>L_A</math> in two ways. For the main <math>f-L_A</math> sweep, we join small inductors in series so that the discrete inductance step is about <math>2~\mu\mathrm{H}</math>. In addition, we use a homebrew compressible coil to demonstrate the sensing mechanism directly. Compressing the coil changes its length and turn density while keeping the cross-sectional area approximately unchanged. We therefore use the <math>f-L_A</math> behavior in Fig. S11(a) of the supplementary material as the main experimental benchmark. Reproducing this trend is sufficient for demonstrating that our setup captures the intended nonlinear voltage-activated, self-stabilized oscillating response, although it is not a full reproduction of every diagnostic reported in Ref. [1].<ref name="Bai2024" /><ref name="Bai2024SM" />

== Team Members ==
Wang Peikun — E1538091@u.nus.edu 
Zhu Ziyang — E1583446@u.nus.edu 
Yuan Si-Yu — siyu_yuan@u.nus.edu 
Li Xunyu — xunyu@u.nus.edu

== Background Knowledge ==

=== Ordinary LC and RLC resonators ===
The basic building block of the experiment is an electrical resonator. An ideal LC resonator contains an inductor <math>L</math> and a capacitor <math>C</math>. Energy repeatedly moves between the magnetic field of the inductor and the electric field of the capacitor. The natural angular frequency is

<math display="block">
\omega_0=\frac{1}{\sqrt{LC}},
</math>

or, in ordinary frequency units,

<math display="block">
f_0=\frac{1}{2\pi\sqrt{LC}}.
</math>

This formula is important for the whole project because it explains why changing an inductance changes the measured frequency. A larger <math>L</math> gives a lower resonance frequency, while a smaller <math>L</math> gives a higher resonance frequency.

Real circuits are not ideal. Wires, inductors, capacitors, breadboard contacts, and measuring probes all dissipate energy. This loss is often represented by a resistor <math>R</math>, giving an RLC resonator. In a passive RLC circuit, the oscillation amplitude decays with time because energy is continuously lost as heat. If a circuit is to maintain a steady activated oscillation, the loss must be compensated by a gain element.

=== Gain, loss, and negative resistance ===
In electronics, gain can be represented as an effective negative resistance. A normal positive resistance removes energy from the resonator. A negative resistance does the opposite: it supplies energy to the resonator. In this experiment, resonator <math>A</math> contains an active op-amp branch that behaves like a negative resistance at small amplitude, while resonator <math>B</math> contains ordinary positive loss.

The gain cannot remain negative and constant forever. If it did, any activated oscillation would be amplified without bound until the op-amp clipped or the circuit failed. The important nonlinear ingredient is '''saturable gain''': the active branch gives strong gain at small amplitude but weaker gain at larger amplitude. The stable oscillation amplitude is selected when the supplied gain balances the total loss. This nonlinear steady state is why the experiment is a nonlinear dynamical problem rather than only a passive resonance measurement.

=== What "non-Hermitian" means in this circuit ===
In ordinary conservative physics, a Hermitian Hamiltonian describes a closed system whose total energy is conserved. Its eigenfrequencies are real, and its eigenmodes form a complete orthogonal basis. This is the familiar setting of many quantum-mechanics courses.

A circuit with gain and loss is an open system. Energy can enter through the op-amp branch and leave through resistive loss. The effective dynamical matrix is therefore generally '''non-Hermitian'''. In a non-Hermitian system, eigenfrequencies can be complex:

<math display="block">
\omega=\omega_\mathrm{r}+i\omega_\mathrm{i}.
</math>

The real part <math>\omega_\mathrm{r}</math> gives the oscillation frequency, while the imaginary part <math>\omega_\mathrm{i}</math> describes amplification or decay. A positive imaginary part means the mode is amplified, a negative imaginary part means it decays, and a stable steady oscillation corresponds to zero imaginary part, or equivalently zero net amplification. This is the sense in which the circuit is self-stabilized after voltage activation.

=== Exceptional points ===
In a usual Hermitian system, two modes can have the same eigenfrequency while still remaining two independent modes. This is an ordinary degeneracy. A non-Hermitian system can show a stronger kind of degeneracy called an '''exceptional point''' (EP). At an EP, not only the eigenvalues but also the eigenvectors coalesce. In simple language, two modes collapse into one mode.

This collapse makes EPs interesting for sensing. Near a second-order linear EP, a small perturbation can produce a square-root frequency splitting:

<math display="block">
\Delta f \propto \sqrt{\epsilon},
</math>

where <math>\epsilon</math> is the perturbation. This response is steeper than an ordinary linear response when <math>\epsilon</math> is very small. However, a conventional linear EP also has a serious drawback: the eigenbasis becomes defective, meaning that the system no longer has enough independent eigenvectors. This is associated with strong modal nonorthogonality and enhanced noise sensitivity. The Petermann factor is one common way to quantify this nonorthogonality.

=== Why the nonlinear exceptional point is different ===
The reference paper studies a nonlinear exceptional point. The word "nonlinear" is essential. The circuit does not have a fixed gain coefficient; instead, the gain depends on the oscillation amplitude. The steady state must therefore be solved self-consistently: the frequency, voltage amplitudes, and saturated gain must all agree with one another.

In the theory of Ref. [1], this nonlinear steady-state problem can reduce to a cubic equation for the oscillation frequency. A cubic equation can have a triple root. At the special operating point, one stable physical branch and two auxiliary branches merge, giving a third-order nonlinear exceptional point, written as <math>\mathrm{NEP}_3</math>. The stable physical branch is the one with real oscillation frequency, i.e. without an imaginary part. The auxiliary branches are needed to define the higher-order coalescence mathematically, but they are not physically stable operating states of the circuit. The corresponding frequency response has the form

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3}.
</math>

This cubic-root response is the key sensing signature. It is steeper than both an ordinary linear response and the square-root response of a second-order EP. The important conceptual point of Ref. [1] is that this enhanced response is obtained while the effective steady-state Hamiltonian can still retain a complete basis in dynamics.<ref name="Bai2024" /><ref name="Bai2024SM" />

=== Sensing mechanism ===
The circuit can be understood as a frequency-generating sensor. It converts an inductance change into a frequency change.

The first step is activation. A voltage is supplied to the resonators to excite the circuit into an oscillating state. The active branch in resonator <math>A</math> then supplies energy through an effective negative resistance and amplifies the selected oscillation mode.

The second step is self-stabilization. The gain is saturable, which means it becomes weaker when the voltage amplitude becomes larger. The activated oscillation therefore is not amplified forever. The circuit settles onto the physical steady branch where the imaginary part of the eigenfrequency is zero. At that point, the gain supplied by resonator <math>A</math> balances the loss in the circuit, especially the loss in resonator <math>B</math>.

The third step is frequency selection. Because resonators <math>A</math> and <math>B</math> are coupled, the final oscillation frequency is not simply the resonance frequency of either isolated resonator. It is selected by the whole coupled circuit: the inductors, capacitors, coupling elements, loss, and saturated gain all contribute.

The fourth step is sensing. The inductance <math>L_A</math> controls the bare frequency of resonator <math>A</math>. When <math>L_A</math> changes, the self-consistent frequency selected by the nonlinear circuit also changes. Therefore, tracking the spectral peak <math>f</math> tells us how the effective inductance has changed.

The fifth step is the nonlinear exceptional-point response. Near the nonlinear exceptional point discussed in Ref. [1], the physical steady branch and the two auxiliary branches coalesce in the nonlinear steady-state equation. The frequency shift of the stable branch can then scale as the cube root of the inductance perturbation. This means that a small inductance change can produce a comparatively large frequency change. Our breadboard experiment focuses on verifying the stable <math>f-L_A</math> branch corresponding to Fig. S11(a) of SM-4, which is the experimentally accessible part of this sensing mechanism.

=== Same mechanism in other platforms ===
Several references are included to show that this is a platform-independent sensing mechanism, not only a feature of our breadboard circuit. The common chain is

<math display="block">
\mathrm{physical\ perturbation}
\longrightarrow
\mathrm{parameter\ change}
\longrightarrow
\mathrm{eigenfrequency\ shift\ of\ a\ resonator/cavity\ with\ gain\ and\ loss}
\longrightarrow
\mathrm{higher\ order\ steady\ state\ response}
\longrightarrow
\mathrm{sensing}.
</math>

In our circuit, the physical perturbation is a change in the inductance <math>L_A</math>, and the measured output is the oscillation frequency <math>f</math>. In coupled optical, mechanical, electrical, or hybrid resonators, the perturbed parameter may instead be a cavity detuning, mechanical frequency, phase shift, magnetic-field-dependent transition, or another effective resonator parameter. The shared point is that gain, loss, coupling, and nonlinearity convert the parameter change into a higher-order steady-state spectral response.

The earlier theoretical work on nonlinear exceptional points showed that nonlinear coupled resonators can host higher-order NEPs while retaining a complete dynamical basis.<ref name="Bai2023">K. Bai, J.-Z. Li, T.-R. Liu, L. Fang, D. Wan, and M. Xiao, Nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''130''', 266901 (2023).</ref> The 2024 experiment implemented the same mechanism in coupled electrical resonators and measured the stable nonlinear frequency branch.<ref name="Bai2024" /><ref name="Bai2024SM" /> Silicon micromechanical resonators provide another realization: there, the measured output can be a phase difference, and the nonlinear exceptional-point response can enhance phase-shift sensing.<ref name="Xie2026">Y.-J. Xie, S.-Y. Zhang, M.-N. Zhang, R. Wang, L.-F. Wang, and Q.-A. Huang, Nonlinear exceptional points boost phase-shift sensing in silicon micromechanical resonators, ''Phys. Rev. Applied'' '''25''', 034039 (2026).</ref> Hybrid quantum systems with nonlinear bistability show a related route, where nonlinear transition dynamics can enhance sensing signal-to-noise ratio.<ref name="Wang2026">H. Wang ''et al.'', Exceptional sensitivity near the bistable transition point of a hybrid quantum system, ''Nature Physics'' '''22''', 577-583 (2026).</ref>

These examples support the same mechanism rather than unrelated applications. At the same time, enhanced response alone is not enough to prove better sensing. Noise can shift or weaken the apparent nonlinear exceptional point and can limit the signal-to-noise improvement.<ref name="Zheng2025">X. Zheng and Y. D. Chong, Noise constraints for nonlinear exceptional point sensing, ''Phys. Rev. Lett.'' '''134''', 133801 (2025).</ref> This is why our project makes a careful claim: we demonstrate the frequency-response mechanism on a breadboard, but we do not claim a complete metrological noise advantage.

== Report objectives and scope ==
The purpose of this work is to study the circuit principle behind '''nonlinear-exceptional-point-enhanced inductive sensing''' in a coupled electronic resonator platform. The scope is intentionally narrower than the full experiment reported in Ref. [1]. We are not attempting to redesign the active branch, derive all nonlinear bifurcation theory from first principles, reconstruct the auxiliary steady-state branches, or characterize every parasitic element. Instead, we adopt the framework of Ref. [1] and SM-4 and ask whether the breadboard circuit can reproduce the experimentally accessible frequency response.

The main target of our experiment is the <math>f-L_A</math> curve corresponding to Fig. S11(a) of SM-4. In the reference experiment, the steady-state frequency, voltage ratio <math>|V_A/V_B|</math>, and relative phase <math>\theta_A-\theta_B</math> are all measured as functions of <math>L_A</math>.<ref name="Bai2024SM" /> In our implementation, the frequency-versus-inductance curve is the primary observable. We tune <math>L_A</math> by connecting small inductors in series with approximately <math>2~\mu\mathrm{H}</math> resolution. We also use a homebrew coil in a separate demonstration where mechanical compression or insertion of an iron rod changes the effective inductance. The experimental goal is therefore to verify that the TL071 breadboard circuit settles into a stable oscillating state and that the measured frequency changes with <math>L_A</math> in the same qualitative manner as Fig. S11(a).

From the sensing perspective, the most relevant figure of merit is the local frequency responsivity. If

<math display="block">
|f-f_{\mathrm{NEP}}| = A |\delta L_A|^{1/3},
</math>

then differentiation gives

<math display="block">
S_f \equiv \left|\frac{df}{dL_A}\right| \propto |\delta L_A|^{-2/3}.
</math>

Thus, as the operating point approaches <math>L_{A,\mathrm{NEP}}</math>, the local spectral responsivity increases rapidly. The main practical constraint is that the system must remain on the intended stable branch while measurement noise and slow parameter drift are kept below the scale of the induced frequency shift.

== Setups ==
[[File:Experimental_setup.png|750px|thumb|center|Experimental setup adapted from Ref. [1].]]

=== Circuit topology and component implementation ===
The reference platform consists of two coupled RLC resonators. Both resonators contain the same nominal shunt capacitance <math>C_0</math>, while the inductances <math>L_A</math> and <math>L_B</math> set their bare resonance frequencies. The two resonators are coupled through a parallel resistor-capacitor channel characterized by <math>R_c</math> and <math>C_c</math>. Resonator <math>B</math> contains an explicit positive resistance <math>R_B</math> and therefore plays the role of the lossy resonator. Resonator <math>A</math>, by contrast, contains an active branch whose effective resistance is negative at small amplitude and becomes less negative as the oscillation amplitude increases, thereby providing saturable gain.<ref name="Bai2024" /><ref name="Bai2024SM" />

Our implementation follows this topology but is built on a breadboard rather than a custom PCB. The active branch is implemented with an op-amp negative-resistance circuit using a '''TL071''' operational amplifier. This is an important practical difference from the supplementary experiment, where the active components explicitly listed are a TI '''LM7171''' operational amplifier and an '''Onsemi BAV99L''' diode.<ref name="Bai2024SM" /> The TL071 substitution and the breadboard layout introduce different bandwidth limits, wiring parasitics, and contact resistances, so the numerical operating point is not expected to match the PCB experiment. For this reason, our comparison emphasizes the observable trend of <math>f</math> versus <math>L_A</math>.

The supplementary material reports the following representative component values for the PCB experiment:

<math display="block">
C_0 = 18.5~\mathrm{nF}, \qquad
C_c = 3.9~\mathrm{nF}, \qquad
R_c = 760~\Omega,
</math>

<math display="block">
L_B = 197.7~\mu\mathrm{H}, \qquad
R_B = 1314~\Omega.
</math>

A representative operating point shown in the main figure uses

<math display="block">
L_A = 245.4~\mu\mathrm{H}.
</math>

Our breadboard circuit follows the same functional topology, but its component values and parasitic elements are not identical to the PCB values above. The horizontal axis of our main <math>f-L_A</math> plot is therefore the series-combined <math>L_A</math> selected from our available small inductors, while the measured frequency <math>f</math> is the robust experimental readout.

=== Inductance tuning methods ===
We use two complementary methods to change the inductance seen by resonator <math>A</math>.

The first method is used for the main <math>f-L_A</math> sweep. Small inductors are connected in series so that the total inductance changes in discrete steps of approximately <math>2~\mu\mathrm{H}</math>. If the inductors are sufficiently separated so that their mutual magnetic coupling is small, the total series inductance is approximately the sum of the individual inductances,

<math display="block">
L_A \approx L_{\mathrm{base}}+\sum_i L_i .
</math>

This method is simple and reproducible. Adding one small inductor increases <math>L_A</math>; removing it decreases <math>L_A</math>. The circuit frequency is then measured for each chosen value. This produces a discrete version of the <math>f-L_A</math> curve in Fig. S11(a) of SM-4. The step size is not as fine as a commercial variable inductor, but <math>2~\mu\mathrm{H}</math> spacing is small enough to observe the frequency trend over the accessible range.

The second method is a homebrew compressible variable inductor. This coil is useful because it makes the sensing mechanism visible: a mechanical displacement changes the coil geometry, which changes the inductance, which then changes the oscillation frequency. The basic idea can be understood from the approximate solenoid formula

<math display="block">
L \approx \mu_\mathrm{eff}\frac{N^2 A}{\ell},
</math>

where <math>N</math> is the number of turns, <math>A</math> is the cross-sectional area, <math>\ell</math> is the coil length, and <math>\mu_\mathrm{eff}</math> describes the effective magnetic permeability of the core and surrounding region.<ref name="Chen2014">S.-C. Chen, D.-K. Le, and V.-S. Nguyen, Inductive displacement sensors with a notch filter for an active magnetic bearing system, ''Sensors'' '''14''', 12640-12657 (2014).</ref> During compression, the number of turns <math>N</math> is unchanged and the cross-sectional area <math>A</math> remains approximately unchanged, but the length <math>\ell</math> decreases. Equivalently, the turn density

<math display="block">
n=\frac{N}{\ell}
</math>

increases. The magnetic field produced by a solenoid is proportional to turn density. Therefore, compressing the coil increases the magnetic flux linkage through the turns and increases the inductance. Stretching the coil has the opposite effect.

This mechanical tuning is not identical to the precision tunable inductor used in the reference experiment. A real compressed coil is not an ideal long solenoid: the spacing between turns is finite, the end effects change as the coil length changes, nearby wires and the breadboard add parasitic capacitance, and the leads contribute additional series inductance and resistance. For this reason, the homebrew inductor is best treated as an experimentally characterized tuning element rather than an ideal component whose value is known exactly from geometry. In the present project, it provides a direct mechanical demonstration of how displacement can perturb <math>L_A</math>.

Experimentally, we measured the effective inductance of this homebrew variable inductor at different compression states. The compression process changes the coil length and wire density while leaving the coil area approximately fixed. Therefore, the tuning mechanism is physically the same as changing <math>\ell</math> and <math>n=N/\ell</math> in the solenoid model, rather than changing the number of turns or the coil diameter.

=== Kirchhoff model of the nonlinear circuit ===
A convenient state-space description uses the node voltages <math>V_A(t)</math> and <math>V_B(t)</math> together with the inductor currents <math>I_A(t)</math> and <math>I_B(t)</math>. A node voltage is the voltage measured at one resonator node relative to ground. Kirchhoff's current law says that the algebraic sum of all currents flowing away from a node must be zero. This gives a direct way to translate the circuit diagram into differential equations.

Denoting the amplitude-dependent conductance of the active branch by <math>G_A(|V_A|)</math>, where <math>G_A<0</math> in the small-signal regime, the nodal equations can be written as

<math display="block">
\begin{aligned}
C_0 \frac{dV_A}{dt} + I_A + G_A(|V_A|)\,V_A + \frac{V_A-V_B}{R_c}
+ C_c\frac{d(V_A-V_B)}{dt} &= 0,\\
C_0 \frac{dV_B}{dt} + I_B + \frac{V_B}{R_B} + \frac{V_B-V_A}{R_c}
+ C_c\frac{d(V_B-V_A)}{dt} &= 0,\\
L_A \frac{dI_A}{dt} &= V_A,\\
L_B \frac{dI_B}{dt} &= V_B.
\end{aligned}
</math>

Each term has a simple circuit meaning. The terms <math>C_0 dV_A/dt</math> and <math>C_0 dV_B/dt</math> are capacitor currents. The terms <math>I_A</math> and <math>I_B</math> are currents through the inductors. The term <math>G_A(|V_A|)V_A</math> is the active branch current; because <math>G_A</math> can be negative, this branch can inject energy instead of dissipating it. The term <math>V_B/R_B</math> is the ordinary loss current through the resistor in resonator <math>B</math>. The terms involving <math>R_c</math> and <math>C_c</math> describe current flowing between resonators <math>A</math> and <math>B</math>, so they are the electrical coupling between the two resonators.

These equations show explicitly how the experiment differs from a linear EP circuit. If <math>G_A</math> were a constant negative conductance, the model would reduce to a linear non-Hermitian dimer. Here, however, <math>G_A</math> depends on the voltage amplitude, so the steady state must be determined self-consistently together with the oscillation amplitude itself.

=== Harmonic-balance formulation of the steady state ===
After the transient has died away, the oscilloscope waveform is approximately periodic. The simplest approximation is therefore to keep only the dominant sinusoidal component. This is called a harmonic-balance approximation. For a stable activated oscillating state, one writes

<math display="block">
V_j(t)=\mathrm{Re}\!\left[\tilde V_j e^{i\omega t}\right],
\qquad j=A,B.
</math>

Substituting this ansatz into the circuit equations yields a nonlinear algebraic problem of the form

<math display="block">
\mathbf{Y}(\omega,|\tilde V_A|)
\begin{pmatrix}
\tilde V_A\\
\tilde V_B
\end{pmatrix}
=
\begin{pmatrix}
Y_A(\omega,|\tilde V_A|)+Y_c(\omega) & -Y_c(\omega)\\
-Y_c(\omega) & Y_B(\omega)+Y_c(\omega)
\end{pmatrix}
\begin{pmatrix}
\tilde V_A\\
\tilde V_B
\end{pmatrix}
=0,
</math>

with

<math display="block">
Y_A(\omega,|\tilde V_A|)=G_A(|\tilde V_A|)+i\omega C_0+\frac{1}{i\omega L_A},
</math>

<math display="block">
Y_B(\omega)=\frac{1}{R_B}+i\omega C_0+\frac{1}{i\omega L_B},
\qquad
Y_c(\omega)=\frac{1}{R_c}+i\omega C_c.
</math>

A nontrivial solution requires

<math display="block">
\det \mathbf{Y}(\omega,|\tilde V_A|)=0.
</math>

This determinant condition is the circuit analogue of an eigenfrequency condition. In a passive LC resonator, the resonance condition gives a single frequency <math>\omega_0=1/\sqrt{LC}</math>. In the present two-resonator active circuit, the same idea becomes a matrix equation because the two resonators are coupled. The amplitudes <math>\tilde V_A</math> and <math>\tilde V_B</math> are not arbitrary: a nonzero oscillation exists only when the admittance matrix has a zero eigenvalue.

Because the active admittance depends on amplitude, this condition is nonlinear and supports multiple branches. The supplementary material shows that the steady-state frequency can be reduced to a '''real-coefficient cubic polynomial''' in <math>\omega</math>. The existence of up to three real steady-state branches is therefore not accidental; it is the algebraic origin of the experimentally relevant <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM" />

=== Effective nonlinear-Hamiltonian picture ===
The same physics is often rewritten in coupled-mode form. This notation is common in non-Hermitian physics because it makes the circuit look like two coupled modes. The variables <math>\psi_A</math> and <math>\psi_B</math> represent the complex oscillation amplitudes in resonators <math>A</math> and <math>B</math>. The model is

<math display="block">
i\frac{d}{dt}
\begin{pmatrix}
\psi_A\\
\psi_B
\end{pmatrix}
=
\begin{pmatrix}
\omega_A+i\,g(|\psi_A|) & \kappa\\
\kappa & \omega_B-i\ell
\end{pmatrix}
\begin{pmatrix}
\psi_A\\
\psi_B
\end{pmatrix},
</math>

where <math>\omega_A</math> and <math>\omega_B</math> are the effective modal frequencies, <math>\kappa</math> is the coupling strength, <math>\ell</math> is the loss of resonator <math>B</math>, and <math>g(|\psi_A|)</math> is the saturable gain of resonator <math>A</math>. The steady-state problem then becomes

<math display="block">
H_s(\Psi)\Psi = \Omega \Psi,
</math>

with a real oscillation eigenfrequency <math>\Omega=2\pi f</math> for the stable oscillating branch. The word "real" is important: it means the imaginary part has vanished, so the mode is neither being amplified nor decaying in the steady state. In the cited experiment, this physical stable branch and two auxiliary steady branches coalesce at the singular point, producing a third-order nonlinear exceptional point rather than a conventional second-order linear EP. The auxiliary branches help form the cubic-root singularity but are not themselves stable physical outputs of the circuit.<ref name="Bai2024" /><ref name="Bai2024SM" />

== Measurements ==

=== Acquisition protocol and steady-state selection ===
The measurement is performed in the time domain. In the reference experiment, an arbitrary waveform generator first applies an external start-up signal of 1 V at 70 kHz in order to bring the system into the basin of attraction of the stable oscillating state.<ref name="Bai2024" /> In our breadboard measurement, an external voltage is likewise required to activate the resonators. The measured frequency is extracted from the activated steady waveform recorded on the oscilloscope.

The physical meaning of this procedure is important. The oscillation is not assumed to appear without activation; a supplied voltage is needed to bring the resonators into the relevant dynamical state. The measured oscillation frequency is then interpreted as the frequency selected by the nonlinear steady-state condition.

=== Extraction of the <math>f-L_A</math> observable ===
Ref. [1] does not treat the oscilloscope traces merely as qualitative evidence of oscillation. The supplementary explicitly states that the full reference experiment reports the steady-state frequency, the voltage ratio <math>|V_A/V_B|</math>, and the relative phase <math>\theta_A-\theta_B</math>, all extracted from the dominant peaks of the Fourier spectra of <math>V_A</math> and <math>V_B</math>.<ref name="Bai2024SM" />

Our available observable is the steady-state frequency. For each chosen value of <math>L_A</math>, the resonators are voltage-activated, the circuit reaches a stable oscillating state, and the dominant spectral peak is recorded. If we denote the Fourier transforms by

<math display="block">
\tilde V_A(f)=\mathcal{F}\{V_A(t)\},
\qquad
\tilde V_B(f)=\mathcal{F}\{V_B(t)\},
</math>

then the dominant oscillation frequency is obtained from

<math display="block">
f_0=\arg\max_f |\tilde V_A(f)| \approx \arg\max_f |\tilde V_B(f)|.
</math>

The experimental data set is therefore

<math display="block">
\{L_A,\ f_0(L_A)\},
</math>

which is compared with the frequency branch plotted in Fig. S11(a) of SM-4. In the full reference measurement, the amplitude ratio

<math display="block">
\rho = \frac{|\tilde V_A(f_0)|}{|\tilde V_B(f_0)|}
</math>

and relative phase

<math display="block">
\Delta\phi = \arg \tilde V_A(f_0)-\arg \tilde V_B(f_0)
</math>

provide additional state verification. Those quantities are useful but are not the main basis of our breadboard validation.

In practice, the same frequency can also be estimated directly from the time-domain period <math>T</math> using <math>f_0=1/T</math> when the waveform is clean. Fourier analysis is preferred because it uses a longer section of waveform and identifies the dominant oscillation peak even when there is small amplitude noise or waveform distortion.

=== Practical inductance sweep ===
In SM-4, the inductive tuning is carried out by varying <math>L_A</math>. The supplementary notes that the tunable inductor exhibits less than about 0.4% inductance variation over the frequency range of interest, which is sufficiently small for controlled sweeping near the singular point. To further suppress voltage-dependent drift during the sweep, the reference experiment slightly adjusts the active-branch resistance so as to maintain approximately <math>V_B \approx 2~\mathrm{V}</math> throughout the measurement series.<ref name="Bai2024SM" />

Our sweep uses controlled series combinations of small inductors, giving an approximately <math>2~\mu\mathrm{H}</math> step in <math>L_A</math>. The homebrew compressible coil is then used to show that a mechanical change of the sensing element also changes the inductance. The conclusion drawn from our data is therefore not a precision determination of <math>L_{A,\mathrm{NEP}}</math>, but a validation that the breadboard circuit produces the same kind of frequency response as the stable branch in Fig. S11(a).

=== Long-window spectral analysis and noise inspection ===
The supplementary also discusses long-time measurements over a window of <math>0<t<5~\mathrm{s}</math>. The motivation is that the noise around the stable oscillation is too small to be judged reliably from raw scope traces alone, so it is more informative to inspect the Fourier spectra and the extracted spectral center frequency and linewidth.<ref name="Bai2024SM" />

This is a useful methodological point for a sensing-oriented reproduction. In an NEP-based sensor, one ultimately cares about how precisely the frequency peak can be resolved and tracked. A long-window Fourier analysis is therefore closer to the sensing task than a purely visual time-domain inspection.

== Results ==

=== Result display: steady oscillation ===
[[File:PC5271inductive1.png|750px|thumb|center|Breadboard implementation and representative steady oscillation waveform.]]

This figure shows our circuit implemented on a breadboard and a representative stabilised voltage waveform after the resonators are activated by the supplied voltage. The displayed waveform is the evidence used to identify a stable oscillation frequency.

=== Result display: <math>f-L_A</math> sweep ===
[[File:PC5271inductive3.jpeg|750px|thumb|center|Measured breadboard <math>f-L_A</math> curve.]]

This figure shows the measured oscillation frequency <math>f</math> as a function of <math>L_A</math>. The values of <math>L_A</math> in the main sweep are produced by connecting small inductors in series with approximately <math>2~\mu\mathrm{H}</math> step size.

=== Result display: iron-rod demonstration ===
[[File:PC5271inductive2.png|750px|thumb|center|Displacement measurement by compressing the solenoid and inserting iron rod into the homebrew coil]]

This figure should show the homebrew coil with an inserted iron rod and the corresponding change in the measured oscillation frequency. The experimental variable is the rod position or insertion state, and the measured output is the activated steady oscillation frequency.

== Evaluation ==

=== Stable activated oscillation ===
The first required experimental signature is that, after the resonators are activated by the supplied voltage, the circuit reaches a stable oscillating waveform instead of decaying immediately or being amplified without bound. A stable waveform means that the TL071 active branch supplies enough small-signal negative resistance to compensate loss, while gain saturation prevents unbounded amplification.

Physically, this behavior confirms the role of the nonlinear active branch. At small amplitude, the effective negative resistance in resonator <math>A</math> overcomes dissipation. At larger amplitude, the gain becomes weaker, so the oscillation amplitude is clamped. The system self-stabilizes because the physical steady branch is the one with zero imaginary part of the eigenfrequency. The auxiliary branches near the nonlinear exceptional point are not stable physical outputs; they are mathematical branches of the nonlinear steady-state equation. The experiment is therefore not a passive ring-down measurement; it is a nonlinear steady-state selection process.

=== Evaluation of the <math>f-L_A</math> trend ===
The main experimental comparison is the measured <math>f-L_A</math> trend against Fig. S11(a) of SM-4. The reference experiment reports the stable frequency branch of the nonlinear circuit as <math>L_A</math> is tuned.<ref name="Bai2024SM" /> This branch is the physical one because its eigenfrequency is real in the steady state. In our breadboard experiment, the same observable is measured using the series-inductor sweep.

It is useful to separate two statements. First, any LC resonator changes frequency when <math>L</math> changes, because <math>f_0=1/(2\pi\sqrt{LC})</math>. This alone would not demonstrate the nonlinear exceptional-point mechanism. Second, in the coupled active-lossy circuit, the frequency is not just the bare LC resonance frequency of one isolated inductor and capacitor. It is the frequency selected by the nonlinear steady-state balance of gain, loss, and coupling. The reason Fig. S11(a) matters is that it shows this selected stable branch as <math>L_A</math> is swept.

Our breadboard circuit uses different component values and a different op-amp from the PCB experiment, so exact numerical overlap is not expected. The meaningful evaluation is whether the measured frequency branch follows the same qualitative <math>f-L_A</math> behavior as the stable branch in SM-4. Observing that behavior is enough to show that the constructed circuit realizes the intended nonlinear voltage-activated oscillating mechanism.

=== Evaluation of the displacement demonstration ===
In the iron-rod demonstration, the homebrew coil becomes a displacement-dependent inductive element. As the rod enters the coil, the effective magnetic permeability inside the coil increases. In the solenoid expression

<math display="block">
L(x) \approx \mu_\mathrm{eff}(x)\frac{N^2 A}{\ell},
</math>

the insertion depth <math>x</math> changes <math>\mu_\mathrm{eff}</math>. The coil geometry <math>N</math>, <math>A</math>, and <math>\ell</math> are approximately fixed during this demonstration, so the dominant effect is the change in magnetic core material inside the coil. Inductive displacement sensors commonly use this principle: the position of a ferromagnetic object changes a coil inductance, and the electronics convert that inductance change into an electrical signal.<ref name="Chen2014" />

In our circuit, the electrical signal is the activated steady oscillation frequency rather than a bridge voltage. The sensing chain is

<math display="block">
\mathrm{rod\ displacement}\ x
\longrightarrow
\mathrm{effective\ inductance}\ L_A(x)
\longrightarrow
\mathrm{oscillation\ frequency}\ f(x).
</math>

This demonstration shows how the inductive readout can be turned into a displacement-sensing readout after calibration of the relation between rod position and frequency.

=== Cubic-root perturbation law and sensing meaning ===
Near the singular operating point, the eigenfrequency shift follows the hallmark cubic-root law,

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3}.
</math>

This law is the main sensing-related result of the reference experiment.<ref name="Bai2024" /><ref name="Bai2024SM" /> Compared with the linear response of an ordinary isolated resonator, the cubic-root dependence can produce a larger frequency displacement for the same small perturbation of inductance when the device is biased sufficiently close to <math>L_{A,\mathrm{NEP}}</math>.

A useful way to interpret the result is through the local derivative,

<math display="block">
\left|\frac{df}{dL_A}\right| \propto |\delta L_A|^{-2/3}.
</math>

This increasing local responsivity is why the circuit is attractive for sensing. Our experiment supports this sensing logic by showing that the breadboard circuit has a measurable stable frequency branch as <math>L_A</math> is changed. A full quantitative exponent extraction would require identifying <math>L_{A,\mathrm{NEP}}</math> and <math>f_{\mathrm{NEP}}</math>, then plotting <math>|f-f_{\mathrm{NEP}}|</math> against <math>|L_A-L_{A,\mathrm{NEP}}|</math> on logarithmic axes.

=== Basis completeness and noise robustness ===
The supplementary material evaluates the Petermann factor as a diagnostic of modal nonorthogonality and basis completeness. For a non-Hermitian eigenproblem, one may write

<math display="block">
\mathrm{PF}_n =
\frac{\langle \Phi_n^{L}|\Phi_n^{L}\rangle \langle \Phi_n^{R}|\Phi_n^{R}\rangle}
{\left|\langle \Phi_n^{L}|\Phi_n^{R}\rangle\right|^2},
</math>

where <math>|\Phi_n^{R}\rangle</math> and <math>|\Phi_n^{L}\rangle</math> are the right and left eigenvectors of the effective steady-state Hamiltonian. At a conventional linear EP, this quantity is associated with diverging excess-noise sensitivity. In Ref. [1], the cited analysis finds that the effective steady-state Hamiltonian remains non-defective and the Petermann factor is finite at the <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM" />

This point explains the significance of the reference experiment. The system does not merely realize a large mathematical responsivity; it realizes that responsivity in a way that avoids the most severe basis-collapse problem of a linear EP. Our breadboard experiment does not directly test basis completeness, but the result provides the theoretical reason why reproducing the stable <math>f-L_A</math> branch is physically meaningful.

=== Overall interpretation ===
In this platform, the measurand is encoded as a perturbation of <math>L_A</math>. Any physical mechanism that modifies the effective inductance of resonator <math>A</math>, for example displacement, magnetic coupling, or the presence of a nearby magnetic target, moves the system away from the operating point and shifts the oscillation frequency. The primary readout in our implementation is therefore spectral: we track <math>f</math> as <math>L_A</math> is changed.

Taken together, Ref. [1] establishes the complete sensing logic: a nonlinear active-lossy circuit is tuned to an <math>\mathrm{NEP}_3</math>; after voltage activation, the system settles onto a stable oscillating branch; the dominant spectral peak is tracked as the perturbation readout; and the enhanced cubic-root response is obtained without the defective-basis pathology of a conventional linear EP. Our breadboard result demonstrates the experimentally accessible core of this logic by reproducing the stable frequency response corresponding to Fig. S11(a) of SM-4.

== References ==
<references />

Inductive Sensors of Ultra-high Sensitivity Based on Nonlinear Exceptional Point

2026-04-23T06:21:55Z

Siyu: /* Result display: f-L_A sweep */

== Introduction ==
This project is a implementation of the experiment reported in ''Observation of Nonlinear Exceptional Points with a Complete Basis in Dynamics'', adapted to our available laboratory equipment.<ref name="Bai2024">K. Bai, T.-R. Liu, L. Fang, J.-Z. Li, C. Lin, D. Wan, and M. Xiao, Observation of nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''132''', 073802 (2024).</ref> The circuit contains two coupled resonators. One resonator has gain, meaning that it can supply energy to the oscillation, while the other has loss, meaning that it dissipates energy. A system without energy conservation is called a '''non-Hermitian''' system. Such systems can show unusual degeneracies called '''exceptional points''' (EPs), where both the eigenfrequencies and the eigenmodes merge.

Exceptional points are interesting for sensing because the measured frequency can respond very strongly to a small perturbation. In an inductive sensor, the perturbation is a change in inductance. If a small change in inductance produces a large change in oscillation frequency, the circuit can in principle detect small physical changes. However, ordinary linear EPs also suffer from strong eigenmode nonorthogonality and a defective basis, which can increase noise sensitivity and complicate practical sensing.<ref name="Bai2024" />

The system studied in this project belongs to the recently developed class of '''nonlinear exceptional points''' (NEPs). The physical platform is a pair of coupled circuit resonators. Resonator <math>A</math> is made active by a saturable negative-resistance branch, while resonator <math>B</math> is intentionally lossy. Because the gain in resonator <math>A</math> depends on the oscillation amplitude, the steady-state problem is nonlinear and admits multiple self-consistent branches. At the singular operating point reported in Ref. [1], one stable steady branch and two auxiliary steady branches merge into a third-order nonlinear exceptional point, denoted <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM">Supplemental Material for K. Bai, T.-R. Liu, L. Fang, J.-Z. Li, C. Lin, D. Wan, and M. Xiao, Observation of nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''132''', 073802 (2024).</ref>

The experimentally important consequence is that the oscillation eigenfrequency obeys a cubic-root perturbation law near the singularity,

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3},
</math>

where <math>\delta L_A = L_A-L_{A,\mathrm{NEP}}</math> is the inductive detuning from the exceptional operating point. This response is steeper than the ordinary linear dependence of a conventional resonator and is therefore useful for ultra-sensitive inductive sensing.<ref name="Bai2024" /><ref name="Bai2024SM" />

A second and equally important point is that the cited work does not interpret the singularity as a conventional defective linear EP. Instead, the effective steady-state Hamiltonian remains non-defective in dynamics, and the associated Petermann factor stays finite at the operating point. In other words, the experiment realizes enhanced spectral responsivity without inheriting the strongest basis-collapse pathology of a standard linear EP.<ref name="Bai2024" /><ref name="Bai2024SM" />

In this project, our goal is to reproduce the experimental logic of Ref. [1] and its supplementary material under course-lab conditions. Unlike the PCB implementation in the paper, our circuit is assembled on a breadboard and the active branch uses a '''TL071''' operational amplifier. Our main observable is the steady oscillation frequency <math>f</math> as a function of the chosen inductance <math>L_A</math>. We tune <math>L_A</math> in two ways. For the main <math>f-L_A</math> sweep, we join small inductors in series so that the discrete inductance step is about <math>2~\mu\mathrm{H}</math>. In addition, we use a homebrew compressible coil to demonstrate the sensing mechanism directly. Compressing the coil changes its length and turn density while keeping the cross-sectional area approximately unchanged. We therefore use the <math>f-L_A</math> behavior in Fig. S11(a) of the supplementary material as the main experimental benchmark. Reproducing this trend is sufficient for demonstrating that our setup captures the intended nonlinear voltage-activated, self-stabilized oscillating response, although it is not a full reproduction of every diagnostic reported in Ref. [1].<ref name="Bai2024" /><ref name="Bai2024SM" />

== Team Members ==
Wang Peikun — E1538091@u.nus.edu 
Zhu Ziyang — E1583446@u.nus.edu 
Yuan Si-Yu — siyu_yuan@u.nus.edu 
Li Xunyu — xunyu@u.nus.edu

== Background Knowledge ==

=== Ordinary LC and RLC resonators ===
The basic building block of the experiment is an electrical resonator. An ideal LC resonator contains an inductor <math>L</math> and a capacitor <math>C</math>. Energy repeatedly moves between the magnetic field of the inductor and the electric field of the capacitor. The natural angular frequency is

<math display="block">
\omega_0=\frac{1}{\sqrt{LC}},
</math>

or, in ordinary frequency units,

<math display="block">
f_0=\frac{1}{2\pi\sqrt{LC}}.
</math>

This formula is important for the whole project because it explains why changing an inductance changes the measured frequency. A larger <math>L</math> gives a lower resonance frequency, while a smaller <math>L</math> gives a higher resonance frequency.

Real circuits are not ideal. Wires, inductors, capacitors, breadboard contacts, and measuring probes all dissipate energy. This loss is often represented by a resistor <math>R</math>, giving an RLC resonator. In a passive RLC circuit, the oscillation amplitude decays with time because energy is continuously lost as heat. If a circuit is to maintain a steady activated oscillation, the loss must be compensated by a gain element.

=== Gain, loss, and negative resistance ===
In electronics, gain can be represented as an effective negative resistance. A normal positive resistance removes energy from the resonator. A negative resistance does the opposite: it supplies energy to the resonator. In this experiment, resonator <math>A</math> contains an active op-amp branch that behaves like a negative resistance at small amplitude, while resonator <math>B</math> contains ordinary positive loss.

The gain cannot remain negative and constant forever. If it did, any activated oscillation would be amplified without bound until the op-amp clipped or the circuit failed. The important nonlinear ingredient is '''saturable gain''': the active branch gives strong gain at small amplitude but weaker gain at larger amplitude. The stable oscillation amplitude is selected when the supplied gain balances the total loss. This nonlinear steady state is why the experiment is a nonlinear dynamical problem rather than only a passive resonance measurement.

=== What "non-Hermitian" means in this circuit ===
In ordinary conservative physics, a Hermitian Hamiltonian describes a closed system whose total energy is conserved. Its eigenfrequencies are real, and its eigenmodes form a complete orthogonal basis. This is the familiar setting of many quantum-mechanics courses.

A circuit with gain and loss is an open system. Energy can enter through the op-amp branch and leave through resistive loss. The effective dynamical matrix is therefore generally '''non-Hermitian'''. In a non-Hermitian system, eigenfrequencies can be complex:

<math display="block">
\omega=\omega_\mathrm{r}+i\omega_\mathrm{i}.
</math>

The real part <math>\omega_\mathrm{r}</math> gives the oscillation frequency, while the imaginary part <math>\omega_\mathrm{i}</math> describes amplification or decay. A positive imaginary part means the mode is amplified, a negative imaginary part means it decays, and a stable steady oscillation corresponds to zero imaginary part, or equivalently zero net amplification. This is the sense in which the circuit is self-stabilized after voltage activation.

=== Exceptional points ===
In a usual Hermitian system, two modes can have the same eigenfrequency while still remaining two independent modes. This is an ordinary degeneracy. A non-Hermitian system can show a stronger kind of degeneracy called an '''exceptional point''' (EP). At an EP, not only the eigenvalues but also the eigenvectors coalesce. In simple language, two modes collapse into one mode.

This collapse makes EPs interesting for sensing. Near a second-order linear EP, a small perturbation can produce a square-root frequency splitting:

<math display="block">
\Delta f \propto \sqrt{\epsilon},
</math>

where <math>\epsilon</math> is the perturbation. This response is steeper than an ordinary linear response when <math>\epsilon</math> is very small. However, a conventional linear EP also has a serious drawback: the eigenbasis becomes defective, meaning that the system no longer has enough independent eigenvectors. This is associated with strong modal nonorthogonality and enhanced noise sensitivity. The Petermann factor is one common way to quantify this nonorthogonality.

=== Why the nonlinear exceptional point is different ===
The reference paper studies a nonlinear exceptional point. The word "nonlinear" is essential. The circuit does not have a fixed gain coefficient; instead, the gain depends on the oscillation amplitude. The steady state must therefore be solved self-consistently: the frequency, voltage amplitudes, and saturated gain must all agree with one another.

In the theory of Ref. [1], this nonlinear steady-state problem can reduce to a cubic equation for the oscillation frequency. A cubic equation can have a triple root. At the special operating point, one stable physical branch and two auxiliary branches merge, giving a third-order nonlinear exceptional point, written as <math>\mathrm{NEP}_3</math>. The stable physical branch is the one with real oscillation frequency, i.e. without an imaginary part. The auxiliary branches are needed to define the higher-order coalescence mathematically, but they are not physically stable operating states of the circuit. The corresponding frequency response has the form

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3}.
</math>

This cubic-root response is the key sensing signature. It is steeper than both an ordinary linear response and the square-root response of a second-order EP. The important conceptual point of Ref. [1] is that this enhanced response is obtained while the effective steady-state Hamiltonian can still retain a complete basis in dynamics.<ref name="Bai2024" /><ref name="Bai2024SM" />

=== Sensing mechanism ===
The circuit can be understood as a frequency-generating sensor. It converts an inductance change into a frequency change.

The first step is activation. A voltage is supplied to the resonators to excite the circuit into an oscillating state. The active branch in resonator <math>A</math> then supplies energy through an effective negative resistance and amplifies the selected oscillation mode.

The second step is self-stabilization. The gain is saturable, which means it becomes weaker when the voltage amplitude becomes larger. The activated oscillation therefore is not amplified forever. The circuit settles onto the physical steady branch where the imaginary part of the eigenfrequency is zero. At that point, the gain supplied by resonator <math>A</math> balances the loss in the circuit, especially the loss in resonator <math>B</math>.

The third step is frequency selection. Because resonators <math>A</math> and <math>B</math> are coupled, the final oscillation frequency is not simply the resonance frequency of either isolated resonator. It is selected by the whole coupled circuit: the inductors, capacitors, coupling elements, loss, and saturated gain all contribute.

The fourth step is sensing. The inductance <math>L_A</math> controls the bare frequency of resonator <math>A</math>. When <math>L_A</math> changes, the self-consistent frequency selected by the nonlinear circuit also changes. Therefore, tracking the spectral peak <math>f</math> tells us how the effective inductance has changed.

The fifth step is the nonlinear exceptional-point response. Near the nonlinear exceptional point discussed in Ref. [1], the physical steady branch and the two auxiliary branches coalesce in the nonlinear steady-state equation. The frequency shift of the stable branch can then scale as the cube root of the inductance perturbation. This means that a small inductance change can produce a comparatively large frequency change. Our breadboard experiment focuses on verifying the stable <math>f-L_A</math> branch corresponding to Fig. S11(a) of SM-4, which is the experimentally accessible part of this sensing mechanism.

=== Same mechanism in other platforms ===
Several references are included to show that this is a platform-independent sensing mechanism, not only a feature of our breadboard circuit. The common chain is

<math display="block">
\mathrm{physical\ perturbation}
\longrightarrow
\mathrm{parameter\ change}
\longrightarrow
\mathrm{eigenfrequency\ shift\ of\ a\ resonator/cavity\ with\ gain\ and\ loss}
\longrightarrow
\mathrm{higher\ order\ steady\ state\ response}
\longrightarrow
\mathrm{sensing}.
</math>

In our circuit, the physical perturbation is a change in the inductance <math>L_A</math>, and the measured output is the oscillation frequency <math>f</math>. In coupled optical, mechanical, electrical, or hybrid resonators, the perturbed parameter may instead be a cavity detuning, mechanical frequency, phase shift, magnetic-field-dependent transition, or another effective resonator parameter. The shared point is that gain, loss, coupling, and nonlinearity convert the parameter change into a higher-order steady-state spectral response.

The earlier theoretical work on nonlinear exceptional points showed that nonlinear coupled resonators can host higher-order NEPs while retaining a complete dynamical basis.<ref name="Bai2023">K. Bai, J.-Z. Li, T.-R. Liu, L. Fang, D. Wan, and M. Xiao, Nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''130''', 266901 (2023).</ref> The 2024 experiment implemented the same mechanism in coupled electrical resonators and measured the stable nonlinear frequency branch.<ref name="Bai2024" /><ref name="Bai2024SM" /> Silicon micromechanical resonators provide another realization: there, the measured output can be a phase difference, and the nonlinear exceptional-point response can enhance phase-shift sensing.<ref name="Xie2026">Y.-J. Xie, S.-Y. Zhang, M.-N. Zhang, R. Wang, L.-F. Wang, and Q.-A. Huang, Nonlinear exceptional points boost phase-shift sensing in silicon micromechanical resonators, ''Phys. Rev. Applied'' '''25''', 034039 (2026).</ref> Hybrid quantum systems with nonlinear bistability show a related route, where nonlinear transition dynamics can enhance sensing signal-to-noise ratio.<ref name="Wang2026">H. Wang ''et al.'', Exceptional sensitivity near the bistable transition point of a hybrid quantum system, ''Nature Physics'' '''22''', 577-583 (2026).</ref>

These examples support the same mechanism rather than unrelated applications. At the same time, enhanced response alone is not enough to prove better sensing. Noise can shift or weaken the apparent nonlinear exceptional point and can limit the signal-to-noise improvement.<ref name="Zheng2025">X. Zheng and Y. D. Chong, Noise constraints for nonlinear exceptional point sensing, ''Phys. Rev. Lett.'' '''134''', 133801 (2025).</ref> This is why our project makes a careful claim: we demonstrate the frequency-response mechanism on a breadboard, but we do not claim a complete metrological noise advantage.

== Report objectives and scope ==
The purpose of this work is to study the circuit principle behind '''nonlinear-exceptional-point-enhanced inductive sensing''' in a coupled electronic resonator platform. The scope is intentionally narrower than the full experiment reported in Ref. [1]. We are not attempting to redesign the active branch, derive all nonlinear bifurcation theory from first principles, reconstruct the auxiliary steady-state branches, or characterize every parasitic element. Instead, we adopt the framework of Ref. [1] and SM-4 and ask whether the breadboard circuit can reproduce the experimentally accessible frequency response.

The main target of our experiment is the <math>f-L_A</math> curve corresponding to Fig. S11(a) of SM-4. In the reference experiment, the steady-state frequency, voltage ratio <math>|V_A/V_B|</math>, and relative phase <math>\theta_A-\theta_B</math> are all measured as functions of <math>L_A</math>.<ref name="Bai2024SM" /> In our implementation, the frequency-versus-inductance curve is the primary observable. We tune <math>L_A</math> by connecting small inductors in series with approximately <math>2~\mu\mathrm{H}</math> resolution. We also use a homebrew coil in a separate demonstration where mechanical compression or insertion of an iron rod changes the effective inductance. The experimental goal is therefore to verify that the TL071 breadboard circuit settles into a stable oscillating state and that the measured frequency changes with <math>L_A</math> in the same qualitative manner as Fig. S11(a).

From the sensing perspective, the most relevant figure of merit is the local frequency responsivity. If

<math display="block">
|f-f_{\mathrm{NEP}}| = A |\delta L_A|^{1/3},
</math>

then differentiation gives

<math display="block">
S_f \equiv \left|\frac{df}{dL_A}\right| \propto |\delta L_A|^{-2/3}.
</math>

Thus, as the operating point approaches <math>L_{A,\mathrm{NEP}}</math>, the local spectral responsivity increases rapidly. The main practical constraint is that the system must remain on the intended stable branch while measurement noise and slow parameter drift are kept below the scale of the induced frequency shift.

== Setups ==
[[File:Experimental_setup.png|750px|thumb|center|Experimental setup adapted from Ref. [1].]]

=== Circuit topology and component implementation ===
The reference platform consists of two coupled RLC resonators. Both resonators contain the same nominal shunt capacitance <math>C_0</math>, while the inductances <math>L_A</math> and <math>L_B</math> set their bare resonance frequencies. The two resonators are coupled through a parallel resistor-capacitor channel characterized by <math>R_c</math> and <math>C_c</math>. Resonator <math>B</math> contains an explicit positive resistance <math>R_B</math> and therefore plays the role of the lossy resonator. Resonator <math>A</math>, by contrast, contains an active branch whose effective resistance is negative at small amplitude and becomes less negative as the oscillation amplitude increases, thereby providing saturable gain.<ref name="Bai2024" /><ref name="Bai2024SM" />

Our implementation follows this topology but is built on a breadboard rather than a custom PCB. The active branch is implemented with an op-amp negative-resistance circuit using a '''TL071''' operational amplifier. This is an important practical difference from the supplementary experiment, where the active components explicitly listed are a TI '''LM7171''' operational amplifier and an '''Onsemi BAV99L''' diode.<ref name="Bai2024SM" /> The TL071 substitution and the breadboard layout introduce different bandwidth limits, wiring parasitics, and contact resistances, so the numerical operating point is not expected to match the PCB experiment. For this reason, our comparison emphasizes the observable trend of <math>f</math> versus <math>L_A</math>.

The supplementary material reports the following representative component values for the PCB experiment:

<math display="block">
C_0 = 18.5~\mathrm{nF}, \qquad
C_c = 3.9~\mathrm{nF}, \qquad
R_c = 760~\Omega,
</math>

<math display="block">
L_B = 197.7~\mu\mathrm{H}, \qquad
R_B = 1314~\Omega.
</math>

A representative operating point shown in the main figure uses

<math display="block">
L_A = 245.4~\mu\mathrm{H}.
</math>

Our breadboard circuit follows the same functional topology, but its component values and parasitic elements are not identical to the PCB values above. The horizontal axis of our main <math>f-L_A</math> plot is therefore the series-combined <math>L_A</math> selected from our available small inductors, while the measured frequency <math>f</math> is the robust experimental readout.

=== Inductance tuning methods ===
We use two complementary methods to change the inductance seen by resonator <math>A</math>.

The first method is used for the main <math>f-L_A</math> sweep. Small inductors are connected in series so that the total inductance changes in discrete steps of approximately <math>2~\mu\mathrm{H}</math>. If the inductors are sufficiently separated so that their mutual magnetic coupling is small, the total series inductance is approximately the sum of the individual inductances,

<math display="block">
L_A \approx L_{\mathrm{base}}+\sum_i L_i .
</math>

This method is simple and reproducible. Adding one small inductor increases <math>L_A</math>; removing it decreases <math>L_A</math>. The circuit frequency is then measured for each chosen value. This produces a discrete version of the <math>f-L_A</math> curve in Fig. S11(a) of SM-4. The step size is not as fine as a commercial variable inductor, but <math>2~\mu\mathrm{H}</math> spacing is small enough to observe the frequency trend over the accessible range.

The second method is a homebrew compressible variable inductor. This coil is useful because it makes the sensing mechanism visible: a mechanical displacement changes the coil geometry, which changes the inductance, which then changes the oscillation frequency. The basic idea can be understood from the approximate solenoid formula

<math display="block">
L \approx \mu_\mathrm{eff}\frac{N^2 A}{\ell},
</math>

where <math>N</math> is the number of turns, <math>A</math> is the cross-sectional area, <math>\ell</math> is the coil length, and <math>\mu_\mathrm{eff}</math> describes the effective magnetic permeability of the core and surrounding region.<ref name="Chen2014">S.-C. Chen, D.-K. Le, and V.-S. Nguyen, Inductive displacement sensors with a notch filter for an active magnetic bearing system, ''Sensors'' '''14''', 12640-12657 (2014).</ref> During compression, the number of turns <math>N</math> is unchanged and the cross-sectional area <math>A</math> remains approximately unchanged, but the length <math>\ell</math> decreases. Equivalently, the turn density

<math display="block">
n=\frac{N}{\ell}
</math>

increases. The magnetic field produced by a solenoid is proportional to turn density. Therefore, compressing the coil increases the magnetic flux linkage through the turns and increases the inductance. Stretching the coil has the opposite effect.

This mechanical tuning is not identical to the precision tunable inductor used in the reference experiment. A real compressed coil is not an ideal long solenoid: the spacing between turns is finite, the end effects change as the coil length changes, nearby wires and the breadboard add parasitic capacitance, and the leads contribute additional series inductance and resistance. For this reason, the homebrew inductor is best treated as an experimentally characterized tuning element rather than an ideal component whose value is known exactly from geometry. In the present project, it provides a direct mechanical demonstration of how displacement can perturb <math>L_A</math>.

Experimentally, we measured the effective inductance of this homebrew variable inductor at different compression states. The compression process changes the coil length and wire density while leaving the coil area approximately fixed. Therefore, the tuning mechanism is physically the same as changing <math>\ell</math> and <math>n=N/\ell</math> in the solenoid model, rather than changing the number of turns or the coil diameter.

=== Kirchhoff model of the nonlinear circuit ===
A convenient state-space description uses the node voltages <math>V_A(t)</math> and <math>V_B(t)</math> together with the inductor currents <math>I_A(t)</math> and <math>I_B(t)</math>. A node voltage is the voltage measured at one resonator node relative to ground. Kirchhoff's current law says that the algebraic sum of all currents flowing away from a node must be zero. This gives a direct way to translate the circuit diagram into differential equations.

Denoting the amplitude-dependent conductance of the active branch by <math>G_A(|V_A|)</math>, where <math>G_A<0</math> in the small-signal regime, the nodal equations can be written as

<math display="block">
\begin{aligned}
C_0 \frac{dV_A}{dt} + I_A + G_A(|V_A|)\,V_A + \frac{V_A-V_B}{R_c}
+ C_c\frac{d(V_A-V_B)}{dt} &= 0,\\
C_0 \frac{dV_B}{dt} + I_B + \frac{V_B}{R_B} + \frac{V_B-V_A}{R_c}
+ C_c\frac{d(V_B-V_A)}{dt} &= 0,\\
L_A \frac{dI_A}{dt} &= V_A,\\
L_B \frac{dI_B}{dt} &= V_B.
\end{aligned}
</math>

Each term has a simple circuit meaning. The terms <math>C_0 dV_A/dt</math> and <math>C_0 dV_B/dt</math> are capacitor currents. The terms <math>I_A</math> and <math>I_B</math> are currents through the inductors. The term <math>G_A(|V_A|)V_A</math> is the active branch current; because <math>G_A</math> can be negative, this branch can inject energy instead of dissipating it. The term <math>V_B/R_B</math> is the ordinary loss current through the resistor in resonator <math>B</math>. The terms involving <math>R_c</math> and <math>C_c</math> describe current flowing between resonators <math>A</math> and <math>B</math>, so they are the electrical coupling between the two resonators.

These equations show explicitly how the experiment differs from a linear EP circuit. If <math>G_A</math> were a constant negative conductance, the model would reduce to a linear non-Hermitian dimer. Here, however, <math>G_A</math> depends on the voltage amplitude, so the steady state must be determined self-consistently together with the oscillation amplitude itself.

=== Harmonic-balance formulation of the steady state ===
After the transient has died away, the oscilloscope waveform is approximately periodic. The simplest approximation is therefore to keep only the dominant sinusoidal component. This is called a harmonic-balance approximation. For a stable activated oscillating state, one writes

<math display="block">
V_j(t)=\mathrm{Re}\!\left[\tilde V_j e^{i\omega t}\right],
\qquad j=A,B.
</math>

Substituting this ansatz into the circuit equations yields a nonlinear algebraic problem of the form

<math display="block">
\mathbf{Y}(\omega,|\tilde V_A|)
\begin{pmatrix}
\tilde V_A\\
\tilde V_B
\end{pmatrix}
=
\begin{pmatrix}
Y_A(\omega,|\tilde V_A|)+Y_c(\omega) & -Y_c(\omega)\\
-Y_c(\omega) & Y_B(\omega)+Y_c(\omega)
\end{pmatrix}
\begin{pmatrix}
\tilde V_A\\
\tilde V_B
\end{pmatrix}
=0,
</math>

with

<math display="block">
Y_A(\omega,|\tilde V_A|)=G_A(|\tilde V_A|)+i\omega C_0+\frac{1}{i\omega L_A},
</math>

<math display="block">
Y_B(\omega)=\frac{1}{R_B}+i\omega C_0+\frac{1}{i\omega L_B},
\qquad
Y_c(\omega)=\frac{1}{R_c}+i\omega C_c.
</math>

A nontrivial solution requires

<math display="block">
\det \mathbf{Y}(\omega,|\tilde V_A|)=0.
</math>

This determinant condition is the circuit analogue of an eigenfrequency condition. In a passive LC resonator, the resonance condition gives a single frequency <math>\omega_0=1/\sqrt{LC}</math>. In the present two-resonator active circuit, the same idea becomes a matrix equation because the two resonators are coupled. The amplitudes <math>\tilde V_A</math> and <math>\tilde V_B</math> are not arbitrary: a nonzero oscillation exists only when the admittance matrix has a zero eigenvalue.

Because the active admittance depends on amplitude, this condition is nonlinear and supports multiple branches. The supplementary material shows that the steady-state frequency can be reduced to a '''real-coefficient cubic polynomial''' in <math>\omega</math>. The existence of up to three real steady-state branches is therefore not accidental; it is the algebraic origin of the experimentally relevant <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM" />

=== Effective nonlinear-Hamiltonian picture ===
The same physics is often rewritten in coupled-mode form. This notation is common in non-Hermitian physics because it makes the circuit look like two coupled modes. The variables <math>\psi_A</math> and <math>\psi_B</math> represent the complex oscillation amplitudes in resonators <math>A</math> and <math>B</math>. The model is

<math display="block">
i\frac{d}{dt}
\begin{pmatrix}
\psi_A\\
\psi_B
\end{pmatrix}
=
\begin{pmatrix}
\omega_A+i\,g(|\psi_A|) & \kappa\\
\kappa & \omega_B-i\ell
\end{pmatrix}
\begin{pmatrix}
\psi_A\\
\psi_B
\end{pmatrix},
</math>

where <math>\omega_A</math> and <math>\omega_B</math> are the effective modal frequencies, <math>\kappa</math> is the coupling strength, <math>\ell</math> is the loss of resonator <math>B</math>, and <math>g(|\psi_A|)</math> is the saturable gain of resonator <math>A</math>. The steady-state problem then becomes

<math display="block">
H_s(\Psi)\Psi = \Omega \Psi,
</math>

with a real oscillation eigenfrequency <math>\Omega=2\pi f</math> for the stable oscillating branch. The word "real" is important: it means the imaginary part has vanished, so the mode is neither being amplified nor decaying in the steady state. In the cited experiment, this physical stable branch and two auxiliary steady branches coalesce at the singular point, producing a third-order nonlinear exceptional point rather than a conventional second-order linear EP. The auxiliary branches help form the cubic-root singularity but are not themselves stable physical outputs of the circuit.<ref name="Bai2024" /><ref name="Bai2024SM" />

== Measurements ==

=== Acquisition protocol and steady-state selection ===
The measurement is performed in the time domain. In the reference experiment, an arbitrary waveform generator first applies an external start-up signal of 1 V at 70 kHz in order to bring the system into the basin of attraction of the stable oscillating state.<ref name="Bai2024" /> In our breadboard measurement, an external voltage is likewise required to activate the resonators. The measured frequency is extracted from the activated steady waveform recorded on the oscilloscope.

The physical meaning of this procedure is important. The oscillation is not assumed to appear without activation; a supplied voltage is needed to bring the resonators into the relevant dynamical state. The measured oscillation frequency is then interpreted as the frequency selected by the nonlinear steady-state condition.

=== Extraction of the <math>f-L_A</math> observable ===
Ref. [1] does not treat the oscilloscope traces merely as qualitative evidence of oscillation. The supplementary explicitly states that the full reference experiment reports the steady-state frequency, the voltage ratio <math>|V_A/V_B|</math>, and the relative phase <math>\theta_A-\theta_B</math>, all extracted from the dominant peaks of the Fourier spectra of <math>V_A</math> and <math>V_B</math>.<ref name="Bai2024SM" />

Our available observable is the steady-state frequency. For each chosen value of <math>L_A</math>, the resonators are voltage-activated, the circuit reaches a stable oscillating state, and the dominant spectral peak is recorded. If we denote the Fourier transforms by

<math display="block">
\tilde V_A(f)=\mathcal{F}\{V_A(t)\},
\qquad
\tilde V_B(f)=\mathcal{F}\{V_B(t)\},
</math>

then the dominant oscillation frequency is obtained from

<math display="block">
f_0=\arg\max_f |\tilde V_A(f)| \approx \arg\max_f |\tilde V_B(f)|.
</math>

The experimental data set is therefore

<math display="block">
\{L_A,\ f_0(L_A)\},
</math>

which is compared with the frequency branch plotted in Fig. S11(a) of SM-4. In the full reference measurement, the amplitude ratio

<math display="block">
\rho = \frac{|\tilde V_A(f_0)|}{|\tilde V_B(f_0)|}
</math>

and relative phase

<math display="block">
\Delta\phi = \arg \tilde V_A(f_0)-\arg \tilde V_B(f_0)
</math>

provide additional state verification. Those quantities are useful but are not the main basis of our breadboard validation.

In practice, the same frequency can also be estimated directly from the time-domain period <math>T</math> using <math>f_0=1/T</math> when the waveform is clean. Fourier analysis is preferred because it uses a longer section of waveform and identifies the dominant oscillation peak even when there is small amplitude noise or waveform distortion.

=== Practical inductance sweep ===
In SM-4, the inductive tuning is carried out by varying <math>L_A</math>. The supplementary notes that the tunable inductor exhibits less than about 0.4% inductance variation over the frequency range of interest, which is sufficiently small for controlled sweeping near the singular point. To further suppress voltage-dependent drift during the sweep, the reference experiment slightly adjusts the active-branch resistance so as to maintain approximately <math>V_B \approx 2~\mathrm{V}</math> throughout the measurement series.<ref name="Bai2024SM" />

Our sweep uses controlled series combinations of small inductors, giving an approximately <math>2~\mu\mathrm{H}</math> step in <math>L_A</math>. The homebrew compressible coil is then used to show that a mechanical change of the sensing element also changes the inductance. The conclusion drawn from our data is therefore not a precision determination of <math>L_{A,\mathrm{NEP}}</math>, but a validation that the breadboard circuit produces the same kind of frequency response as the stable branch in Fig. S11(a).

=== Long-window spectral analysis and noise inspection ===
The supplementary also discusses long-time measurements over a window of <math>0<t<5~\mathrm{s}</math>. The motivation is that the noise around the stable oscillation is too small to be judged reliably from raw scope traces alone, so it is more informative to inspect the Fourier spectra and the extracted spectral center frequency and linewidth.<ref name="Bai2024SM" />

This is a useful methodological point for a sensing-oriented reproduction. In an NEP-based sensor, one ultimately cares about how precisely the frequency peak can be resolved and tracked. A long-window Fourier analysis is therefore closer to the sensing task than a purely visual time-domain inspection.

== Results ==

=== Result display: steady oscillation ===
[[File:PC5271inductive1.png|750px|thumb|center|Breadboard implementation and representative steady oscillation waveform.]]

This figure shows our circuit implemented on a breadboard and a representative stabilised voltage waveform after the resonators are activated by the supplied voltage. The displayed waveform is the evidence used to identify a stable oscillation frequency.

=== Result display: <math>f-L_A</math> sweep ===
[[File:PC5271inductive3.jpeg|750px|thumb|center|Measured breadboard <math>f-L_A</math> curve.]]

This figure shows the measured oscillation frequency <math>f</math> as a function of <math>L_A</math>. The values of <math>L_A</math> in the main sweep are produced by connecting small inductors in series with approximately <math>2~\mu\mathrm{H}</math> step size.<ref name="Bai2024SM" />

=== Result display: iron-rod demonstration ===
[[File:PC5271inductive2.png|750px|thumb|center|Displacement measurement by compressing the solenoid and inserting iron rod into the homebrew coil]]

This figure should show the homebrew coil with an inserted iron rod and the corresponding change in the measured oscillation frequency. The experimental variable is the rod position or insertion state, and the measured output is the activated steady oscillation frequency.

== Evaluation ==

=== Stable activated oscillation ===
The first required experimental signature is that, after the resonators are activated by the supplied voltage, the circuit reaches a stable oscillating waveform instead of decaying immediately or being amplified without bound. A stable waveform means that the TL071 active branch supplies enough small-signal negative resistance to compensate loss, while gain saturation prevents unbounded amplification.

Physically, this behavior confirms the role of the nonlinear active branch. At small amplitude, the effective negative resistance in resonator <math>A</math> overcomes dissipation. At larger amplitude, the gain becomes weaker, so the oscillation amplitude is clamped. The system self-stabilizes because the physical steady branch is the one with zero imaginary part of the eigenfrequency. The auxiliary branches near the nonlinear exceptional point are not stable physical outputs; they are mathematical branches of the nonlinear steady-state equation. The experiment is therefore not a passive ring-down measurement; it is a nonlinear steady-state selection process.

=== Evaluation of the <math>f-L_A</math> trend ===
The main experimental comparison is the measured <math>f-L_A</math> trend against Fig. S11(a) of SM-4. The reference experiment reports the stable frequency branch of the nonlinear circuit as <math>L_A</math> is tuned.<ref name="Bai2024SM" /> This branch is the physical one because its eigenfrequency is real in the steady state. In our breadboard experiment, the same observable is measured using the series-inductor sweep.

It is useful to separate two statements. First, any LC resonator changes frequency when <math>L</math> changes, because <math>f_0=1/(2\pi\sqrt{LC})</math>. This alone would not demonstrate the nonlinear exceptional-point mechanism. Second, in the coupled active-lossy circuit, the frequency is not just the bare LC resonance frequency of one isolated inductor and capacitor. It is the frequency selected by the nonlinear steady-state balance of gain, loss, and coupling. The reason Fig. S11(a) matters is that it shows this selected stable branch as <math>L_A</math> is swept.

Our breadboard circuit uses different component values and a different op-amp from the PCB experiment, so exact numerical overlap is not expected. The meaningful evaluation is whether the measured frequency branch follows the same qualitative <math>f-L_A</math> behavior as the stable branch in SM-4. Observing that behavior is enough to show that the constructed circuit realizes the intended nonlinear voltage-activated oscillating mechanism.

=== Evaluation of the displacement demonstration ===
In the iron-rod demonstration, the homebrew coil becomes a displacement-dependent inductive element. As the rod enters the coil, the effective magnetic permeability inside the coil increases. In the solenoid expression

<math display="block">
L(x) \approx \mu_\mathrm{eff}(x)\frac{N^2 A}{\ell},
</math>

the insertion depth <math>x</math> changes <math>\mu_\mathrm{eff}</math>. The coil geometry <math>N</math>, <math>A</math>, and <math>\ell</math> are approximately fixed during this demonstration, so the dominant effect is the change in magnetic core material inside the coil. Inductive displacement sensors commonly use this principle: the position of a ferromagnetic object changes a coil inductance, and the electronics convert that inductance change into an electrical signal.<ref name="Chen2014" />

In our circuit, the electrical signal is the activated steady oscillation frequency rather than a bridge voltage. The sensing chain is

<math display="block">
\mathrm{rod\ displacement}\ x
\longrightarrow
\mathrm{effective\ inductance}\ L_A(x)
\longrightarrow
\mathrm{oscillation\ frequency}\ f(x).
</math>

This demonstration shows how the inductive readout can be turned into a displacement-sensing readout after calibration of the relation between rod position and frequency.

=== Cubic-root perturbation law and sensing meaning ===
Near the singular operating point, the eigenfrequency shift follows the hallmark cubic-root law,

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3}.
</math>

This law is the main sensing-related result of the reference experiment.<ref name="Bai2024" /><ref name="Bai2024SM" /> Compared with the linear response of an ordinary isolated resonator, the cubic-root dependence can produce a larger frequency displacement for the same small perturbation of inductance when the device is biased sufficiently close to <math>L_{A,\mathrm{NEP}}</math>.

A useful way to interpret the result is through the local derivative,

<math display="block">
\left|\frac{df}{dL_A}\right| \propto |\delta L_A|^{-2/3}.
</math>

This increasing local responsivity is why the circuit is attractive for sensing. Our experiment supports this sensing logic by showing that the breadboard circuit has a measurable stable frequency branch as <math>L_A</math> is changed. A full quantitative exponent extraction would require identifying <math>L_{A,\mathrm{NEP}}</math> and <math>f_{\mathrm{NEP}}</math>, then plotting <math>|f-f_{\mathrm{NEP}}|</math> against <math>|L_A-L_{A,\mathrm{NEP}}|</math> on logarithmic axes.

=== Basis completeness and noise robustness ===
The supplementary material evaluates the Petermann factor as a diagnostic of modal nonorthogonality and basis completeness. For a non-Hermitian eigenproblem, one may write

<math display="block">
\mathrm{PF}_n =
\frac{\langle \Phi_n^{L}|\Phi_n^{L}\rangle \langle \Phi_n^{R}|\Phi_n^{R}\rangle}
{\left|\langle \Phi_n^{L}|\Phi_n^{R}\rangle\right|^2},
</math>

where <math>|\Phi_n^{R}\rangle</math> and <math>|\Phi_n^{L}\rangle</math> are the right and left eigenvectors of the effective steady-state Hamiltonian. At a conventional linear EP, this quantity is associated with diverging excess-noise sensitivity. In Ref. [1], the cited analysis finds that the effective steady-state Hamiltonian remains non-defective and the Petermann factor is finite at the <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM" />

This point explains the significance of the reference experiment. The system does not merely realize a large mathematical responsivity; it realizes that responsivity in a way that avoids the most severe basis-collapse problem of a linear EP. Our breadboard experiment does not directly test basis completeness, but the result provides the theoretical reason why reproducing the stable <math>f-L_A</math> branch is physically meaningful.

=== Overall interpretation ===
In this platform, the measurand is encoded as a perturbation of <math>L_A</math>. Any physical mechanism that modifies the effective inductance of resonator <math>A</math>, for example displacement, magnetic coupling, or the presence of a nearby magnetic target, moves the system away from the operating point and shifts the oscillation frequency. The primary readout in our implementation is therefore spectral: we track <math>f</math> as <math>L_A</math> is changed.

Taken together, Ref. [1] establishes the complete sensing logic: a nonlinear active-lossy circuit is tuned to an <math>\mathrm{NEP}_3</math>; after voltage activation, the system settles onto a stable oscillating branch; the dominant spectral peak is tracked as the perturbation readout; and the enhanced cubic-root response is obtained without the defective-basis pathology of a conventional linear EP. Our breadboard result demonstrates the experimentally accessible core of this logic by reproducing the stable frequency response corresponding to Fig. S11(a) of SM-4.

== References ==
<references />

Inductive Sensors of Ultra-high Sensitivity Based on Nonlinear Exceptional Point

2026-04-23T06:20:37Z

Siyu: /* Result display: steady oscillation */

== Introduction ==
This project is a implementation of the experiment reported in ''Observation of Nonlinear Exceptional Points with a Complete Basis in Dynamics'', adapted to our available laboratory equipment.<ref name="Bai2024">K. Bai, T.-R. Liu, L. Fang, J.-Z. Li, C. Lin, D. Wan, and M. Xiao, Observation of nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''132''', 073802 (2024).</ref> The circuit contains two coupled resonators. One resonator has gain, meaning that it can supply energy to the oscillation, while the other has loss, meaning that it dissipates energy. A system without energy conservation is called a '''non-Hermitian''' system. Such systems can show unusual degeneracies called '''exceptional points''' (EPs), where both the eigenfrequencies and the eigenmodes merge.

Exceptional points are interesting for sensing because the measured frequency can respond very strongly to a small perturbation. In an inductive sensor, the perturbation is a change in inductance. If a small change in inductance produces a large change in oscillation frequency, the circuit can in principle detect small physical changes. However, ordinary linear EPs also suffer from strong eigenmode nonorthogonality and a defective basis, which can increase noise sensitivity and complicate practical sensing.<ref name="Bai2024" />

The system studied in this project belongs to the recently developed class of '''nonlinear exceptional points''' (NEPs). The physical platform is a pair of coupled circuit resonators. Resonator <math>A</math> is made active by a saturable negative-resistance branch, while resonator <math>B</math> is intentionally lossy. Because the gain in resonator <math>A</math> depends on the oscillation amplitude, the steady-state problem is nonlinear and admits multiple self-consistent branches. At the singular operating point reported in Ref. [1], one stable steady branch and two auxiliary steady branches merge into a third-order nonlinear exceptional point, denoted <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM">Supplemental Material for K. Bai, T.-R. Liu, L. Fang, J.-Z. Li, C. Lin, D. Wan, and M. Xiao, Observation of nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''132''', 073802 (2024).</ref>

The experimentally important consequence is that the oscillation eigenfrequency obeys a cubic-root perturbation law near the singularity,

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3},
</math>

where <math>\delta L_A = L_A-L_{A,\mathrm{NEP}}</math> is the inductive detuning from the exceptional operating point. This response is steeper than the ordinary linear dependence of a conventional resonator and is therefore useful for ultra-sensitive inductive sensing.<ref name="Bai2024" /><ref name="Bai2024SM" />

A second and equally important point is that the cited work does not interpret the singularity as a conventional defective linear EP. Instead, the effective steady-state Hamiltonian remains non-defective in dynamics, and the associated Petermann factor stays finite at the operating point. In other words, the experiment realizes enhanced spectral responsivity without inheriting the strongest basis-collapse pathology of a standard linear EP.<ref name="Bai2024" /><ref name="Bai2024SM" />

In this project, our goal is to reproduce the experimental logic of Ref. [1] and its supplementary material under course-lab conditions. Unlike the PCB implementation in the paper, our circuit is assembled on a breadboard and the active branch uses a '''TL071''' operational amplifier. Our main observable is the steady oscillation frequency <math>f</math> as a function of the chosen inductance <math>L_A</math>. We tune <math>L_A</math> in two ways. For the main <math>f-L_A</math> sweep, we join small inductors in series so that the discrete inductance step is about <math>2~\mu\mathrm{H}</math>. In addition, we use a homebrew compressible coil to demonstrate the sensing mechanism directly. Compressing the coil changes its length and turn density while keeping the cross-sectional area approximately unchanged. We therefore use the <math>f-L_A</math> behavior in Fig. S11(a) of the supplementary material as the main experimental benchmark. Reproducing this trend is sufficient for demonstrating that our setup captures the intended nonlinear voltage-activated, self-stabilized oscillating response, although it is not a full reproduction of every diagnostic reported in Ref. [1].<ref name="Bai2024" /><ref name="Bai2024SM" />

== Team Members ==
Wang Peikun — E1538091@u.nus.edu 
Zhu Ziyang — E1583446@u.nus.edu 
Yuan Si-Yu — siyu_yuan@u.nus.edu 
Li Xunyu — xunyu@u.nus.edu

== Background Knowledge ==

=== Ordinary LC and RLC resonators ===
The basic building block of the experiment is an electrical resonator. An ideal LC resonator contains an inductor <math>L</math> and a capacitor <math>C</math>. Energy repeatedly moves between the magnetic field of the inductor and the electric field of the capacitor. The natural angular frequency is

<math display="block">
\omega_0=\frac{1}{\sqrt{LC}},
</math>

or, in ordinary frequency units,

<math display="block">
f_0=\frac{1}{2\pi\sqrt{LC}}.
</math>

This formula is important for the whole project because it explains why changing an inductance changes the measured frequency. A larger <math>L</math> gives a lower resonance frequency, while a smaller <math>L</math> gives a higher resonance frequency.

Real circuits are not ideal. Wires, inductors, capacitors, breadboard contacts, and measuring probes all dissipate energy. This loss is often represented by a resistor <math>R</math>, giving an RLC resonator. In a passive RLC circuit, the oscillation amplitude decays with time because energy is continuously lost as heat. If a circuit is to maintain a steady activated oscillation, the loss must be compensated by a gain element.

=== Gain, loss, and negative resistance ===
In electronics, gain can be represented as an effective negative resistance. A normal positive resistance removes energy from the resonator. A negative resistance does the opposite: it supplies energy to the resonator. In this experiment, resonator <math>A</math> contains an active op-amp branch that behaves like a negative resistance at small amplitude, while resonator <math>B</math> contains ordinary positive loss.

The gain cannot remain negative and constant forever. If it did, any activated oscillation would be amplified without bound until the op-amp clipped or the circuit failed. The important nonlinear ingredient is '''saturable gain''': the active branch gives strong gain at small amplitude but weaker gain at larger amplitude. The stable oscillation amplitude is selected when the supplied gain balances the total loss. This nonlinear steady state is why the experiment is a nonlinear dynamical problem rather than only a passive resonance measurement.

=== What "non-Hermitian" means in this circuit ===
In ordinary conservative physics, a Hermitian Hamiltonian describes a closed system whose total energy is conserved. Its eigenfrequencies are real, and its eigenmodes form a complete orthogonal basis. This is the familiar setting of many quantum-mechanics courses.

A circuit with gain and loss is an open system. Energy can enter through the op-amp branch and leave through resistive loss. The effective dynamical matrix is therefore generally '''non-Hermitian'''. In a non-Hermitian system, eigenfrequencies can be complex:

<math display="block">
\omega=\omega_\mathrm{r}+i\omega_\mathrm{i}.
</math>

The real part <math>\omega_\mathrm{r}</math> gives the oscillation frequency, while the imaginary part <math>\omega_\mathrm{i}</math> describes amplification or decay. A positive imaginary part means the mode is amplified, a negative imaginary part means it decays, and a stable steady oscillation corresponds to zero imaginary part, or equivalently zero net amplification. This is the sense in which the circuit is self-stabilized after voltage activation.

=== Exceptional points ===
In a usual Hermitian system, two modes can have the same eigenfrequency while still remaining two independent modes. This is an ordinary degeneracy. A non-Hermitian system can show a stronger kind of degeneracy called an '''exceptional point''' (EP). At an EP, not only the eigenvalues but also the eigenvectors coalesce. In simple language, two modes collapse into one mode.

This collapse makes EPs interesting for sensing. Near a second-order linear EP, a small perturbation can produce a square-root frequency splitting:

<math display="block">
\Delta f \propto \sqrt{\epsilon},
</math>

where <math>\epsilon</math> is the perturbation. This response is steeper than an ordinary linear response when <math>\epsilon</math> is very small. However, a conventional linear EP also has a serious drawback: the eigenbasis becomes defective, meaning that the system no longer has enough independent eigenvectors. This is associated with strong modal nonorthogonality and enhanced noise sensitivity. The Petermann factor is one common way to quantify this nonorthogonality.

=== Why the nonlinear exceptional point is different ===
The reference paper studies a nonlinear exceptional point. The word "nonlinear" is essential. The circuit does not have a fixed gain coefficient; instead, the gain depends on the oscillation amplitude. The steady state must therefore be solved self-consistently: the frequency, voltage amplitudes, and saturated gain must all agree with one another.

In the theory of Ref. [1], this nonlinear steady-state problem can reduce to a cubic equation for the oscillation frequency. A cubic equation can have a triple root. At the special operating point, one stable physical branch and two auxiliary branches merge, giving a third-order nonlinear exceptional point, written as <math>\mathrm{NEP}_3</math>. The stable physical branch is the one with real oscillation frequency, i.e. without an imaginary part. The auxiliary branches are needed to define the higher-order coalescence mathematically, but they are not physically stable operating states of the circuit. The corresponding frequency response has the form

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3}.
</math>

This cubic-root response is the key sensing signature. It is steeper than both an ordinary linear response and the square-root response of a second-order EP. The important conceptual point of Ref. [1] is that this enhanced response is obtained while the effective steady-state Hamiltonian can still retain a complete basis in dynamics.<ref name="Bai2024" /><ref name="Bai2024SM" />

=== Sensing mechanism ===
The circuit can be understood as a frequency-generating sensor. It converts an inductance change into a frequency change.

The first step is activation. A voltage is supplied to the resonators to excite the circuit into an oscillating state. The active branch in resonator <math>A</math> then supplies energy through an effective negative resistance and amplifies the selected oscillation mode.

The second step is self-stabilization. The gain is saturable, which means it becomes weaker when the voltage amplitude becomes larger. The activated oscillation therefore is not amplified forever. The circuit settles onto the physical steady branch where the imaginary part of the eigenfrequency is zero. At that point, the gain supplied by resonator <math>A</math> balances the loss in the circuit, especially the loss in resonator <math>B</math>.

The third step is frequency selection. Because resonators <math>A</math> and <math>B</math> are coupled, the final oscillation frequency is not simply the resonance frequency of either isolated resonator. It is selected by the whole coupled circuit: the inductors, capacitors, coupling elements, loss, and saturated gain all contribute.

The fourth step is sensing. The inductance <math>L_A</math> controls the bare frequency of resonator <math>A</math>. When <math>L_A</math> changes, the self-consistent frequency selected by the nonlinear circuit also changes. Therefore, tracking the spectral peak <math>f</math> tells us how the effective inductance has changed.

The fifth step is the nonlinear exceptional-point response. Near the nonlinear exceptional point discussed in Ref. [1], the physical steady branch and the two auxiliary branches coalesce in the nonlinear steady-state equation. The frequency shift of the stable branch can then scale as the cube root of the inductance perturbation. This means that a small inductance change can produce a comparatively large frequency change. Our breadboard experiment focuses on verifying the stable <math>f-L_A</math> branch corresponding to Fig. S11(a) of SM-4, which is the experimentally accessible part of this sensing mechanism.

=== Same mechanism in other platforms ===
Several references are included to show that this is a platform-independent sensing mechanism, not only a feature of our breadboard circuit. The common chain is

<math display="block">
\mathrm{physical\ perturbation}
\longrightarrow
\mathrm{parameter\ change}
\longrightarrow
\mathrm{eigenfrequency\ shift\ of\ a\ resonator/cavity\ with\ gain\ and\ loss}
\longrightarrow
\mathrm{higher\ order\ steady\ state\ response}
\longrightarrow
\mathrm{sensing}.
</math>

In our circuit, the physical perturbation is a change in the inductance <math>L_A</math>, and the measured output is the oscillation frequency <math>f</math>. In coupled optical, mechanical, electrical, or hybrid resonators, the perturbed parameter may instead be a cavity detuning, mechanical frequency, phase shift, magnetic-field-dependent transition, or another effective resonator parameter. The shared point is that gain, loss, coupling, and nonlinearity convert the parameter change into a higher-order steady-state spectral response.

The earlier theoretical work on nonlinear exceptional points showed that nonlinear coupled resonators can host higher-order NEPs while retaining a complete dynamical basis.<ref name="Bai2023">K. Bai, J.-Z. Li, T.-R. Liu, L. Fang, D. Wan, and M. Xiao, Nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''130''', 266901 (2023).</ref> The 2024 experiment implemented the same mechanism in coupled electrical resonators and measured the stable nonlinear frequency branch.<ref name="Bai2024" /><ref name="Bai2024SM" /> Silicon micromechanical resonators provide another realization: there, the measured output can be a phase difference, and the nonlinear exceptional-point response can enhance phase-shift sensing.<ref name="Xie2026">Y.-J. Xie, S.-Y. Zhang, M.-N. Zhang, R. Wang, L.-F. Wang, and Q.-A. Huang, Nonlinear exceptional points boost phase-shift sensing in silicon micromechanical resonators, ''Phys. Rev. Applied'' '''25''', 034039 (2026).</ref> Hybrid quantum systems with nonlinear bistability show a related route, where nonlinear transition dynamics can enhance sensing signal-to-noise ratio.<ref name="Wang2026">H. Wang ''et al.'', Exceptional sensitivity near the bistable transition point of a hybrid quantum system, ''Nature Physics'' '''22''', 577-583 (2026).</ref>

These examples support the same mechanism rather than unrelated applications. At the same time, enhanced response alone is not enough to prove better sensing. Noise can shift or weaken the apparent nonlinear exceptional point and can limit the signal-to-noise improvement.<ref name="Zheng2025">X. Zheng and Y. D. Chong, Noise constraints for nonlinear exceptional point sensing, ''Phys. Rev. Lett.'' '''134''', 133801 (2025).</ref> This is why our project makes a careful claim: we demonstrate the frequency-response mechanism on a breadboard, but we do not claim a complete metrological noise advantage.

== Report objectives and scope ==
The purpose of this work is to study the circuit principle behind '''nonlinear-exceptional-point-enhanced inductive sensing''' in a coupled electronic resonator platform. The scope is intentionally narrower than the full experiment reported in Ref. [1]. We are not attempting to redesign the active branch, derive all nonlinear bifurcation theory from first principles, reconstruct the auxiliary steady-state branches, or characterize every parasitic element. Instead, we adopt the framework of Ref. [1] and SM-4 and ask whether the breadboard circuit can reproduce the experimentally accessible frequency response.

The main target of our experiment is the <math>f-L_A</math> curve corresponding to Fig. S11(a) of SM-4. In the reference experiment, the steady-state frequency, voltage ratio <math>|V_A/V_B|</math>, and relative phase <math>\theta_A-\theta_B</math> are all measured as functions of <math>L_A</math>.<ref name="Bai2024SM" /> In our implementation, the frequency-versus-inductance curve is the primary observable. We tune <math>L_A</math> by connecting small inductors in series with approximately <math>2~\mu\mathrm{H}</math> resolution. We also use a homebrew coil in a separate demonstration where mechanical compression or insertion of an iron rod changes the effective inductance. The experimental goal is therefore to verify that the TL071 breadboard circuit settles into a stable oscillating state and that the measured frequency changes with <math>L_A</math> in the same qualitative manner as Fig. S11(a).

From the sensing perspective, the most relevant figure of merit is the local frequency responsivity. If

<math display="block">
|f-f_{\mathrm{NEP}}| = A |\delta L_A|^{1/3},
</math>

then differentiation gives

<math display="block">
S_f \equiv \left|\frac{df}{dL_A}\right| \propto |\delta L_A|^{-2/3}.
</math>

Thus, as the operating point approaches <math>L_{A,\mathrm{NEP}}</math>, the local spectral responsivity increases rapidly. The main practical constraint is that the system must remain on the intended stable branch while measurement noise and slow parameter drift are kept below the scale of the induced frequency shift.

== Setups ==
[[File:Experimental_setup.png|750px|thumb|center|Experimental setup adapted from Ref. [1].]]

=== Circuit topology and component implementation ===
The reference platform consists of two coupled RLC resonators. Both resonators contain the same nominal shunt capacitance <math>C_0</math>, while the inductances <math>L_A</math> and <math>L_B</math> set their bare resonance frequencies. The two resonators are coupled through a parallel resistor-capacitor channel characterized by <math>R_c</math> and <math>C_c</math>. Resonator <math>B</math> contains an explicit positive resistance <math>R_B</math> and therefore plays the role of the lossy resonator. Resonator <math>A</math>, by contrast, contains an active branch whose effective resistance is negative at small amplitude and becomes less negative as the oscillation amplitude increases, thereby providing saturable gain.<ref name="Bai2024" /><ref name="Bai2024SM" />

Our implementation follows this topology but is built on a breadboard rather than a custom PCB. The active branch is implemented with an op-amp negative-resistance circuit using a '''TL071''' operational amplifier. This is an important practical difference from the supplementary experiment, where the active components explicitly listed are a TI '''LM7171''' operational amplifier and an '''Onsemi BAV99L''' diode.<ref name="Bai2024SM" /> The TL071 substitution and the breadboard layout introduce different bandwidth limits, wiring parasitics, and contact resistances, so the numerical operating point is not expected to match the PCB experiment. For this reason, our comparison emphasizes the observable trend of <math>f</math> versus <math>L_A</math>.

The supplementary material reports the following representative component values for the PCB experiment:

<math display="block">
C_0 = 18.5~\mathrm{nF}, \qquad
C_c = 3.9~\mathrm{nF}, \qquad
R_c = 760~\Omega,
</math>

<math display="block">
L_B = 197.7~\mu\mathrm{H}, \qquad
R_B = 1314~\Omega.
</math>

A representative operating point shown in the main figure uses

<math display="block">
L_A = 245.4~\mu\mathrm{H}.
</math>

Our breadboard circuit follows the same functional topology, but its component values and parasitic elements are not identical to the PCB values above. The horizontal axis of our main <math>f-L_A</math> plot is therefore the series-combined <math>L_A</math> selected from our available small inductors, while the measured frequency <math>f</math> is the robust experimental readout.

=== Inductance tuning methods ===
We use two complementary methods to change the inductance seen by resonator <math>A</math>.

The first method is used for the main <math>f-L_A</math> sweep. Small inductors are connected in series so that the total inductance changes in discrete steps of approximately <math>2~\mu\mathrm{H}</math>. If the inductors are sufficiently separated so that their mutual magnetic coupling is small, the total series inductance is approximately the sum of the individual inductances,

<math display="block">
L_A \approx L_{\mathrm{base}}+\sum_i L_i .
</math>

This method is simple and reproducible. Adding one small inductor increases <math>L_A</math>; removing it decreases <math>L_A</math>. The circuit frequency is then measured for each chosen value. This produces a discrete version of the <math>f-L_A</math> curve in Fig. S11(a) of SM-4. The step size is not as fine as a commercial variable inductor, but <math>2~\mu\mathrm{H}</math> spacing is small enough to observe the frequency trend over the accessible range.

The second method is a homebrew compressible variable inductor. This coil is useful because it makes the sensing mechanism visible: a mechanical displacement changes the coil geometry, which changes the inductance, which then changes the oscillation frequency. The basic idea can be understood from the approximate solenoid formula

<math display="block">
L \approx \mu_\mathrm{eff}\frac{N^2 A}{\ell},
</math>

where <math>N</math> is the number of turns, <math>A</math> is the cross-sectional area, <math>\ell</math> is the coil length, and <math>\mu_\mathrm{eff}</math> describes the effective magnetic permeability of the core and surrounding region.<ref name="Chen2014">S.-C. Chen, D.-K. Le, and V.-S. Nguyen, Inductive displacement sensors with a notch filter for an active magnetic bearing system, ''Sensors'' '''14''', 12640-12657 (2014).</ref> During compression, the number of turns <math>N</math> is unchanged and the cross-sectional area <math>A</math> remains approximately unchanged, but the length <math>\ell</math> decreases. Equivalently, the turn density

<math display="block">
n=\frac{N}{\ell}
</math>

increases. The magnetic field produced by a solenoid is proportional to turn density. Therefore, compressing the coil increases the magnetic flux linkage through the turns and increases the inductance. Stretching the coil has the opposite effect.

This mechanical tuning is not identical to the precision tunable inductor used in the reference experiment. A real compressed coil is not an ideal long solenoid: the spacing between turns is finite, the end effects change as the coil length changes, nearby wires and the breadboard add parasitic capacitance, and the leads contribute additional series inductance and resistance. For this reason, the homebrew inductor is best treated as an experimentally characterized tuning element rather than an ideal component whose value is known exactly from geometry. In the present project, it provides a direct mechanical demonstration of how displacement can perturb <math>L_A</math>.

Experimentally, we measured the effective inductance of this homebrew variable inductor at different compression states. The compression process changes the coil length and wire density while leaving the coil area approximately fixed. Therefore, the tuning mechanism is physically the same as changing <math>\ell</math> and <math>n=N/\ell</math> in the solenoid model, rather than changing the number of turns or the coil diameter.

=== Kirchhoff model of the nonlinear circuit ===
A convenient state-space description uses the node voltages <math>V_A(t)</math> and <math>V_B(t)</math> together with the inductor currents <math>I_A(t)</math> and <math>I_B(t)</math>. A node voltage is the voltage measured at one resonator node relative to ground. Kirchhoff's current law says that the algebraic sum of all currents flowing away from a node must be zero. This gives a direct way to translate the circuit diagram into differential equations.

Denoting the amplitude-dependent conductance of the active branch by <math>G_A(|V_A|)</math>, where <math>G_A<0</math> in the small-signal regime, the nodal equations can be written as

<math display="block">
\begin{aligned}
C_0 \frac{dV_A}{dt} + I_A + G_A(|V_A|)\,V_A + \frac{V_A-V_B}{R_c}
+ C_c\frac{d(V_A-V_B)}{dt} &= 0,\\
C_0 \frac{dV_B}{dt} + I_B + \frac{V_B}{R_B} + \frac{V_B-V_A}{R_c}
+ C_c\frac{d(V_B-V_A)}{dt} &= 0,\\
L_A \frac{dI_A}{dt} &= V_A,\\
L_B \frac{dI_B}{dt} &= V_B.
\end{aligned}
</math>

Each term has a simple circuit meaning. The terms <math>C_0 dV_A/dt</math> and <math>C_0 dV_B/dt</math> are capacitor currents. The terms <math>I_A</math> and <math>I_B</math> are currents through the inductors. The term <math>G_A(|V_A|)V_A</math> is the active branch current; because <math>G_A</math> can be negative, this branch can inject energy instead of dissipating it. The term <math>V_B/R_B</math> is the ordinary loss current through the resistor in resonator <math>B</math>. The terms involving <math>R_c</math> and <math>C_c</math> describe current flowing between resonators <math>A</math> and <math>B</math>, so they are the electrical coupling between the two resonators.

These equations show explicitly how the experiment differs from a linear EP circuit. If <math>G_A</math> were a constant negative conductance, the model would reduce to a linear non-Hermitian dimer. Here, however, <math>G_A</math> depends on the voltage amplitude, so the steady state must be determined self-consistently together with the oscillation amplitude itself.

=== Harmonic-balance formulation of the steady state ===
After the transient has died away, the oscilloscope waveform is approximately periodic. The simplest approximation is therefore to keep only the dominant sinusoidal component. This is called a harmonic-balance approximation. For a stable activated oscillating state, one writes

<math display="block">
V_j(t)=\mathrm{Re}\!\left[\tilde V_j e^{i\omega t}\right],
\qquad j=A,B.
</math>

Substituting this ansatz into the circuit equations yields a nonlinear algebraic problem of the form

<math display="block">
\mathbf{Y}(\omega,|\tilde V_A|)
\begin{pmatrix}
\tilde V_A\\
\tilde V_B
\end{pmatrix}
=
\begin{pmatrix}
Y_A(\omega,|\tilde V_A|)+Y_c(\omega) & -Y_c(\omega)\\
-Y_c(\omega) & Y_B(\omega)+Y_c(\omega)
\end{pmatrix}
\begin{pmatrix}
\tilde V_A\\
\tilde V_B
\end{pmatrix}
=0,
</math>

with

<math display="block">
Y_A(\omega,|\tilde V_A|)=G_A(|\tilde V_A|)+i\omega C_0+\frac{1}{i\omega L_A},
</math>

<math display="block">
Y_B(\omega)=\frac{1}{R_B}+i\omega C_0+\frac{1}{i\omega L_B},
\qquad
Y_c(\omega)=\frac{1}{R_c}+i\omega C_c.
</math>

A nontrivial solution requires

<math display="block">
\det \mathbf{Y}(\omega,|\tilde V_A|)=0.
</math>

This determinant condition is the circuit analogue of an eigenfrequency condition. In a passive LC resonator, the resonance condition gives a single frequency <math>\omega_0=1/\sqrt{LC}</math>. In the present two-resonator active circuit, the same idea becomes a matrix equation because the two resonators are coupled. The amplitudes <math>\tilde V_A</math> and <math>\tilde V_B</math> are not arbitrary: a nonzero oscillation exists only when the admittance matrix has a zero eigenvalue.

Because the active admittance depends on amplitude, this condition is nonlinear and supports multiple branches. The supplementary material shows that the steady-state frequency can be reduced to a '''real-coefficient cubic polynomial''' in <math>\omega</math>. The existence of up to three real steady-state branches is therefore not accidental; it is the algebraic origin of the experimentally relevant <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM" />

=== Effective nonlinear-Hamiltonian picture ===
The same physics is often rewritten in coupled-mode form. This notation is common in non-Hermitian physics because it makes the circuit look like two coupled modes. The variables <math>\psi_A</math> and <math>\psi_B</math> represent the complex oscillation amplitudes in resonators <math>A</math> and <math>B</math>. The model is

<math display="block">
i\frac{d}{dt}
\begin{pmatrix}
\psi_A\\
\psi_B
\end{pmatrix}
=
\begin{pmatrix}
\omega_A+i\,g(|\psi_A|) & \kappa\\
\kappa & \omega_B-i\ell
\end{pmatrix}
\begin{pmatrix}
\psi_A\\
\psi_B
\end{pmatrix},
</math>

where <math>\omega_A</math> and <math>\omega_B</math> are the effective modal frequencies, <math>\kappa</math> is the coupling strength, <math>\ell</math> is the loss of resonator <math>B</math>, and <math>g(|\psi_A|)</math> is the saturable gain of resonator <math>A</math>. The steady-state problem then becomes

<math display="block">
H_s(\Psi)\Psi = \Omega \Psi,
</math>

with a real oscillation eigenfrequency <math>\Omega=2\pi f</math> for the stable oscillating branch. The word "real" is important: it means the imaginary part has vanished, so the mode is neither being amplified nor decaying in the steady state. In the cited experiment, this physical stable branch and two auxiliary steady branches coalesce at the singular point, producing a third-order nonlinear exceptional point rather than a conventional second-order linear EP. The auxiliary branches help form the cubic-root singularity but are not themselves stable physical outputs of the circuit.<ref name="Bai2024" /><ref name="Bai2024SM" />

== Measurements ==

=== Acquisition protocol and steady-state selection ===
The measurement is performed in the time domain. In the reference experiment, an arbitrary waveform generator first applies an external start-up signal of 1 V at 70 kHz in order to bring the system into the basin of attraction of the stable oscillating state.<ref name="Bai2024" /> In our breadboard measurement, an external voltage is likewise required to activate the resonators. The measured frequency is extracted from the activated steady waveform recorded on the oscilloscope.

The physical meaning of this procedure is important. The oscillation is not assumed to appear without activation; a supplied voltage is needed to bring the resonators into the relevant dynamical state. The measured oscillation frequency is then interpreted as the frequency selected by the nonlinear steady-state condition.

=== Extraction of the <math>f-L_A</math> observable ===
Ref. [1] does not treat the oscilloscope traces merely as qualitative evidence of oscillation. The supplementary explicitly states that the full reference experiment reports the steady-state frequency, the voltage ratio <math>|V_A/V_B|</math>, and the relative phase <math>\theta_A-\theta_B</math>, all extracted from the dominant peaks of the Fourier spectra of <math>V_A</math> and <math>V_B</math>.<ref name="Bai2024SM" />

Our available observable is the steady-state frequency. For each chosen value of <math>L_A</math>, the resonators are voltage-activated, the circuit reaches a stable oscillating state, and the dominant spectral peak is recorded. If we denote the Fourier transforms by

<math display="block">
\tilde V_A(f)=\mathcal{F}\{V_A(t)\},
\qquad
\tilde V_B(f)=\mathcal{F}\{V_B(t)\},
</math>

then the dominant oscillation frequency is obtained from

<math display="block">
f_0=\arg\max_f |\tilde V_A(f)| \approx \arg\max_f |\tilde V_B(f)|.
</math>

The experimental data set is therefore

<math display="block">
\{L_A,\ f_0(L_A)\},
</math>

which is compared with the frequency branch plotted in Fig. S11(a) of SM-4. In the full reference measurement, the amplitude ratio

<math display="block">
\rho = \frac{|\tilde V_A(f_0)|}{|\tilde V_B(f_0)|}
</math>

and relative phase

<math display="block">
\Delta\phi = \arg \tilde V_A(f_0)-\arg \tilde V_B(f_0)
</math>

provide additional state verification. Those quantities are useful but are not the main basis of our breadboard validation.

In practice, the same frequency can also be estimated directly from the time-domain period <math>T</math> using <math>f_0=1/T</math> when the waveform is clean. Fourier analysis is preferred because it uses a longer section of waveform and identifies the dominant oscillation peak even when there is small amplitude noise or waveform distortion.

=== Practical inductance sweep ===
In SM-4, the inductive tuning is carried out by varying <math>L_A</math>. The supplementary notes that the tunable inductor exhibits less than about 0.4% inductance variation over the frequency range of interest, which is sufficiently small for controlled sweeping near the singular point. To further suppress voltage-dependent drift during the sweep, the reference experiment slightly adjusts the active-branch resistance so as to maintain approximately <math>V_B \approx 2~\mathrm{V}</math> throughout the measurement series.<ref name="Bai2024SM" />

Our sweep uses controlled series combinations of small inductors, giving an approximately <math>2~\mu\mathrm{H}</math> step in <math>L_A</math>. The homebrew compressible coil is then used to show that a mechanical change of the sensing element also changes the inductance. The conclusion drawn from our data is therefore not a precision determination of <math>L_{A,\mathrm{NEP}}</math>, but a validation that the breadboard circuit produces the same kind of frequency response as the stable branch in Fig. S11(a).

=== Long-window spectral analysis and noise inspection ===
The supplementary also discusses long-time measurements over a window of <math>0<t<5~\mathrm{s}</math>. The motivation is that the noise around the stable oscillation is too small to be judged reliably from raw scope traces alone, so it is more informative to inspect the Fourier spectra and the extracted spectral center frequency and linewidth.<ref name="Bai2024SM" />

This is a useful methodological point for a sensing-oriented reproduction. In an NEP-based sensor, one ultimately cares about how precisely the frequency peak can be resolved and tracked. A long-window Fourier analysis is therefore closer to the sensing task than a purely visual time-domain inspection.

== Results ==

=== Result display: steady oscillation ===
[[File:PC5271inductive1.png|750px|thumb|center|Breadboard implementation and representative steady oscillation waveform.]]

This figure shows our circuit implemented on a breadboard and a representative stabilised voltage waveform after the resonators are activated by the supplied voltage. The displayed waveform is the evidence used to identify a stable oscillation frequency.

=== Result display: <math>f-L_A</math> sweep ===
[[File:PC5271inductive3.jpeg|750px|thumb|center|Measured breadboard <math>f-L_A</math> curve.]]

This figure should show the measured oscillation frequency <math>f</math> as a function of <math>L_A</math>. The values of <math>L_A</math> in the main sweep are produced by connecting small inductors in series with approximately <math>2~\mu\mathrm{H}</math> step size. The reference curve for comparison is the stable frequency branch in Fig. S11(a) of SM-4.<ref name="Bai2024SM" />

=== Result display: iron-rod demonstration ===
[[File:PC5271inductive2.png|750px|thumb|center|Displacement measurement by compressing the solenoid and inserting iron rod into the homebrew coil]]

This figure should show the homebrew coil with an inserted iron rod and the corresponding change in the measured oscillation frequency. The experimental variable is the rod position or insertion state, and the measured output is the activated steady oscillation frequency.

== Evaluation ==

=== Stable activated oscillation ===
The first required experimental signature is that, after the resonators are activated by the supplied voltage, the circuit reaches a stable oscillating waveform instead of decaying immediately or being amplified without bound. A stable waveform means that the TL071 active branch supplies enough small-signal negative resistance to compensate loss, while gain saturation prevents unbounded amplification.

Physically, this behavior confirms the role of the nonlinear active branch. At small amplitude, the effective negative resistance in resonator <math>A</math> overcomes dissipation. At larger amplitude, the gain becomes weaker, so the oscillation amplitude is clamped. The system self-stabilizes because the physical steady branch is the one with zero imaginary part of the eigenfrequency. The auxiliary branches near the nonlinear exceptional point are not stable physical outputs; they are mathematical branches of the nonlinear steady-state equation. The experiment is therefore not a passive ring-down measurement; it is a nonlinear steady-state selection process.

=== Evaluation of the <math>f-L_A</math> trend ===
The main experimental comparison is the measured <math>f-L_A</math> trend against Fig. S11(a) of SM-4. The reference experiment reports the stable frequency branch of the nonlinear circuit as <math>L_A</math> is tuned.<ref name="Bai2024SM" /> This branch is the physical one because its eigenfrequency is real in the steady state. In our breadboard experiment, the same observable is measured using the series-inductor sweep.

It is useful to separate two statements. First, any LC resonator changes frequency when <math>L</math> changes, because <math>f_0=1/(2\pi\sqrt{LC})</math>. This alone would not demonstrate the nonlinear exceptional-point mechanism. Second, in the coupled active-lossy circuit, the frequency is not just the bare LC resonance frequency of one isolated inductor and capacitor. It is the frequency selected by the nonlinear steady-state balance of gain, loss, and coupling. The reason Fig. S11(a) matters is that it shows this selected stable branch as <math>L_A</math> is swept.

Our breadboard circuit uses different component values and a different op-amp from the PCB experiment, so exact numerical overlap is not expected. The meaningful evaluation is whether the measured frequency branch follows the same qualitative <math>f-L_A</math> behavior as the stable branch in SM-4. Observing that behavior is enough to show that the constructed circuit realizes the intended nonlinear voltage-activated oscillating mechanism.

=== Evaluation of the displacement demonstration ===
In the iron-rod demonstration, the homebrew coil becomes a displacement-dependent inductive element. As the rod enters the coil, the effective magnetic permeability inside the coil increases. In the solenoid expression

<math display="block">
L(x) \approx \mu_\mathrm{eff}(x)\frac{N^2 A}{\ell},
</math>

the insertion depth <math>x</math> changes <math>\mu_\mathrm{eff}</math>. The coil geometry <math>N</math>, <math>A</math>, and <math>\ell</math> are approximately fixed during this demonstration, so the dominant effect is the change in magnetic core material inside the coil. Inductive displacement sensors commonly use this principle: the position of a ferromagnetic object changes a coil inductance, and the electronics convert that inductance change into an electrical signal.<ref name="Chen2014" />

In our circuit, the electrical signal is the activated steady oscillation frequency rather than a bridge voltage. The sensing chain is

<math display="block">
\mathrm{rod\ displacement}\ x
\longrightarrow
\mathrm{effective\ inductance}\ L_A(x)
\longrightarrow
\mathrm{oscillation\ frequency}\ f(x).
</math>

This demonstration shows how the inductive readout can be turned into a displacement-sensing readout after calibration of the relation between rod position and frequency.

=== Cubic-root perturbation law and sensing meaning ===
Near the singular operating point, the eigenfrequency shift follows the hallmark cubic-root law,

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3}.
</math>

This law is the main sensing-related result of the reference experiment.<ref name="Bai2024" /><ref name="Bai2024SM" /> Compared with the linear response of an ordinary isolated resonator, the cubic-root dependence can produce a larger frequency displacement for the same small perturbation of inductance when the device is biased sufficiently close to <math>L_{A,\mathrm{NEP}}</math>.

A useful way to interpret the result is through the local derivative,

<math display="block">
\left|\frac{df}{dL_A}\right| \propto |\delta L_A|^{-2/3}.
</math>

This increasing local responsivity is why the circuit is attractive for sensing. Our experiment supports this sensing logic by showing that the breadboard circuit has a measurable stable frequency branch as <math>L_A</math> is changed. A full quantitative exponent extraction would require identifying <math>L_{A,\mathrm{NEP}}</math> and <math>f_{\mathrm{NEP}}</math>, then plotting <math>|f-f_{\mathrm{NEP}}|</math> against <math>|L_A-L_{A,\mathrm{NEP}}|</math> on logarithmic axes.

=== Basis completeness and noise robustness ===
The supplementary material evaluates the Petermann factor as a diagnostic of modal nonorthogonality and basis completeness. For a non-Hermitian eigenproblem, one may write

<math display="block">
\mathrm{PF}_n =
\frac{\langle \Phi_n^{L}|\Phi_n^{L}\rangle \langle \Phi_n^{R}|\Phi_n^{R}\rangle}
{\left|\langle \Phi_n^{L}|\Phi_n^{R}\rangle\right|^2},
</math>

where <math>|\Phi_n^{R}\rangle</math> and <math>|\Phi_n^{L}\rangle</math> are the right and left eigenvectors of the effective steady-state Hamiltonian. At a conventional linear EP, this quantity is associated with diverging excess-noise sensitivity. In Ref. [1], the cited analysis finds that the effective steady-state Hamiltonian remains non-defective and the Petermann factor is finite at the <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM" />

This point explains the significance of the reference experiment. The system does not merely realize a large mathematical responsivity; it realizes that responsivity in a way that avoids the most severe basis-collapse problem of a linear EP. Our breadboard experiment does not directly test basis completeness, but the result provides the theoretical reason why reproducing the stable <math>f-L_A</math> branch is physically meaningful.

=== Overall interpretation ===
In this platform, the measurand is encoded as a perturbation of <math>L_A</math>. Any physical mechanism that modifies the effective inductance of resonator <math>A</math>, for example displacement, magnetic coupling, or the presence of a nearby magnetic target, moves the system away from the operating point and shifts the oscillation frequency. The primary readout in our implementation is therefore spectral: we track <math>f</math> as <math>L_A</math> is changed.

Taken together, Ref. [1] establishes the complete sensing logic: a nonlinear active-lossy circuit is tuned to an <math>\mathrm{NEP}_3</math>; after voltage activation, the system settles onto a stable oscillating branch; the dominant spectral peak is tracked as the perturbation readout; and the enhanced cubic-root response is obtained without the defective-basis pathology of a conventional linear EP. Our breadboard result demonstrates the experimentally accessible core of this logic by reproducing the stable frequency response corresponding to Fig. S11(a) of SM-4.

== References ==
<references />

Inductive Sensors of Ultra-high Sensitivity Based on Nonlinear Exceptional Point

2026-04-23T06:10:46Z

Siyu: /* Introduction */

== Introduction ==
This project is a implementation of the experiment reported in ''Observation of Nonlinear Exceptional Points with a Complete Basis in Dynamics'', adapted to our available laboratory equipment.<ref name="Bai2024">K. Bai, T.-R. Liu, L. Fang, J.-Z. Li, C. Lin, D. Wan, and M. Xiao, Observation of nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''132''', 073802 (2024).</ref> The circuit contains two coupled resonators. One resonator has gain, meaning that it can supply energy to the oscillation, while the other has loss, meaning that it dissipates energy. A system without energy conservation is called a '''non-Hermitian''' system. Such systems can show unusual degeneracies called '''exceptional points''' (EPs), where both the eigenfrequencies and the eigenmodes merge.

Exceptional points are interesting for sensing because the measured frequency can respond very strongly to a small perturbation. In an inductive sensor, the perturbation is a change in inductance. If a small change in inductance produces a large change in oscillation frequency, the circuit can in principle detect small physical changes. However, ordinary linear EPs also suffer from strong eigenmode nonorthogonality and a defective basis, which can increase noise sensitivity and complicate practical sensing.<ref name="Bai2024" />

The system studied in this project belongs to the recently developed class of '''nonlinear exceptional points''' (NEPs). The physical platform is a pair of coupled circuit resonators. Resonator <math>A</math> is made active by a saturable negative-resistance branch, while resonator <math>B</math> is intentionally lossy. Because the gain in resonator <math>A</math> depends on the oscillation amplitude, the steady-state problem is nonlinear and admits multiple self-consistent branches. At the singular operating point reported in Ref. [1], one stable steady branch and two auxiliary steady branches merge into a third-order nonlinear exceptional point, denoted <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM">Supplemental Material for K. Bai, T.-R. Liu, L. Fang, J.-Z. Li, C. Lin, D. Wan, and M. Xiao, Observation of nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''132''', 073802 (2024).</ref>

The experimentally important consequence is that the oscillation eigenfrequency obeys a cubic-root perturbation law near the singularity,

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3},
</math>

where <math>\delta L_A = L_A-L_{A,\mathrm{NEP}}</math> is the inductive detuning from the exceptional operating point. This response is steeper than the ordinary linear dependence of a conventional resonator and is therefore useful for ultra-sensitive inductive sensing.<ref name="Bai2024" /><ref name="Bai2024SM" />

A second and equally important point is that the cited work does not interpret the singularity as a conventional defective linear EP. Instead, the effective steady-state Hamiltonian remains non-defective in dynamics, and the associated Petermann factor stays finite at the operating point. In other words, the experiment realizes enhanced spectral responsivity without inheriting the strongest basis-collapse pathology of a standard linear EP.<ref name="Bai2024" /><ref name="Bai2024SM" />

In this project, our goal is to reproduce the experimental logic of Ref. [1] and its supplementary material under course-lab conditions. Unlike the PCB implementation in the paper, our circuit is assembled on a breadboard and the active branch uses a '''TL071''' operational amplifier. Our main observable is the steady oscillation frequency <math>f</math> as a function of the chosen inductance <math>L_A</math>. We tune <math>L_A</math> in two ways. For the main <math>f-L_A</math> sweep, we join small inductors in series so that the discrete inductance step is about <math>2~\mu\mathrm{H}</math>. In addition, we use a homebrew compressible coil to demonstrate the sensing mechanism directly. Compressing the coil changes its length and turn density while keeping the cross-sectional area approximately unchanged. We therefore use the <math>f-L_A</math> behavior in Fig. S11(a) of the supplementary material as the main experimental benchmark. Reproducing this trend is sufficient for demonstrating that our setup captures the intended nonlinear voltage-activated, self-stabilized oscillating response, although it is not a full reproduction of every diagnostic reported in Ref. [1].<ref name="Bai2024" /><ref name="Bai2024SM" />

== Team Members ==
Wang Peikun — E1538091@u.nus.edu 
Zhu Ziyang — E1583446@u.nus.edu 
Yuan Si-Yu — siyu_yuan@u.nus.edu 
Li Xunyu — xunyu@u.nus.edu

== Background Knowledge ==

=== Ordinary LC and RLC resonators ===
The basic building block of the experiment is an electrical resonator. An ideal LC resonator contains an inductor <math>L</math> and a capacitor <math>C</math>. Energy repeatedly moves between the magnetic field of the inductor and the electric field of the capacitor. The natural angular frequency is

<math display="block">
\omega_0=\frac{1}{\sqrt{LC}},
</math>

or, in ordinary frequency units,

<math display="block">
f_0=\frac{1}{2\pi\sqrt{LC}}.
</math>

This formula is important for the whole project because it explains why changing an inductance changes the measured frequency. A larger <math>L</math> gives a lower resonance frequency, while a smaller <math>L</math> gives a higher resonance frequency.

Real circuits are not ideal. Wires, inductors, capacitors, breadboard contacts, and measuring probes all dissipate energy. This loss is often represented by a resistor <math>R</math>, giving an RLC resonator. In a passive RLC circuit, the oscillation amplitude decays with time because energy is continuously lost as heat. If a circuit is to maintain a steady activated oscillation, the loss must be compensated by a gain element.

=== Gain, loss, and negative resistance ===
In electronics, gain can be represented as an effective negative resistance. A normal positive resistance removes energy from the resonator. A negative resistance does the opposite: it supplies energy to the resonator. In this experiment, resonator <math>A</math> contains an active op-amp branch that behaves like a negative resistance at small amplitude, while resonator <math>B</math> contains ordinary positive loss.

The gain cannot remain negative and constant forever. If it did, any activated oscillation would be amplified without bound until the op-amp clipped or the circuit failed. The important nonlinear ingredient is '''saturable gain''': the active branch gives strong gain at small amplitude but weaker gain at larger amplitude. The stable oscillation amplitude is selected when the supplied gain balances the total loss. This nonlinear steady state is why the experiment is a nonlinear dynamical problem rather than only a passive resonance measurement.

=== What "non-Hermitian" means in this circuit ===
In ordinary conservative physics, a Hermitian Hamiltonian describes a closed system whose total energy is conserved. Its eigenfrequencies are real, and its eigenmodes form a complete orthogonal basis. This is the familiar setting of many quantum-mechanics courses.

A circuit with gain and loss is an open system. Energy can enter through the op-amp branch and leave through resistive loss. The effective dynamical matrix is therefore generally '''non-Hermitian'''. In a non-Hermitian system, eigenfrequencies can be complex:

<math display="block">
\omega=\omega_\mathrm{r}+i\omega_\mathrm{i}.
</math>

The real part <math>\omega_\mathrm{r}</math> gives the oscillation frequency, while the imaginary part <math>\omega_\mathrm{i}</math> describes amplification or decay. A positive imaginary part means the mode is amplified, a negative imaginary part means it decays, and a stable steady oscillation corresponds to zero imaginary part, or equivalently zero net amplification. This is the sense in which the circuit is self-stabilized after voltage activation.

=== Exceptional points ===
In a usual Hermitian system, two modes can have the same eigenfrequency while still remaining two independent modes. This is an ordinary degeneracy. A non-Hermitian system can show a stronger kind of degeneracy called an '''exceptional point''' (EP). At an EP, not only the eigenvalues but also the eigenvectors coalesce. In simple language, two modes collapse into one mode.

This collapse makes EPs interesting for sensing. Near a second-order linear EP, a small perturbation can produce a square-root frequency splitting:

<math display="block">
\Delta f \propto \sqrt{\epsilon},
</math>

where <math>\epsilon</math> is the perturbation. This response is steeper than an ordinary linear response when <math>\epsilon</math> is very small. However, a conventional linear EP also has a serious drawback: the eigenbasis becomes defective, meaning that the system no longer has enough independent eigenvectors. This is associated with strong modal nonorthogonality and enhanced noise sensitivity. The Petermann factor is one common way to quantify this nonorthogonality.

=== Why the nonlinear exceptional point is different ===
The reference paper studies a nonlinear exceptional point. The word "nonlinear" is essential. The circuit does not have a fixed gain coefficient; instead, the gain depends on the oscillation amplitude. The steady state must therefore be solved self-consistently: the frequency, voltage amplitudes, and saturated gain must all agree with one another.

In the theory of Ref. [1], this nonlinear steady-state problem can reduce to a cubic equation for the oscillation frequency. A cubic equation can have a triple root. At the special operating point, one stable physical branch and two auxiliary branches merge, giving a third-order nonlinear exceptional point, written as <math>\mathrm{NEP}_3</math>. The stable physical branch is the one with real oscillation frequency, i.e. without an imaginary part. The auxiliary branches are needed to define the higher-order coalescence mathematically, but they are not physically stable operating states of the circuit. The corresponding frequency response has the form

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3}.
</math>

This cubic-root response is the key sensing signature. It is steeper than both an ordinary linear response and the square-root response of a second-order EP. The important conceptual point of Ref. [1] is that this enhanced response is obtained while the effective steady-state Hamiltonian can still retain a complete basis in dynamics.<ref name="Bai2024" /><ref name="Bai2024SM" />

=== Sensing mechanism ===
The circuit can be understood as a frequency-generating sensor. It converts an inductance change into a frequency change.

The first step is activation. A voltage is supplied to the resonators to excite the circuit into an oscillating state. The active branch in resonator <math>A</math> then supplies energy through an effective negative resistance and amplifies the selected oscillation mode.

The second step is self-stabilization. The gain is saturable, which means it becomes weaker when the voltage amplitude becomes larger. The activated oscillation therefore is not amplified forever. The circuit settles onto the physical steady branch where the imaginary part of the eigenfrequency is zero. At that point, the gain supplied by resonator <math>A</math> balances the loss in the circuit, especially the loss in resonator <math>B</math>.

The third step is frequency selection. Because resonators <math>A</math> and <math>B</math> are coupled, the final oscillation frequency is not simply the resonance frequency of either isolated resonator. It is selected by the whole coupled circuit: the inductors, capacitors, coupling elements, loss, and saturated gain all contribute.

The fourth step is sensing. The inductance <math>L_A</math> controls the bare frequency of resonator <math>A</math>. When <math>L_A</math> changes, the self-consistent frequency selected by the nonlinear circuit also changes. Therefore, tracking the spectral peak <math>f</math> tells us how the effective inductance has changed.

The fifth step is the nonlinear exceptional-point response. Near the nonlinear exceptional point discussed in Ref. [1], the physical steady branch and the two auxiliary branches coalesce in the nonlinear steady-state equation. The frequency shift of the stable branch can then scale as the cube root of the inductance perturbation. This means that a small inductance change can produce a comparatively large frequency change. Our breadboard experiment focuses on verifying the stable <math>f-L_A</math> branch corresponding to Fig. S11(a) of SM-4, which is the experimentally accessible part of this sensing mechanism.

=== Same mechanism in other platforms ===
Several references are included to show that this is a platform-independent sensing mechanism, not only a feature of our breadboard circuit. The common chain is

<math display="block">
\mathrm{physical\ perturbation}
\longrightarrow
\mathrm{parameter\ change}
\longrightarrow
\mathrm{eigenfrequency\ shift\ of\ a\ resonator/cavity\ with\ gain\ and\ loss}
\longrightarrow
\mathrm{higher\ order\ steady\ state\ response}
\longrightarrow
\mathrm{sensing}.
</math>

In our circuit, the physical perturbation is a change in the inductance <math>L_A</math>, and the measured output is the oscillation frequency <math>f</math>. In coupled optical, mechanical, electrical, or hybrid resonators, the perturbed parameter may instead be a cavity detuning, mechanical frequency, phase shift, magnetic-field-dependent transition, or another effective resonator parameter. The shared point is that gain, loss, coupling, and nonlinearity convert the parameter change into a higher-order steady-state spectral response.

The earlier theoretical work on nonlinear exceptional points showed that nonlinear coupled resonators can host higher-order NEPs while retaining a complete dynamical basis.<ref name="Bai2023">K. Bai, J.-Z. Li, T.-R. Liu, L. Fang, D. Wan, and M. Xiao, Nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''130''', 266901 (2023).</ref> The 2024 experiment implemented the same mechanism in coupled electrical resonators and measured the stable nonlinear frequency branch.<ref name="Bai2024" /><ref name="Bai2024SM" /> Silicon micromechanical resonators provide another realization: there, the measured output can be a phase difference, and the nonlinear exceptional-point response can enhance phase-shift sensing.<ref name="Xie2026">Y.-J. Xie, S.-Y. Zhang, M.-N. Zhang, R. Wang, L.-F. Wang, and Q.-A. Huang, Nonlinear exceptional points boost phase-shift sensing in silicon micromechanical resonators, ''Phys. Rev. Applied'' '''25''', 034039 (2026).</ref> Hybrid quantum systems with nonlinear bistability show a related route, where nonlinear transition dynamics can enhance sensing signal-to-noise ratio.<ref name="Wang2026">H. Wang ''et al.'', Exceptional sensitivity near the bistable transition point of a hybrid quantum system, ''Nature Physics'' '''22''', 577-583 (2026).</ref>

These examples support the same mechanism rather than unrelated applications. At the same time, enhanced response alone is not enough to prove better sensing. Noise can shift or weaken the apparent nonlinear exceptional point and can limit the signal-to-noise improvement.<ref name="Zheng2025">X. Zheng and Y. D. Chong, Noise constraints for nonlinear exceptional point sensing, ''Phys. Rev. Lett.'' '''134''', 133801 (2025).</ref> This is why our project makes a careful claim: we demonstrate the frequency-response mechanism on a breadboard, but we do not claim a complete metrological noise advantage.

== Report objectives and scope ==
The purpose of this work is to study the circuit principle behind '''nonlinear-exceptional-point-enhanced inductive sensing''' in a coupled electronic resonator platform. The scope is intentionally narrower than the full experiment reported in Ref. [1]. We are not attempting to redesign the active branch, derive all nonlinear bifurcation theory from first principles, reconstruct the auxiliary steady-state branches, or characterize every parasitic element. Instead, we adopt the framework of Ref. [1] and SM-4 and ask whether the breadboard circuit can reproduce the experimentally accessible frequency response.

The main target of our experiment is the <math>f-L_A</math> curve corresponding to Fig. S11(a) of SM-4. In the reference experiment, the steady-state frequency, voltage ratio <math>|V_A/V_B|</math>, and relative phase <math>\theta_A-\theta_B</math> are all measured as functions of <math>L_A</math>.<ref name="Bai2024SM" /> In our implementation, the frequency-versus-inductance curve is the primary observable. We tune <math>L_A</math> by connecting small inductors in series with approximately <math>2~\mu\mathrm{H}</math> resolution. We also use a homebrew coil in a separate demonstration where mechanical compression or insertion of an iron rod changes the effective inductance. The experimental goal is therefore to verify that the TL071 breadboard circuit settles into a stable oscillating state and that the measured frequency changes with <math>L_A</math> in the same qualitative manner as Fig. S11(a).

From the sensing perspective, the most relevant figure of merit is the local frequency responsivity. If

<math display="block">
|f-f_{\mathrm{NEP}}| = A |\delta L_A|^{1/3},
</math>

then differentiation gives

<math display="block">
S_f \equiv \left|\frac{df}{dL_A}\right| \propto |\delta L_A|^{-2/3}.
</math>

Thus, as the operating point approaches <math>L_{A,\mathrm{NEP}}</math>, the local spectral responsivity increases rapidly. The main practical constraint is that the system must remain on the intended stable branch while measurement noise and slow parameter drift are kept below the scale of the induced frequency shift.

== Setups ==
[[File:Experimental_setup.png|750px|thumb|center|Experimental setup adapted from Ref. [1].]]

=== Circuit topology and component implementation ===
The reference platform consists of two coupled RLC resonators. Both resonators contain the same nominal shunt capacitance <math>C_0</math>, while the inductances <math>L_A</math> and <math>L_B</math> set their bare resonance frequencies. The two resonators are coupled through a parallel resistor-capacitor channel characterized by <math>R_c</math> and <math>C_c</math>. Resonator <math>B</math> contains an explicit positive resistance <math>R_B</math> and therefore plays the role of the lossy resonator. Resonator <math>A</math>, by contrast, contains an active branch whose effective resistance is negative at small amplitude and becomes less negative as the oscillation amplitude increases, thereby providing saturable gain.<ref name="Bai2024" /><ref name="Bai2024SM" />

Our implementation follows this topology but is built on a breadboard rather than a custom PCB. The active branch is implemented with an op-amp negative-resistance circuit using a '''TL071''' operational amplifier. This is an important practical difference from the supplementary experiment, where the active components explicitly listed are a TI '''LM7171''' operational amplifier and an '''Onsemi BAV99L''' diode.<ref name="Bai2024SM" /> The TL071 substitution and the breadboard layout introduce different bandwidth limits, wiring parasitics, and contact resistances, so the numerical operating point is not expected to match the PCB experiment. For this reason, our comparison emphasizes the observable trend of <math>f</math> versus <math>L_A</math>.

The supplementary material reports the following representative component values for the PCB experiment:

<math display="block">
C_0 = 18.5~\mathrm{nF}, \qquad
C_c = 3.9~\mathrm{nF}, \qquad
R_c = 760~\Omega,
</math>

<math display="block">
L_B = 197.7~\mu\mathrm{H}, \qquad
R_B = 1314~\Omega.
</math>

A representative operating point shown in the main figure uses

<math display="block">
L_A = 245.4~\mu\mathrm{H}.
</math>

Our breadboard circuit follows the same functional topology, but its component values and parasitic elements are not identical to the PCB values above. The horizontal axis of our main <math>f-L_A</math> plot is therefore the series-combined <math>L_A</math> selected from our available small inductors, while the measured frequency <math>f</math> is the robust experimental readout.

=== Inductance tuning methods ===
We use two complementary methods to change the inductance seen by resonator <math>A</math>.

The first method is used for the main <math>f-L_A</math> sweep. Small inductors are connected in series so that the total inductance changes in discrete steps of approximately <math>2~\mu\mathrm{H}</math>. If the inductors are sufficiently separated so that their mutual magnetic coupling is small, the total series inductance is approximately the sum of the individual inductances,

<math display="block">
L_A \approx L_{\mathrm{base}}+\sum_i L_i .
</math>

This method is simple and reproducible. Adding one small inductor increases <math>L_A</math>; removing it decreases <math>L_A</math>. The circuit frequency is then measured for each chosen value. This produces a discrete version of the <math>f-L_A</math> curve in Fig. S11(a) of SM-4. The step size is not as fine as a commercial variable inductor, but <math>2~\mu\mathrm{H}</math> spacing is small enough to observe the frequency trend over the accessible range.

The second method is a homebrew compressible variable inductor. This coil is useful because it makes the sensing mechanism visible: a mechanical displacement changes the coil geometry, which changes the inductance, which then changes the oscillation frequency. The basic idea can be understood from the approximate solenoid formula

<math display="block">
L \approx \mu_\mathrm{eff}\frac{N^2 A}{\ell},
</math>

where <math>N</math> is the number of turns, <math>A</math> is the cross-sectional area, <math>\ell</math> is the coil length, and <math>\mu_\mathrm{eff}</math> describes the effective magnetic permeability of the core and surrounding region.<ref name="Chen2014">S.-C. Chen, D.-K. Le, and V.-S. Nguyen, Inductive displacement sensors with a notch filter for an active magnetic bearing system, ''Sensors'' '''14''', 12640-12657 (2014).</ref> During compression, the number of turns <math>N</math> is unchanged and the cross-sectional area <math>A</math> remains approximately unchanged, but the length <math>\ell</math> decreases. Equivalently, the turn density

<math display="block">
n=\frac{N}{\ell}
</math>

increases. The magnetic field produced by a solenoid is proportional to turn density. Therefore, compressing the coil increases the magnetic flux linkage through the turns and increases the inductance. Stretching the coil has the opposite effect.

This mechanical tuning is not identical to the precision tunable inductor used in the reference experiment. A real compressed coil is not an ideal long solenoid: the spacing between turns is finite, the end effects change as the coil length changes, nearby wires and the breadboard add parasitic capacitance, and the leads contribute additional series inductance and resistance. For this reason, the homebrew inductor is best treated as an experimentally characterized tuning element rather than an ideal component whose value is known exactly from geometry. In the present project, it provides a direct mechanical demonstration of how displacement can perturb <math>L_A</math>.

Experimentally, we measured the effective inductance of this homebrew variable inductor at different compression states. The compression process changes the coil length and wire density while leaving the coil area approximately fixed. Therefore, the tuning mechanism is physically the same as changing <math>\ell</math> and <math>n=N/\ell</math> in the solenoid model, rather than changing the number of turns or the coil diameter.

=== Kirchhoff model of the nonlinear circuit ===
A convenient state-space description uses the node voltages <math>V_A(t)</math> and <math>V_B(t)</math> together with the inductor currents <math>I_A(t)</math> and <math>I_B(t)</math>. A node voltage is the voltage measured at one resonator node relative to ground. Kirchhoff's current law says that the algebraic sum of all currents flowing away from a node must be zero. This gives a direct way to translate the circuit diagram into differential equations.

Denoting the amplitude-dependent conductance of the active branch by <math>G_A(|V_A|)</math>, where <math>G_A<0</math> in the small-signal regime, the nodal equations can be written as

<math display="block">
\begin{aligned}
C_0 \frac{dV_A}{dt} + I_A + G_A(|V_A|)\,V_A + \frac{V_A-V_B}{R_c}
+ C_c\frac{d(V_A-V_B)}{dt} &= 0,\\
C_0 \frac{dV_B}{dt} + I_B + \frac{V_B}{R_B} + \frac{V_B-V_A}{R_c}
+ C_c\frac{d(V_B-V_A)}{dt} &= 0,\\
L_A \frac{dI_A}{dt} &= V_A,\\
L_B \frac{dI_B}{dt} &= V_B.
\end{aligned}
</math>

Each term has a simple circuit meaning. The terms <math>C_0 dV_A/dt</math> and <math>C_0 dV_B/dt</math> are capacitor currents. The terms <math>I_A</math> and <math>I_B</math> are currents through the inductors. The term <math>G_A(|V_A|)V_A</math> is the active branch current; because <math>G_A</math> can be negative, this branch can inject energy instead of dissipating it. The term <math>V_B/R_B</math> is the ordinary loss current through the resistor in resonator <math>B</math>. The terms involving <math>R_c</math> and <math>C_c</math> describe current flowing between resonators <math>A</math> and <math>B</math>, so they are the electrical coupling between the two resonators.

These equations show explicitly how the experiment differs from a linear EP circuit. If <math>G_A</math> were a constant negative conductance, the model would reduce to a linear non-Hermitian dimer. Here, however, <math>G_A</math> depends on the voltage amplitude, so the steady state must be determined self-consistently together with the oscillation amplitude itself.

=== Harmonic-balance formulation of the steady state ===
After the transient has died away, the oscilloscope waveform is approximately periodic. The simplest approximation is therefore to keep only the dominant sinusoidal component. This is called a harmonic-balance approximation. For a stable activated oscillating state, one writes

<math display="block">
V_j(t)=\mathrm{Re}\!\left[\tilde V_j e^{i\omega t}\right],
\qquad j=A,B.
</math>

Substituting this ansatz into the circuit equations yields a nonlinear algebraic problem of the form

<math display="block">
\mathbf{Y}(\omega,|\tilde V_A|)
\begin{pmatrix}
\tilde V_A\\
\tilde V_B
\end{pmatrix}
=
\begin{pmatrix}
Y_A(\omega,|\tilde V_A|)+Y_c(\omega) & -Y_c(\omega)\\
-Y_c(\omega) & Y_B(\omega)+Y_c(\omega)
\end{pmatrix}
\begin{pmatrix}
\tilde V_A\\
\tilde V_B
\end{pmatrix}
=0,
</math>

with

<math display="block">
Y_A(\omega,|\tilde V_A|)=G_A(|\tilde V_A|)+i\omega C_0+\frac{1}{i\omega L_A},
</math>

<math display="block">
Y_B(\omega)=\frac{1}{R_B}+i\omega C_0+\frac{1}{i\omega L_B},
\qquad
Y_c(\omega)=\frac{1}{R_c}+i\omega C_c.
</math>

A nontrivial solution requires

<math display="block">
\det \mathbf{Y}(\omega,|\tilde V_A|)=0.
</math>

This determinant condition is the circuit analogue of an eigenfrequency condition. In a passive LC resonator, the resonance condition gives a single frequency <math>\omega_0=1/\sqrt{LC}</math>. In the present two-resonator active circuit, the same idea becomes a matrix equation because the two resonators are coupled. The amplitudes <math>\tilde V_A</math> and <math>\tilde V_B</math> are not arbitrary: a nonzero oscillation exists only when the admittance matrix has a zero eigenvalue.

Because the active admittance depends on amplitude, this condition is nonlinear and supports multiple branches. The supplementary material shows that the steady-state frequency can be reduced to a '''real-coefficient cubic polynomial''' in <math>\omega</math>. The existence of up to three real steady-state branches is therefore not accidental; it is the algebraic origin of the experimentally relevant <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM" />

=== Effective nonlinear-Hamiltonian picture ===
The same physics is often rewritten in coupled-mode form. This notation is common in non-Hermitian physics because it makes the circuit look like two coupled modes. The variables <math>\psi_A</math> and <math>\psi_B</math> represent the complex oscillation amplitudes in resonators <math>A</math> and <math>B</math>. The model is

<math display="block">
i\frac{d}{dt}
\begin{pmatrix}
\psi_A\\
\psi_B
\end{pmatrix}
=
\begin{pmatrix}
\omega_A+i\,g(|\psi_A|) & \kappa\\
\kappa & \omega_B-i\ell
\end{pmatrix}
\begin{pmatrix}
\psi_A\\
\psi_B
\end{pmatrix},
</math>

where <math>\omega_A</math> and <math>\omega_B</math> are the effective modal frequencies, <math>\kappa</math> is the coupling strength, <math>\ell</math> is the loss of resonator <math>B</math>, and <math>g(|\psi_A|)</math> is the saturable gain of resonator <math>A</math>. The steady-state problem then becomes

<math display="block">
H_s(\Psi)\Psi = \Omega \Psi,
</math>

with a real oscillation eigenfrequency <math>\Omega=2\pi f</math> for the stable oscillating branch. The word "real" is important: it means the imaginary part has vanished, so the mode is neither being amplified nor decaying in the steady state. In the cited experiment, this physical stable branch and two auxiliary steady branches coalesce at the singular point, producing a third-order nonlinear exceptional point rather than a conventional second-order linear EP. The auxiliary branches help form the cubic-root singularity but are not themselves stable physical outputs of the circuit.<ref name="Bai2024" /><ref name="Bai2024SM" />

== Measurements ==

=== Acquisition protocol and steady-state selection ===
The measurement is performed in the time domain. In the reference experiment, an arbitrary waveform generator first applies an external start-up signal of 1 V at 70 kHz in order to bring the system into the basin of attraction of the stable oscillating state.<ref name="Bai2024" /> In our breadboard measurement, an external voltage is likewise required to activate the resonators. The measured frequency is extracted from the activated steady waveform recorded on the oscilloscope.

The physical meaning of this procedure is important. The oscillation is not assumed to appear without activation; a supplied voltage is needed to bring the resonators into the relevant dynamical state. The measured oscillation frequency is then interpreted as the frequency selected by the nonlinear steady-state condition.

=== Extraction of the <math>f-L_A</math> observable ===
Ref. [1] does not treat the oscilloscope traces merely as qualitative evidence of oscillation. The supplementary explicitly states that the full reference experiment reports the steady-state frequency, the voltage ratio <math>|V_A/V_B|</math>, and the relative phase <math>\theta_A-\theta_B</math>, all extracted from the dominant peaks of the Fourier spectra of <math>V_A</math> and <math>V_B</math>.<ref name="Bai2024SM" />

Our available observable is the steady-state frequency. For each chosen value of <math>L_A</math>, the resonators are voltage-activated, the circuit reaches a stable oscillating state, and the dominant spectral peak is recorded. If we denote the Fourier transforms by

<math display="block">
\tilde V_A(f)=\mathcal{F}\{V_A(t)\},
\qquad
\tilde V_B(f)=\mathcal{F}\{V_B(t)\},
</math>

then the dominant oscillation frequency is obtained from

<math display="block">
f_0=\arg\max_f |\tilde V_A(f)| \approx \arg\max_f |\tilde V_B(f)|.
</math>

The experimental data set is therefore

<math display="block">
\{L_A,\ f_0(L_A)\},
</math>

which is compared with the frequency branch plotted in Fig. S11(a) of SM-4. In the full reference measurement, the amplitude ratio

<math display="block">
\rho = \frac{|\tilde V_A(f_0)|}{|\tilde V_B(f_0)|}
</math>

and relative phase

<math display="block">
\Delta\phi = \arg \tilde V_A(f_0)-\arg \tilde V_B(f_0)
</math>

provide additional state verification. Those quantities are useful but are not the main basis of our breadboard validation.

In practice, the same frequency can also be estimated directly from the time-domain period <math>T</math> using <math>f_0=1/T</math> when the waveform is clean. Fourier analysis is preferred because it uses a longer section of waveform and identifies the dominant oscillation peak even when there is small amplitude noise or waveform distortion.

=== Practical inductance sweep ===
In SM-4, the inductive tuning is carried out by varying <math>L_A</math>. The supplementary notes that the tunable inductor exhibits less than about 0.4% inductance variation over the frequency range of interest, which is sufficiently small for controlled sweeping near the singular point. To further suppress voltage-dependent drift during the sweep, the reference experiment slightly adjusts the active-branch resistance so as to maintain approximately <math>V_B \approx 2~\mathrm{V}</math> throughout the measurement series.<ref name="Bai2024SM" />

Our sweep uses controlled series combinations of small inductors, giving an approximately <math>2~\mu\mathrm{H}</math> step in <math>L_A</math>. The homebrew compressible coil is then used to show that a mechanical change of the sensing element also changes the inductance. The conclusion drawn from our data is therefore not a precision determination of <math>L_{A,\mathrm{NEP}}</math>, but a validation that the breadboard circuit produces the same kind of frequency response as the stable branch in Fig. S11(a).

=== Long-window spectral analysis and noise inspection ===
The supplementary also discusses long-time measurements over a window of <math>0<t<5~\mathrm{s}</math>. The motivation is that the noise around the stable oscillation is too small to be judged reliably from raw scope traces alone, so it is more informative to inspect the Fourier spectra and the extracted spectral center frequency and linewidth.<ref name="Bai2024SM" />

This is a useful methodological point for a sensing-oriented reproduction. In an NEP-based sensor, one ultimately cares about how precisely the frequency peak can be resolved and tracked. A long-window Fourier analysis is therefore closer to the sensing task than a purely visual time-domain inspection.

== Results ==

=== Result display: steady oscillation ===
[[File:PC5271inductive1.png|750px|thumb|center|Breadboard implementation and representative steady oscillation waveform]]

This figure should show the breadboard circuit and a representative voltage waveform after the resonators are activated by the supplied voltage. The displayed waveform is the evidence used to identify a stable oscillation frequency.

=== Result display: <math>f-L_A</math> sweep ===
[[File:PC5271inductive3.jpeg|750px|thumb|center|Measured breadboard <math>f-L_A</math> curve.]]

This figure should show the measured oscillation frequency <math>f</math> as a function of <math>L_A</math>. The values of <math>L_A</math> in the main sweep are produced by connecting small inductors in series with approximately <math>2~\mu\mathrm{H}</math> step size. The reference curve for comparison is the stable frequency branch in Fig. S11(a) of SM-4.<ref name="Bai2024SM" />

=== Result display: iron-rod demonstration ===
[[File:PC5271inductive2.png|750px|thumb|center|Displacement measurement by compressing the solenoid and inserting iron rod into the homebrew coil]]

This figure should show the homebrew coil with an inserted iron rod and the corresponding change in the measured oscillation frequency. The experimental variable is the rod position or insertion state, and the measured output is the activated steady oscillation frequency.

== Evaluation ==

=== Stable activated oscillation ===
The first required experimental signature is that, after the resonators are activated by the supplied voltage, the circuit reaches a stable oscillating waveform instead of decaying immediately or being amplified without bound. A stable waveform means that the TL071 active branch supplies enough small-signal negative resistance to compensate loss, while gain saturation prevents unbounded amplification.

Physically, this behavior confirms the role of the nonlinear active branch. At small amplitude, the effective negative resistance in resonator <math>A</math> overcomes dissipation. At larger amplitude, the gain becomes weaker, so the oscillation amplitude is clamped. The system self-stabilizes because the physical steady branch is the one with zero imaginary part of the eigenfrequency. The auxiliary branches near the nonlinear exceptional point are not stable physical outputs; they are mathematical branches of the nonlinear steady-state equation. The experiment is therefore not a passive ring-down measurement; it is a nonlinear steady-state selection process.

=== Evaluation of the <math>f-L_A</math> trend ===
The main experimental comparison is the measured <math>f-L_A</math> trend against Fig. S11(a) of SM-4. The reference experiment reports the stable frequency branch of the nonlinear circuit as <math>L_A</math> is tuned.<ref name="Bai2024SM" /> This branch is the physical one because its eigenfrequency is real in the steady state. In our breadboard experiment, the same observable is measured using the series-inductor sweep.

It is useful to separate two statements. First, any LC resonator changes frequency when <math>L</math> changes, because <math>f_0=1/(2\pi\sqrt{LC})</math>. This alone would not demonstrate the nonlinear exceptional-point mechanism. Second, in the coupled active-lossy circuit, the frequency is not just the bare LC resonance frequency of one isolated inductor and capacitor. It is the frequency selected by the nonlinear steady-state balance of gain, loss, and coupling. The reason Fig. S11(a) matters is that it shows this selected stable branch as <math>L_A</math> is swept.

Our breadboard circuit uses different component values and a different op-amp from the PCB experiment, so exact numerical overlap is not expected. The meaningful evaluation is whether the measured frequency branch follows the same qualitative <math>f-L_A</math> behavior as the stable branch in SM-4. Observing that behavior is enough to show that the constructed circuit realizes the intended nonlinear voltage-activated oscillating mechanism.

=== Evaluation of the displacement demonstration ===
In the iron-rod demonstration, the homebrew coil becomes a displacement-dependent inductive element. As the rod enters the coil, the effective magnetic permeability inside the coil increases. In the solenoid expression

<math display="block">
L(x) \approx \mu_\mathrm{eff}(x)\frac{N^2 A}{\ell},
</math>

the insertion depth <math>x</math> changes <math>\mu_\mathrm{eff}</math>. The coil geometry <math>N</math>, <math>A</math>, and <math>\ell</math> are approximately fixed during this demonstration, so the dominant effect is the change in magnetic core material inside the coil. Inductive displacement sensors commonly use this principle: the position of a ferromagnetic object changes a coil inductance, and the electronics convert that inductance change into an electrical signal.<ref name="Chen2014" />

In our circuit, the electrical signal is the activated steady oscillation frequency rather than a bridge voltage. The sensing chain is

<math display="block">
\mathrm{rod\ displacement}\ x
\longrightarrow
\mathrm{effective\ inductance}\ L_A(x)
\longrightarrow
\mathrm{oscillation\ frequency}\ f(x).
</math>

This demonstration shows how the inductive readout can be turned into a displacement-sensing readout after calibration of the relation between rod position and frequency.

=== Cubic-root perturbation law and sensing meaning ===
Near the singular operating point, the eigenfrequency shift follows the hallmark cubic-root law,

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3}.
</math>

This law is the main sensing-related result of the reference experiment.<ref name="Bai2024" /><ref name="Bai2024SM" /> Compared with the linear response of an ordinary isolated resonator, the cubic-root dependence can produce a larger frequency displacement for the same small perturbation of inductance when the device is biased sufficiently close to <math>L_{A,\mathrm{NEP}}</math>.

A useful way to interpret the result is through the local derivative,

<math display="block">
\left|\frac{df}{dL_A}\right| \propto |\delta L_A|^{-2/3}.
</math>

This increasing local responsivity is why the circuit is attractive for sensing. Our experiment supports this sensing logic by showing that the breadboard circuit has a measurable stable frequency branch as <math>L_A</math> is changed. A full quantitative exponent extraction would require identifying <math>L_{A,\mathrm{NEP}}</math> and <math>f_{\mathrm{NEP}}</math>, then plotting <math>|f-f_{\mathrm{NEP}}|</math> against <math>|L_A-L_{A,\mathrm{NEP}}|</math> on logarithmic axes.

=== Basis completeness and noise robustness ===
The supplementary material evaluates the Petermann factor as a diagnostic of modal nonorthogonality and basis completeness. For a non-Hermitian eigenproblem, one may write

<math display="block">
\mathrm{PF}_n =
\frac{\langle \Phi_n^{L}|\Phi_n^{L}\rangle \langle \Phi_n^{R}|\Phi_n^{R}\rangle}
{\left|\langle \Phi_n^{L}|\Phi_n^{R}\rangle\right|^2},
</math>

where <math>|\Phi_n^{R}\rangle</math> and <math>|\Phi_n^{L}\rangle</math> are the right and left eigenvectors of the effective steady-state Hamiltonian. At a conventional linear EP, this quantity is associated with diverging excess-noise sensitivity. In Ref. [1], the cited analysis finds that the effective steady-state Hamiltonian remains non-defective and the Petermann factor is finite at the <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM" />

This point explains the significance of the reference experiment. The system does not merely realize a large mathematical responsivity; it realizes that responsivity in a way that avoids the most severe basis-collapse problem of a linear EP. Our breadboard experiment does not directly test basis completeness, but the result provides the theoretical reason why reproducing the stable <math>f-L_A</math> branch is physically meaningful.

=== Overall interpretation ===
In this platform, the measurand is encoded as a perturbation of <math>L_A</math>. Any physical mechanism that modifies the effective inductance of resonator <math>A</math>, for example displacement, magnetic coupling, or the presence of a nearby magnetic target, moves the system away from the operating point and shifts the oscillation frequency. The primary readout in our implementation is therefore spectral: we track <math>f</math> as <math>L_A</math> is changed.

Taken together, Ref. [1] establishes the complete sensing logic: a nonlinear active-lossy circuit is tuned to an <math>\mathrm{NEP}_3</math>; after voltage activation, the system settles onto a stable oscillating branch; the dominant spectral peak is tracked as the perturbation readout; and the enhanced cubic-root response is obtained without the defective-basis pathology of a conventional linear EP. Our breadboard result demonstrates the experimentally accessible core of this logic by reproducing the stable frequency response corresponding to Fig. S11(a) of SM-4.

== References ==
<references />

Inductive Sensors of Ultra-high Sensitivity Based on Nonlinear Exceptional Point

2026-04-23T06:09:26Z

Siyu: /* Introduction */

== Introduction ==
This project is a implementation of the experiment reported in ''Observation of Nonlinear Exceptional Points with a Complete Basis in Dynamics'', adapted to our available laboratory equipment.<ref name="Bai2024">K. Bai, T.-R. Liu, L. Fang, J.-Z. Li, C. Lin, D. Wan, and M. Xiao, Observation of nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''132''', 073802 (2024).</ref> The circuit contains two coupled resonators. One resonator has gain, meaning that it can supply energy to the oscillation, while the other has loss, meaning that it dissipates energy. A system with both gain and loss is called a '''non-Hermitian''' system. Such systems can show unusual degeneracies called '''exceptional points''' (EPs), where both the eigenfrequencies and the eigenmodes merge.

Exceptional points are interesting for sensing because the measured frequency can respond very strongly to a small perturbation. In an inductive sensor, the perturbation is a change in inductance. If a small change in inductance produces a large change in oscillation frequency, the circuit can in principle detect small physical changes. However, ordinary linear EPs also suffer from strong eigenmode nonorthogonality and a defective basis, which can increase noise sensitivity and complicate practical sensing.<ref name="Bai2024" />

The system studied in this project belongs to the recently developed class of '''nonlinear exceptional points''' (NEPs). The physical platform is a pair of coupled circuit resonators. Resonator <math>A</math> is made active by a saturable negative-resistance branch, while resonator <math>B</math> is intentionally lossy. Because the gain in resonator <math>A</math> depends on the oscillation amplitude, the steady-state problem is nonlinear and admits multiple self-consistent branches. At the singular operating point reported in Ref. [1], one stable steady branch and two auxiliary steady branches merge into a third-order nonlinear exceptional point, denoted <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM">Supplemental Material for K. Bai, T.-R. Liu, L. Fang, J.-Z. Li, C. Lin, D. Wan, and M. Xiao, Observation of nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''132''', 073802 (2024).</ref>

The experimentally important consequence is that the oscillation eigenfrequency obeys a cubic-root perturbation law near the singularity,

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3},
</math>

where <math>\delta L_A = L_A-L_{A,\mathrm{NEP}}</math> is the inductive detuning from the exceptional operating point. This response is steeper than the ordinary linear dependence of a conventional resonator and is therefore useful for ultra-sensitive inductive sensing.<ref name="Bai2024" /><ref name="Bai2024SM" />

A second and equally important point is that the cited work does not interpret the singularity as a conventional defective linear EP. Instead, the effective steady-state Hamiltonian remains non-defective in dynamics, and the associated Petermann factor stays finite at the operating point. In other words, the experiment realizes enhanced spectral responsivity without inheriting the strongest basis-collapse pathology of a standard linear EP.<ref name="Bai2024" /><ref name="Bai2024SM" />

In this project, our goal is to reproduce the experimental logic of Ref. [1] and its supplementary material under course-lab conditions. Unlike the PCB implementation in the paper, our circuit is assembled on a breadboard and the active branch uses a '''TL071''' operational amplifier. Our main observable is the steady oscillation frequency <math>f</math> as a function of the chosen inductance <math>L_A</math>. We tune <math>L_A</math> in two ways. For the main <math>f-L_A</math> sweep, we join small inductors in series so that the discrete inductance step is about <math>2~\mu\mathrm{H}</math>. In addition, we use a homebrew compressible coil to demonstrate the sensing mechanism directly. Compressing the coil changes its length and turn density while keeping the cross-sectional area approximately unchanged. We therefore use the <math>f-L_A</math> behavior in Fig. S11(a) of the supplementary material as the main experimental benchmark. Reproducing this trend is sufficient for demonstrating that our setup captures the intended nonlinear voltage-activated, self-stabilized oscillating response, although it is not a full reproduction of every diagnostic reported in Ref. [1].<ref name="Bai2024" /><ref name="Bai2024SM" />

== Team Members ==
Wang Peikun — E1538091@u.nus.edu 
Zhu Ziyang — E1583446@u.nus.edu 
Yuan Si-Yu — siyu_yuan@u.nus.edu 
Li Xunyu — xunyu@u.nus.edu

== Background Knowledge ==

=== Ordinary LC and RLC resonators ===
The basic building block of the experiment is an electrical resonator. An ideal LC resonator contains an inductor <math>L</math> and a capacitor <math>C</math>. Energy repeatedly moves between the magnetic field of the inductor and the electric field of the capacitor. The natural angular frequency is

<math display="block">
\omega_0=\frac{1}{\sqrt{LC}},
</math>

or, in ordinary frequency units,

<math display="block">
f_0=\frac{1}{2\pi\sqrt{LC}}.
</math>

This formula is important for the whole project because it explains why changing an inductance changes the measured frequency. A larger <math>L</math> gives a lower resonance frequency, while a smaller <math>L</math> gives a higher resonance frequency.

Real circuits are not ideal. Wires, inductors, capacitors, breadboard contacts, and measuring probes all dissipate energy. This loss is often represented by a resistor <math>R</math>, giving an RLC resonator. In a passive RLC circuit, the oscillation amplitude decays with time because energy is continuously lost as heat. If a circuit is to maintain a steady activated oscillation, the loss must be compensated by a gain element.

=== Gain, loss, and negative resistance ===
In electronics, gain can be represented as an effective negative resistance. A normal positive resistance removes energy from the resonator. A negative resistance does the opposite: it supplies energy to the resonator. In this experiment, resonator <math>A</math> contains an active op-amp branch that behaves like a negative resistance at small amplitude, while resonator <math>B</math> contains ordinary positive loss.

The gain cannot remain negative and constant forever. If it did, any activated oscillation would be amplified without bound until the op-amp clipped or the circuit failed. The important nonlinear ingredient is '''saturable gain''': the active branch gives strong gain at small amplitude but weaker gain at larger amplitude. The stable oscillation amplitude is selected when the supplied gain balances the total loss. This nonlinear steady state is why the experiment is a nonlinear dynamical problem rather than only a passive resonance measurement.

=== What "non-Hermitian" means in this circuit ===
In ordinary conservative physics, a Hermitian Hamiltonian describes a closed system whose total energy is conserved. Its eigenfrequencies are real, and its eigenmodes form a complete orthogonal basis. This is the familiar setting of many quantum-mechanics courses.

A circuit with gain and loss is an open system. Energy can enter through the op-amp branch and leave through resistive loss. The effective dynamical matrix is therefore generally '''non-Hermitian'''. In a non-Hermitian system, eigenfrequencies can be complex:

<math display="block">
\omega=\omega_\mathrm{r}+i\omega_\mathrm{i}.
</math>

The real part <math>\omega_\mathrm{r}</math> gives the oscillation frequency, while the imaginary part <math>\omega_\mathrm{i}</math> describes amplification or decay. A positive imaginary part means the mode is amplified, a negative imaginary part means it decays, and a stable steady oscillation corresponds to zero imaginary part, or equivalently zero net amplification. This is the sense in which the circuit is self-stabilized after voltage activation.

=== Exceptional points ===
In a usual Hermitian system, two modes can have the same eigenfrequency while still remaining two independent modes. This is an ordinary degeneracy. A non-Hermitian system can show a stronger kind of degeneracy called an '''exceptional point''' (EP). At an EP, not only the eigenvalues but also the eigenvectors coalesce. In simple language, two modes collapse into one mode.

This collapse makes EPs interesting for sensing. Near a second-order linear EP, a small perturbation can produce a square-root frequency splitting:

<math display="block">
\Delta f \propto \sqrt{\epsilon},
</math>

where <math>\epsilon</math> is the perturbation. This response is steeper than an ordinary linear response when <math>\epsilon</math> is very small. However, a conventional linear EP also has a serious drawback: the eigenbasis becomes defective, meaning that the system no longer has enough independent eigenvectors. This is associated with strong modal nonorthogonality and enhanced noise sensitivity. The Petermann factor is one common way to quantify this nonorthogonality.

=== Why the nonlinear exceptional point is different ===
The reference paper studies a nonlinear exceptional point. The word "nonlinear" is essential. The circuit does not have a fixed gain coefficient; instead, the gain depends on the oscillation amplitude. The steady state must therefore be solved self-consistently: the frequency, voltage amplitudes, and saturated gain must all agree with one another.

In the theory of Ref. [1], this nonlinear steady-state problem can reduce to a cubic equation for the oscillation frequency. A cubic equation can have a triple root. At the special operating point, one stable physical branch and two auxiliary branches merge, giving a third-order nonlinear exceptional point, written as <math>\mathrm{NEP}_3</math>. The stable physical branch is the one with real oscillation frequency, i.e. without an imaginary part. The auxiliary branches are needed to define the higher-order coalescence mathematically, but they are not physically stable operating states of the circuit. The corresponding frequency response has the form

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3}.
</math>

This cubic-root response is the key sensing signature. It is steeper than both an ordinary linear response and the square-root response of a second-order EP. The important conceptual point of Ref. [1] is that this enhanced response is obtained while the effective steady-state Hamiltonian can still retain a complete basis in dynamics.<ref name="Bai2024" /><ref name="Bai2024SM" />

=== Sensing mechanism ===
The circuit can be understood as a frequency-generating sensor. It converts an inductance change into a frequency change.

The first step is activation. A voltage is supplied to the resonators to excite the circuit into an oscillating state. The active branch in resonator <math>A</math> then supplies energy through an effective negative resistance and amplifies the selected oscillation mode.

The second step is self-stabilization. The gain is saturable, which means it becomes weaker when the voltage amplitude becomes larger. The activated oscillation therefore is not amplified forever. The circuit settles onto the physical steady branch where the imaginary part of the eigenfrequency is zero. At that point, the gain supplied by resonator <math>A</math> balances the loss in the circuit, especially the loss in resonator <math>B</math>.

The third step is frequency selection. Because resonators <math>A</math> and <math>B</math> are coupled, the final oscillation frequency is not simply the resonance frequency of either isolated resonator. It is selected by the whole coupled circuit: the inductors, capacitors, coupling elements, loss, and saturated gain all contribute.

The fourth step is sensing. The inductance <math>L_A</math> controls the bare frequency of resonator <math>A</math>. When <math>L_A</math> changes, the self-consistent frequency selected by the nonlinear circuit also changes. Therefore, tracking the spectral peak <math>f</math> tells us how the effective inductance has changed.

The fifth step is the nonlinear exceptional-point response. Near the nonlinear exceptional point discussed in Ref. [1], the physical steady branch and the two auxiliary branches coalesce in the nonlinear steady-state equation. The frequency shift of the stable branch can then scale as the cube root of the inductance perturbation. This means that a small inductance change can produce a comparatively large frequency change. Our breadboard experiment focuses on verifying the stable <math>f-L_A</math> branch corresponding to Fig. S11(a) of SM-4, which is the experimentally accessible part of this sensing mechanism.

=== Same mechanism in other platforms ===
Several references are included to show that this is a platform-independent sensing mechanism, not only a feature of our breadboard circuit. The common chain is

<math display="block">
\mathrm{physical\ perturbation}
\longrightarrow
\mathrm{parameter\ change}
\longrightarrow
\mathrm{eigenfrequency\ shift\ of\ a\ resonator/cavity\ with\ gain\ and\ loss}
\longrightarrow
\mathrm{higher\ order\ steady\ state\ response}
\longrightarrow
\mathrm{sensing}.
</math>

In our circuit, the physical perturbation is a change in the inductance <math>L_A</math>, and the measured output is the oscillation frequency <math>f</math>. In coupled optical, mechanical, electrical, or hybrid resonators, the perturbed parameter may instead be a cavity detuning, mechanical frequency, phase shift, magnetic-field-dependent transition, or another effective resonator parameter. The shared point is that gain, loss, coupling, and nonlinearity convert the parameter change into a higher-order steady-state spectral response.

The earlier theoretical work on nonlinear exceptional points showed that nonlinear coupled resonators can host higher-order NEPs while retaining a complete dynamical basis.<ref name="Bai2023">K. Bai, J.-Z. Li, T.-R. Liu, L. Fang, D. Wan, and M. Xiao, Nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''130''', 266901 (2023).</ref> The 2024 experiment implemented the same mechanism in coupled electrical resonators and measured the stable nonlinear frequency branch.<ref name="Bai2024" /><ref name="Bai2024SM" /> Silicon micromechanical resonators provide another realization: there, the measured output can be a phase difference, and the nonlinear exceptional-point response can enhance phase-shift sensing.<ref name="Xie2026">Y.-J. Xie, S.-Y. Zhang, M.-N. Zhang, R. Wang, L.-F. Wang, and Q.-A. Huang, Nonlinear exceptional points boost phase-shift sensing in silicon micromechanical resonators, ''Phys. Rev. Applied'' '''25''', 034039 (2026).</ref> Hybrid quantum systems with nonlinear bistability show a related route, where nonlinear transition dynamics can enhance sensing signal-to-noise ratio.<ref name="Wang2026">H. Wang ''et al.'', Exceptional sensitivity near the bistable transition point of a hybrid quantum system, ''Nature Physics'' '''22''', 577-583 (2026).</ref>

These examples support the same mechanism rather than unrelated applications. At the same time, enhanced response alone is not enough to prove better sensing. Noise can shift or weaken the apparent nonlinear exceptional point and can limit the signal-to-noise improvement.<ref name="Zheng2025">X. Zheng and Y. D. Chong, Noise constraints for nonlinear exceptional point sensing, ''Phys. Rev. Lett.'' '''134''', 133801 (2025).</ref> This is why our project makes a careful claim: we demonstrate the frequency-response mechanism on a breadboard, but we do not claim a complete metrological noise advantage.

== Report objectives and scope ==
The purpose of this work is to study the circuit principle behind '''nonlinear-exceptional-point-enhanced inductive sensing''' in a coupled electronic resonator platform. The scope is intentionally narrower than the full experiment reported in Ref. [1]. We are not attempting to redesign the active branch, derive all nonlinear bifurcation theory from first principles, reconstruct the auxiliary steady-state branches, or characterize every parasitic element. Instead, we adopt the framework of Ref. [1] and SM-4 and ask whether the breadboard circuit can reproduce the experimentally accessible frequency response.

The main target of our experiment is the <math>f-L_A</math> curve corresponding to Fig. S11(a) of SM-4. In the reference experiment, the steady-state frequency, voltage ratio <math>|V_A/V_B|</math>, and relative phase <math>\theta_A-\theta_B</math> are all measured as functions of <math>L_A</math>.<ref name="Bai2024SM" /> In our implementation, the frequency-versus-inductance curve is the primary observable. We tune <math>L_A</math> by connecting small inductors in series with approximately <math>2~\mu\mathrm{H}</math> resolution. We also use a homebrew coil in a separate demonstration where mechanical compression or insertion of an iron rod changes the effective inductance. The experimental goal is therefore to verify that the TL071 breadboard circuit settles into a stable oscillating state and that the measured frequency changes with <math>L_A</math> in the same qualitative manner as Fig. S11(a).

From the sensing perspective, the most relevant figure of merit is the local frequency responsivity. If

<math display="block">
|f-f_{\mathrm{NEP}}| = A |\delta L_A|^{1/3},
</math>

then differentiation gives

<math display="block">
S_f \equiv \left|\frac{df}{dL_A}\right| \propto |\delta L_A|^{-2/3}.
</math>

Thus, as the operating point approaches <math>L_{A,\mathrm{NEP}}</math>, the local spectral responsivity increases rapidly. The main practical constraint is that the system must remain on the intended stable branch while measurement noise and slow parameter drift are kept below the scale of the induced frequency shift.

== Setups ==
[[File:Experimental_setup.png|750px|thumb|center|Experimental setup adapted from Ref. [1].]]

=== Circuit topology and component implementation ===
The reference platform consists of two coupled RLC resonators. Both resonators contain the same nominal shunt capacitance <math>C_0</math>, while the inductances <math>L_A</math> and <math>L_B</math> set their bare resonance frequencies. The two resonators are coupled through a parallel resistor-capacitor channel characterized by <math>R_c</math> and <math>C_c</math>. Resonator <math>B</math> contains an explicit positive resistance <math>R_B</math> and therefore plays the role of the lossy resonator. Resonator <math>A</math>, by contrast, contains an active branch whose effective resistance is negative at small amplitude and becomes less negative as the oscillation amplitude increases, thereby providing saturable gain.<ref name="Bai2024" /><ref name="Bai2024SM" />

Our implementation follows this topology but is built on a breadboard rather than a custom PCB. The active branch is implemented with an op-amp negative-resistance circuit using a '''TL071''' operational amplifier. This is an important practical difference from the supplementary experiment, where the active components explicitly listed are a TI '''LM7171''' operational amplifier and an '''Onsemi BAV99L''' diode.<ref name="Bai2024SM" /> The TL071 substitution and the breadboard layout introduce different bandwidth limits, wiring parasitics, and contact resistances, so the numerical operating point is not expected to match the PCB experiment. For this reason, our comparison emphasizes the observable trend of <math>f</math> versus <math>L_A</math>.

The supplementary material reports the following representative component values for the PCB experiment:

<math display="block">
C_0 = 18.5~\mathrm{nF}, \qquad
C_c = 3.9~\mathrm{nF}, \qquad
R_c = 760~\Omega,
</math>

<math display="block">
L_B = 197.7~\mu\mathrm{H}, \qquad
R_B = 1314~\Omega.
</math>

A representative operating point shown in the main figure uses

<math display="block">
L_A = 245.4~\mu\mathrm{H}.
</math>

Our breadboard circuit follows the same functional topology, but its component values and parasitic elements are not identical to the PCB values above. The horizontal axis of our main <math>f-L_A</math> plot is therefore the series-combined <math>L_A</math> selected from our available small inductors, while the measured frequency <math>f</math> is the robust experimental readout.

=== Inductance tuning methods ===
We use two complementary methods to change the inductance seen by resonator <math>A</math>.

The first method is used for the main <math>f-L_A</math> sweep. Small inductors are connected in series so that the total inductance changes in discrete steps of approximately <math>2~\mu\mathrm{H}</math>. If the inductors are sufficiently separated so that their mutual magnetic coupling is small, the total series inductance is approximately the sum of the individual inductances,

<math display="block">
L_A \approx L_{\mathrm{base}}+\sum_i L_i .
</math>

This method is simple and reproducible. Adding one small inductor increases <math>L_A</math>; removing it decreases <math>L_A</math>. The circuit frequency is then measured for each chosen value. This produces a discrete version of the <math>f-L_A</math> curve in Fig. S11(a) of SM-4. The step size is not as fine as a commercial variable inductor, but <math>2~\mu\mathrm{H}</math> spacing is small enough to observe the frequency trend over the accessible range.

The second method is a homebrew compressible variable inductor. This coil is useful because it makes the sensing mechanism visible: a mechanical displacement changes the coil geometry, which changes the inductance, which then changes the oscillation frequency. The basic idea can be understood from the approximate solenoid formula

<math display="block">
L \approx \mu_\mathrm{eff}\frac{N^2 A}{\ell},
</math>

where <math>N</math> is the number of turns, <math>A</math> is the cross-sectional area, <math>\ell</math> is the coil length, and <math>\mu_\mathrm{eff}</math> describes the effective magnetic permeability of the core and surrounding region.<ref name="Chen2014">S.-C. Chen, D.-K. Le, and V.-S. Nguyen, Inductive displacement sensors with a notch filter for an active magnetic bearing system, ''Sensors'' '''14''', 12640-12657 (2014).</ref> During compression, the number of turns <math>N</math> is unchanged and the cross-sectional area <math>A</math> remains approximately unchanged, but the length <math>\ell</math> decreases. Equivalently, the turn density

<math display="block">
n=\frac{N}{\ell}
</math>

increases. The magnetic field produced by a solenoid is proportional to turn density. Therefore, compressing the coil increases the magnetic flux linkage through the turns and increases the inductance. Stretching the coil has the opposite effect.

This mechanical tuning is not identical to the precision tunable inductor used in the reference experiment. A real compressed coil is not an ideal long solenoid: the spacing between turns is finite, the end effects change as the coil length changes, nearby wires and the breadboard add parasitic capacitance, and the leads contribute additional series inductance and resistance. For this reason, the homebrew inductor is best treated as an experimentally characterized tuning element rather than an ideal component whose value is known exactly from geometry. In the present project, it provides a direct mechanical demonstration of how displacement can perturb <math>L_A</math>.

Experimentally, we measured the effective inductance of this homebrew variable inductor at different compression states. The compression process changes the coil length and wire density while leaving the coil area approximately fixed. Therefore, the tuning mechanism is physically the same as changing <math>\ell</math> and <math>n=N/\ell</math> in the solenoid model, rather than changing the number of turns or the coil diameter.

=== Kirchhoff model of the nonlinear circuit ===
A convenient state-space description uses the node voltages <math>V_A(t)</math> and <math>V_B(t)</math> together with the inductor currents <math>I_A(t)</math> and <math>I_B(t)</math>. A node voltage is the voltage measured at one resonator node relative to ground. Kirchhoff's current law says that the algebraic sum of all currents flowing away from a node must be zero. This gives a direct way to translate the circuit diagram into differential equations.

Denoting the amplitude-dependent conductance of the active branch by <math>G_A(|V_A|)</math>, where <math>G_A<0</math> in the small-signal regime, the nodal equations can be written as

<math display="block">
\begin{aligned}
C_0 \frac{dV_A}{dt} + I_A + G_A(|V_A|)\,V_A + \frac{V_A-V_B}{R_c}
+ C_c\frac{d(V_A-V_B)}{dt} &= 0,\\
C_0 \frac{dV_B}{dt} + I_B + \frac{V_B}{R_B} + \frac{V_B-V_A}{R_c}
+ C_c\frac{d(V_B-V_A)}{dt} &= 0,\\
L_A \frac{dI_A}{dt} &= V_A,\\
L_B \frac{dI_B}{dt} &= V_B.
\end{aligned}
</math>

Each term has a simple circuit meaning. The terms <math>C_0 dV_A/dt</math> and <math>C_0 dV_B/dt</math> are capacitor currents. The terms <math>I_A</math> and <math>I_B</math> are currents through the inductors. The term <math>G_A(|V_A|)V_A</math> is the active branch current; because <math>G_A</math> can be negative, this branch can inject energy instead of dissipating it. The term <math>V_B/R_B</math> is the ordinary loss current through the resistor in resonator <math>B</math>. The terms involving <math>R_c</math> and <math>C_c</math> describe current flowing between resonators <math>A</math> and <math>B</math>, so they are the electrical coupling between the two resonators.

These equations show explicitly how the experiment differs from a linear EP circuit. If <math>G_A</math> were a constant negative conductance, the model would reduce to a linear non-Hermitian dimer. Here, however, <math>G_A</math> depends on the voltage amplitude, so the steady state must be determined self-consistently together with the oscillation amplitude itself.

=== Harmonic-balance formulation of the steady state ===
After the transient has died away, the oscilloscope waveform is approximately periodic. The simplest approximation is therefore to keep only the dominant sinusoidal component. This is called a harmonic-balance approximation. For a stable activated oscillating state, one writes

<math display="block">
V_j(t)=\mathrm{Re}\!\left[\tilde V_j e^{i\omega t}\right],
\qquad j=A,B.
</math>

Substituting this ansatz into the circuit equations yields a nonlinear algebraic problem of the form

<math display="block">
\mathbf{Y}(\omega,|\tilde V_A|)
\begin{pmatrix}
\tilde V_A\\
\tilde V_B
\end{pmatrix}
=
\begin{pmatrix}
Y_A(\omega,|\tilde V_A|)+Y_c(\omega) & -Y_c(\omega)\\
-Y_c(\omega) & Y_B(\omega)+Y_c(\omega)
\end{pmatrix}
\begin{pmatrix}
\tilde V_A\\
\tilde V_B
\end{pmatrix}
=0,
</math>

with

<math display="block">
Y_A(\omega,|\tilde V_A|)=G_A(|\tilde V_A|)+i\omega C_0+\frac{1}{i\omega L_A},
</math>

<math display="block">
Y_B(\omega)=\frac{1}{R_B}+i\omega C_0+\frac{1}{i\omega L_B},
\qquad
Y_c(\omega)=\frac{1}{R_c}+i\omega C_c.
</math>

A nontrivial solution requires

<math display="block">
\det \mathbf{Y}(\omega,|\tilde V_A|)=0.
</math>

This determinant condition is the circuit analogue of an eigenfrequency condition. In a passive LC resonator, the resonance condition gives a single frequency <math>\omega_0=1/\sqrt{LC}</math>. In the present two-resonator active circuit, the same idea becomes a matrix equation because the two resonators are coupled. The amplitudes <math>\tilde V_A</math> and <math>\tilde V_B</math> are not arbitrary: a nonzero oscillation exists only when the admittance matrix has a zero eigenvalue.

Because the active admittance depends on amplitude, this condition is nonlinear and supports multiple branches. The supplementary material shows that the steady-state frequency can be reduced to a '''real-coefficient cubic polynomial''' in <math>\omega</math>. The existence of up to three real steady-state branches is therefore not accidental; it is the algebraic origin of the experimentally relevant <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM" />

=== Effective nonlinear-Hamiltonian picture ===
The same physics is often rewritten in coupled-mode form. This notation is common in non-Hermitian physics because it makes the circuit look like two coupled modes. The variables <math>\psi_A</math> and <math>\psi_B</math> represent the complex oscillation amplitudes in resonators <math>A</math> and <math>B</math>. The model is

<math display="block">
i\frac{d}{dt}
\begin{pmatrix}
\psi_A\\
\psi_B
\end{pmatrix}
=
\begin{pmatrix}
\omega_A+i\,g(|\psi_A|) & \kappa\\
\kappa & \omega_B-i\ell
\end{pmatrix}
\begin{pmatrix}
\psi_A\\
\psi_B
\end{pmatrix},
</math>

where <math>\omega_A</math> and <math>\omega_B</math> are the effective modal frequencies, <math>\kappa</math> is the coupling strength, <math>\ell</math> is the loss of resonator <math>B</math>, and <math>g(|\psi_A|)</math> is the saturable gain of resonator <math>A</math>. The steady-state problem then becomes

<math display="block">
H_s(\Psi)\Psi = \Omega \Psi,
</math>

with a real oscillation eigenfrequency <math>\Omega=2\pi f</math> for the stable oscillating branch. The word "real" is important: it means the imaginary part has vanished, so the mode is neither being amplified nor decaying in the steady state. In the cited experiment, this physical stable branch and two auxiliary steady branches coalesce at the singular point, producing a third-order nonlinear exceptional point rather than a conventional second-order linear EP. The auxiliary branches help form the cubic-root singularity but are not themselves stable physical outputs of the circuit.<ref name="Bai2024" /><ref name="Bai2024SM" />

== Measurements ==

=== Acquisition protocol and steady-state selection ===
The measurement is performed in the time domain. In the reference experiment, an arbitrary waveform generator first applies an external start-up signal of 1 V at 70 kHz in order to bring the system into the basin of attraction of the stable oscillating state.<ref name="Bai2024" /> In our breadboard measurement, an external voltage is likewise required to activate the resonators. The measured frequency is extracted from the activated steady waveform recorded on the oscilloscope.

The physical meaning of this procedure is important. The oscillation is not assumed to appear without activation; a supplied voltage is needed to bring the resonators into the relevant dynamical state. The measured oscillation frequency is then interpreted as the frequency selected by the nonlinear steady-state condition.

=== Extraction of the <math>f-L_A</math> observable ===
Ref. [1] does not treat the oscilloscope traces merely as qualitative evidence of oscillation. The supplementary explicitly states that the full reference experiment reports the steady-state frequency, the voltage ratio <math>|V_A/V_B|</math>, and the relative phase <math>\theta_A-\theta_B</math>, all extracted from the dominant peaks of the Fourier spectra of <math>V_A</math> and <math>V_B</math>.<ref name="Bai2024SM" />

Our available observable is the steady-state frequency. For each chosen value of <math>L_A</math>, the resonators are voltage-activated, the circuit reaches a stable oscillating state, and the dominant spectral peak is recorded. If we denote the Fourier transforms by

<math display="block">
\tilde V_A(f)=\mathcal{F}\{V_A(t)\},
\qquad
\tilde V_B(f)=\mathcal{F}\{V_B(t)\},
</math>

then the dominant oscillation frequency is obtained from

<math display="block">
f_0=\arg\max_f |\tilde V_A(f)| \approx \arg\max_f |\tilde V_B(f)|.
</math>

The experimental data set is therefore

<math display="block">
\{L_A,\ f_0(L_A)\},
</math>

which is compared with the frequency branch plotted in Fig. S11(a) of SM-4. In the full reference measurement, the amplitude ratio

<math display="block">
\rho = \frac{|\tilde V_A(f_0)|}{|\tilde V_B(f_0)|}
</math>

and relative phase

<math display="block">
\Delta\phi = \arg \tilde V_A(f_0)-\arg \tilde V_B(f_0)
</math>

provide additional state verification. Those quantities are useful but are not the main basis of our breadboard validation.

In practice, the same frequency can also be estimated directly from the time-domain period <math>T</math> using <math>f_0=1/T</math> when the waveform is clean. Fourier analysis is preferred because it uses a longer section of waveform and identifies the dominant oscillation peak even when there is small amplitude noise or waveform distortion.

=== Practical inductance sweep ===
In SM-4, the inductive tuning is carried out by varying <math>L_A</math>. The supplementary notes that the tunable inductor exhibits less than about 0.4% inductance variation over the frequency range of interest, which is sufficiently small for controlled sweeping near the singular point. To further suppress voltage-dependent drift during the sweep, the reference experiment slightly adjusts the active-branch resistance so as to maintain approximately <math>V_B \approx 2~\mathrm{V}</math> throughout the measurement series.<ref name="Bai2024SM" />

Our sweep uses controlled series combinations of small inductors, giving an approximately <math>2~\mu\mathrm{H}</math> step in <math>L_A</math>. The homebrew compressible coil is then used to show that a mechanical change of the sensing element also changes the inductance. The conclusion drawn from our data is therefore not a precision determination of <math>L_{A,\mathrm{NEP}}</math>, but a validation that the breadboard circuit produces the same kind of frequency response as the stable branch in Fig. S11(a).

=== Long-window spectral analysis and noise inspection ===
The supplementary also discusses long-time measurements over a window of <math>0<t<5~\mathrm{s}</math>. The motivation is that the noise around the stable oscillation is too small to be judged reliably from raw scope traces alone, so it is more informative to inspect the Fourier spectra and the extracted spectral center frequency and linewidth.<ref name="Bai2024SM" />

This is a useful methodological point for a sensing-oriented reproduction. In an NEP-based sensor, one ultimately cares about how precisely the frequency peak can be resolved and tracked. A long-window Fourier analysis is therefore closer to the sensing task than a purely visual time-domain inspection.

== Results ==

=== Result display: steady oscillation ===
[[File:PC5271inductive1.png|750px|thumb|center|Breadboard implementation and representative steady oscillation waveform]]

This figure should show the breadboard circuit and a representative voltage waveform after the resonators are activated by the supplied voltage. The displayed waveform is the evidence used to identify a stable oscillation frequency.

=== Result display: <math>f-L_A</math> sweep ===
[[File:PC5271inductive3.jpeg|750px|thumb|center|Measured breadboard <math>f-L_A</math> curve.]]

This figure should show the measured oscillation frequency <math>f</math> as a function of <math>L_A</math>. The values of <math>L_A</math> in the main sweep are produced by connecting small inductors in series with approximately <math>2~\mu\mathrm{H}</math> step size. The reference curve for comparison is the stable frequency branch in Fig. S11(a) of SM-4.<ref name="Bai2024SM" />

=== Result display: iron-rod demonstration ===
[[File:PC5271inductive2.png|750px|thumb|center|Displacement measurement by compressing the solenoid and inserting iron rod into the homebrew coil]]

This figure should show the homebrew coil with an inserted iron rod and the corresponding change in the measured oscillation frequency. The experimental variable is the rod position or insertion state, and the measured output is the activated steady oscillation frequency.

== Evaluation ==

=== Stable activated oscillation ===
The first required experimental signature is that, after the resonators are activated by the supplied voltage, the circuit reaches a stable oscillating waveform instead of decaying immediately or being amplified without bound. A stable waveform means that the TL071 active branch supplies enough small-signal negative resistance to compensate loss, while gain saturation prevents unbounded amplification.

Physically, this behavior confirms the role of the nonlinear active branch. At small amplitude, the effective negative resistance in resonator <math>A</math> overcomes dissipation. At larger amplitude, the gain becomes weaker, so the oscillation amplitude is clamped. The system self-stabilizes because the physical steady branch is the one with zero imaginary part of the eigenfrequency. The auxiliary branches near the nonlinear exceptional point are not stable physical outputs; they are mathematical branches of the nonlinear steady-state equation. The experiment is therefore not a passive ring-down measurement; it is a nonlinear steady-state selection process.

=== Evaluation of the <math>f-L_A</math> trend ===
The main experimental comparison is the measured <math>f-L_A</math> trend against Fig. S11(a) of SM-4. The reference experiment reports the stable frequency branch of the nonlinear circuit as <math>L_A</math> is tuned.<ref name="Bai2024SM" /> This branch is the physical one because its eigenfrequency is real in the steady state. In our breadboard experiment, the same observable is measured using the series-inductor sweep.

It is useful to separate two statements. First, any LC resonator changes frequency when <math>L</math> changes, because <math>f_0=1/(2\pi\sqrt{LC})</math>. This alone would not demonstrate the nonlinear exceptional-point mechanism. Second, in the coupled active-lossy circuit, the frequency is not just the bare LC resonance frequency of one isolated inductor and capacitor. It is the frequency selected by the nonlinear steady-state balance of gain, loss, and coupling. The reason Fig. S11(a) matters is that it shows this selected stable branch as <math>L_A</math> is swept.

Our breadboard circuit uses different component values and a different op-amp from the PCB experiment, so exact numerical overlap is not expected. The meaningful evaluation is whether the measured frequency branch follows the same qualitative <math>f-L_A</math> behavior as the stable branch in SM-4. Observing that behavior is enough to show that the constructed circuit realizes the intended nonlinear voltage-activated oscillating mechanism.

=== Evaluation of the displacement demonstration ===
In the iron-rod demonstration, the homebrew coil becomes a displacement-dependent inductive element. As the rod enters the coil, the effective magnetic permeability inside the coil increases. In the solenoid expression

<math display="block">
L(x) \approx \mu_\mathrm{eff}(x)\frac{N^2 A}{\ell},
</math>

the insertion depth <math>x</math> changes <math>\mu_\mathrm{eff}</math>. The coil geometry <math>N</math>, <math>A</math>, and <math>\ell</math> are approximately fixed during this demonstration, so the dominant effect is the change in magnetic core material inside the coil. Inductive displacement sensors commonly use this principle: the position of a ferromagnetic object changes a coil inductance, and the electronics convert that inductance change into an electrical signal.<ref name="Chen2014" />

In our circuit, the electrical signal is the activated steady oscillation frequency rather than a bridge voltage. The sensing chain is

<math display="block">
\mathrm{rod\ displacement}\ x
\longrightarrow
\mathrm{effective\ inductance}\ L_A(x)
\longrightarrow
\mathrm{oscillation\ frequency}\ f(x).
</math>

This demonstration shows how the inductive readout can be turned into a displacement-sensing readout after calibration of the relation between rod position and frequency.

=== Cubic-root perturbation law and sensing meaning ===
Near the singular operating point, the eigenfrequency shift follows the hallmark cubic-root law,

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3}.
</math>

This law is the main sensing-related result of the reference experiment.<ref name="Bai2024" /><ref name="Bai2024SM" /> Compared with the linear response of an ordinary isolated resonator, the cubic-root dependence can produce a larger frequency displacement for the same small perturbation of inductance when the device is biased sufficiently close to <math>L_{A,\mathrm{NEP}}</math>.

A useful way to interpret the result is through the local derivative,

<math display="block">
\left|\frac{df}{dL_A}\right| \propto |\delta L_A|^{-2/3}.
</math>

This increasing local responsivity is why the circuit is attractive for sensing. Our experiment supports this sensing logic by showing that the breadboard circuit has a measurable stable frequency branch as <math>L_A</math> is changed. A full quantitative exponent extraction would require identifying <math>L_{A,\mathrm{NEP}}</math> and <math>f_{\mathrm{NEP}}</math>, then plotting <math>|f-f_{\mathrm{NEP}}|</math> against <math>|L_A-L_{A,\mathrm{NEP}}|</math> on logarithmic axes.

=== Basis completeness and noise robustness ===
The supplementary material evaluates the Petermann factor as a diagnostic of modal nonorthogonality and basis completeness. For a non-Hermitian eigenproblem, one may write

<math display="block">
\mathrm{PF}_n =
\frac{\langle \Phi_n^{L}|\Phi_n^{L}\rangle \langle \Phi_n^{R}|\Phi_n^{R}\rangle}
{\left|\langle \Phi_n^{L}|\Phi_n^{R}\rangle\right|^2},
</math>

where <math>|\Phi_n^{R}\rangle</math> and <math>|\Phi_n^{L}\rangle</math> are the right and left eigenvectors of the effective steady-state Hamiltonian. At a conventional linear EP, this quantity is associated with diverging excess-noise sensitivity. In Ref. [1], the cited analysis finds that the effective steady-state Hamiltonian remains non-defective and the Petermann factor is finite at the <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM" />

This point explains the significance of the reference experiment. The system does not merely realize a large mathematical responsivity; it realizes that responsivity in a way that avoids the most severe basis-collapse problem of a linear EP. Our breadboard experiment does not directly test basis completeness, but the result provides the theoretical reason why reproducing the stable <math>f-L_A</math> branch is physically meaningful.

=== Overall interpretation ===
In this platform, the measurand is encoded as a perturbation of <math>L_A</math>. Any physical mechanism that modifies the effective inductance of resonator <math>A</math>, for example displacement, magnetic coupling, or the presence of a nearby magnetic target, moves the system away from the operating point and shifts the oscillation frequency. The primary readout in our implementation is therefore spectral: we track <math>f</math> as <math>L_A</math> is changed.

Taken together, Ref. [1] establishes the complete sensing logic: a nonlinear active-lossy circuit is tuned to an <math>\mathrm{NEP}_3</math>; after voltage activation, the system settles onto a stable oscillating branch; the dominant spectral peak is tracked as the perturbation readout; and the enhanced cubic-root response is obtained without the defective-basis pathology of a conventional linear EP. Our breadboard result demonstrates the experimentally accessible core of this logic by reproducing the stable frequency response corresponding to Fig. S11(a) of SM-4.

== References ==
<references />

Inductive Sensors of Ultra-high Sensitivity Based on Nonlinear Exceptional Point

2026-04-23T06:07:32Z

Siyu: /* Result display: f-L_A sweep */

== Introduction ==
This project is a repetition of the experiment reported in ''Observation of Nonlinear Exceptional Points with a Complete Basis in Dynamics'', adapted to our available laboratory equipment.<ref name="Bai2024">K. Bai, T.-R. Liu, L. Fang, J.-Z. Li, C. Lin, D. Wan, and M. Xiao, Observation of nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''132''', 073802 (2024).</ref> The circuit contains two coupled resonators. One resonator has gain, meaning that it can supply energy to the oscillation, while the other has loss, meaning that it dissipates energy. A system with both gain and loss is called a '''non-Hermitian''' system. Such systems can show unusual degeneracies called '''exceptional points''' (EPs), where both the eigenfrequencies and the eigenmodes merge.

Exceptional points are interesting for sensing because the measured frequency can respond very strongly to a small perturbation. In an inductive sensor, the perturbation is a change in inductance. If a small change in inductance produces a large change in oscillation frequency, the circuit can in principle detect small physical changes. However, ordinary linear EPs also suffer from strong eigenmode nonorthogonality and a defective basis, which can increase noise sensitivity and complicate practical sensing.<ref name="Bai2024" />

The system studied in this project belongs to the recently developed class of '''nonlinear exceptional points''' (NEPs). The physical platform is a pair of coupled circuit resonators. Resonator <math>A</math> is made active by a saturable negative-resistance branch, while resonator <math>B</math> is intentionally lossy. Because the gain in resonator <math>A</math> depends on the oscillation amplitude, the steady-state problem is nonlinear and admits multiple self-consistent branches. At the singular operating point reported in Ref. [1], one stable steady branch and two auxiliary steady branches merge into a third-order nonlinear exceptional point, denoted <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM">Supplemental Material for K. Bai, T.-R. Liu, L. Fang, J.-Z. Li, C. Lin, D. Wan, and M. Xiao, Observation of nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''132''', 073802 (2024).</ref>

The experimentally important consequence is that the oscillation eigenfrequency obeys a cubic-root perturbation law near the singularity,

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3},
</math>

where <math>\delta L_A = L_A-L_{A,\mathrm{NEP}}</math> is the inductive detuning from the exceptional operating point. This response is steeper than the ordinary linear dependence of a conventional resonator and is therefore useful for ultra-sensitive inductive sensing.<ref name="Bai2024" /><ref name="Bai2024SM" />

A second and equally important point is that the cited work does not interpret the singularity as a conventional defective linear EP. Instead, the effective steady-state Hamiltonian remains non-defective in dynamics, and the associated Petermann factor stays finite at the operating point. In other words, the experiment realizes enhanced spectral responsivity without inheriting the strongest basis-collapse pathology of a standard linear EP.<ref name="Bai2024" /><ref name="Bai2024SM" />

In this project, our goal is to reproduce the experimental logic of Ref. [1] and its supplementary material under course-lab conditions. Unlike the PCB implementation in the paper, our circuit is assembled on a breadboard and the active branch uses a '''TL071''' operational amplifier. Our main observable is the steady oscillation frequency <math>f</math> as a function of the chosen inductance <math>L_A</math>. We tune <math>L_A</math> in two ways. For the main <math>f-L_A</math> sweep, we join small inductors in series so that the discrete inductance step is about <math>2~\mu\mathrm{H}</math>. In addition, we use a homebrew compressible coil to demonstrate the sensing mechanism directly. Compressing the coil changes its length and turn density while keeping the cross-sectional area approximately unchanged. We therefore use the <math>f-L_A</math> behavior in Fig. S11(a) of the supplementary material as the main experimental benchmark. Reproducing this trend is sufficient for demonstrating that our setup captures the intended nonlinear voltage-activated, self-stabilized oscillating response, although it is not a full reproduction of every diagnostic reported in Ref. [1].<ref name="Bai2024" /><ref name="Bai2024SM" />

== Team Members ==
Wang Peikun — E1538091@u.nus.edu 
Zhu Ziyang — E1583446@u.nus.edu 
Yuan Si-Yu — siyu_yuan@u.nus.edu 
Li Xunyu — xunyu@u.nus.edu

== Background Knowledge ==

=== Ordinary LC and RLC resonators ===
The basic building block of the experiment is an electrical resonator. An ideal LC resonator contains an inductor <math>L</math> and a capacitor <math>C</math>. Energy repeatedly moves between the magnetic field of the inductor and the electric field of the capacitor. The natural angular frequency is

<math display="block">
\omega_0=\frac{1}{\sqrt{LC}},
</math>

or, in ordinary frequency units,

<math display="block">
f_0=\frac{1}{2\pi\sqrt{LC}}.
</math>

This formula is important for the whole project because it explains why changing an inductance changes the measured frequency. A larger <math>L</math> gives a lower resonance frequency, while a smaller <math>L</math> gives a higher resonance frequency.

Real circuits are not ideal. Wires, inductors, capacitors, breadboard contacts, and measuring probes all dissipate energy. This loss is often represented by a resistor <math>R</math>, giving an RLC resonator. In a passive RLC circuit, the oscillation amplitude decays with time because energy is continuously lost as heat. If a circuit is to maintain a steady activated oscillation, the loss must be compensated by a gain element.

=== Gain, loss, and negative resistance ===
In electronics, gain can be represented as an effective negative resistance. A normal positive resistance removes energy from the resonator. A negative resistance does the opposite: it supplies energy to the resonator. In this experiment, resonator <math>A</math> contains an active op-amp branch that behaves like a negative resistance at small amplitude, while resonator <math>B</math> contains ordinary positive loss.

The gain cannot remain negative and constant forever. If it did, any activated oscillation would be amplified without bound until the op-amp clipped or the circuit failed. The important nonlinear ingredient is '''saturable gain''': the active branch gives strong gain at small amplitude but weaker gain at larger amplitude. The stable oscillation amplitude is selected when the supplied gain balances the total loss. This nonlinear steady state is why the experiment is a nonlinear dynamical problem rather than only a passive resonance measurement.

=== What "non-Hermitian" means in this circuit ===
In ordinary conservative physics, a Hermitian Hamiltonian describes a closed system whose total energy is conserved. Its eigenfrequencies are real, and its eigenmodes form a complete orthogonal basis. This is the familiar setting of many quantum-mechanics courses.

A circuit with gain and loss is an open system. Energy can enter through the op-amp branch and leave through resistive loss. The effective dynamical matrix is therefore generally '''non-Hermitian'''. In a non-Hermitian system, eigenfrequencies can be complex:

<math display="block">
\omega=\omega_\mathrm{r}+i\omega_\mathrm{i}.
</math>

The real part <math>\omega_\mathrm{r}</math> gives the oscillation frequency, while the imaginary part <math>\omega_\mathrm{i}</math> describes amplification or decay. A positive imaginary part means the mode is amplified, a negative imaginary part means it decays, and a stable steady oscillation corresponds to zero imaginary part, or equivalently zero net amplification. This is the sense in which the circuit is self-stabilized after voltage activation.

=== Exceptional points ===
In a usual Hermitian system, two modes can have the same eigenfrequency while still remaining two independent modes. This is an ordinary degeneracy. A non-Hermitian system can show a stronger kind of degeneracy called an '''exceptional point''' (EP). At an EP, not only the eigenvalues but also the eigenvectors coalesce. In simple language, two modes collapse into one mode.

This collapse makes EPs interesting for sensing. Near a second-order linear EP, a small perturbation can produce a square-root frequency splitting:

<math display="block">
\Delta f \propto \sqrt{\epsilon},
</math>

where <math>\epsilon</math> is the perturbation. This response is steeper than an ordinary linear response when <math>\epsilon</math> is very small. However, a conventional linear EP also has a serious drawback: the eigenbasis becomes defective, meaning that the system no longer has enough independent eigenvectors. This is associated with strong modal nonorthogonality and enhanced noise sensitivity. The Petermann factor is one common way to quantify this nonorthogonality.

=== Why the nonlinear exceptional point is different ===
The reference paper studies a nonlinear exceptional point. The word "nonlinear" is essential. The circuit does not have a fixed gain coefficient; instead, the gain depends on the oscillation amplitude. The steady state must therefore be solved self-consistently: the frequency, voltage amplitudes, and saturated gain must all agree with one another.

In the theory of Ref. [1], this nonlinear steady-state problem can reduce to a cubic equation for the oscillation frequency. A cubic equation can have a triple root. At the special operating point, one stable physical branch and two auxiliary branches merge, giving a third-order nonlinear exceptional point, written as <math>\mathrm{NEP}_3</math>. The stable physical branch is the one with real oscillation frequency, i.e. without an imaginary part. The auxiliary branches are needed to define the higher-order coalescence mathematically, but they are not physically stable operating states of the circuit. The corresponding frequency response has the form

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3}.
</math>

This cubic-root response is the key sensing signature. It is steeper than both an ordinary linear response and the square-root response of a second-order EP. The important conceptual point of Ref. [1] is that this enhanced response is obtained while the effective steady-state Hamiltonian can still retain a complete basis in dynamics.<ref name="Bai2024" /><ref name="Bai2024SM" />

=== Sensing mechanism ===
The circuit can be understood as a frequency-generating sensor. It converts an inductance change into a frequency change.

The first step is activation. A voltage is supplied to the resonators to excite the circuit into an oscillating state. The active branch in resonator <math>A</math> then supplies energy through an effective negative resistance and amplifies the selected oscillation mode.

The second step is self-stabilization. The gain is saturable, which means it becomes weaker when the voltage amplitude becomes larger. The activated oscillation therefore is not amplified forever. The circuit settles onto the physical steady branch where the imaginary part of the eigenfrequency is zero. At that point, the gain supplied by resonator <math>A</math> balances the loss in the circuit, especially the loss in resonator <math>B</math>.

The third step is frequency selection. Because resonators <math>A</math> and <math>B</math> are coupled, the final oscillation frequency is not simply the resonance frequency of either isolated resonator. It is selected by the whole coupled circuit: the inductors, capacitors, coupling elements, loss, and saturated gain all contribute.

The fourth step is sensing. The inductance <math>L_A</math> controls the bare frequency of resonator <math>A</math>. When <math>L_A</math> changes, the self-consistent frequency selected by the nonlinear circuit also changes. Therefore, tracking the spectral peak <math>f</math> tells us how the effective inductance has changed.

The fifth step is the nonlinear exceptional-point response. Near the nonlinear exceptional point discussed in Ref. [1], the physical steady branch and the two auxiliary branches coalesce in the nonlinear steady-state equation. The frequency shift of the stable branch can then scale as the cube root of the inductance perturbation. This means that a small inductance change can produce a comparatively large frequency change. Our breadboard experiment focuses on verifying the stable <math>f-L_A</math> branch corresponding to Fig. S11(a) of SM-4, which is the experimentally accessible part of this sensing mechanism.

=== Same mechanism in other platforms ===
Several references are included to show that this is a platform-independent sensing mechanism, not only a feature of our breadboard circuit. The common chain is

<math display="block">
\mathrm{physical\ perturbation}
\longrightarrow
\mathrm{parameter\ change}
\longrightarrow
\mathrm{eigenfrequency\ shift\ of\ a\ resonator/cavity\ with\ gain\ and\ loss}
\longrightarrow
\mathrm{higher\ order\ steady\ state\ response}
\longrightarrow
\mathrm{sensing}.
</math>

In our circuit, the physical perturbation is a change in the inductance <math>L_A</math>, and the measured output is the oscillation frequency <math>f</math>. In coupled optical, mechanical, electrical, or hybrid resonators, the perturbed parameter may instead be a cavity detuning, mechanical frequency, phase shift, magnetic-field-dependent transition, or another effective resonator parameter. The shared point is that gain, loss, coupling, and nonlinearity convert the parameter change into a higher-order steady-state spectral response.

The earlier theoretical work on nonlinear exceptional points showed that nonlinear coupled resonators can host higher-order NEPs while retaining a complete dynamical basis.<ref name="Bai2023">K. Bai, J.-Z. Li, T.-R. Liu, L. Fang, D. Wan, and M. Xiao, Nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''130''', 266901 (2023).</ref> The 2024 experiment implemented the same mechanism in coupled electrical resonators and measured the stable nonlinear frequency branch.<ref name="Bai2024" /><ref name="Bai2024SM" /> Silicon micromechanical resonators provide another realization: there, the measured output can be a phase difference, and the nonlinear exceptional-point response can enhance phase-shift sensing.<ref name="Xie2026">Y.-J. Xie, S.-Y. Zhang, M.-N. Zhang, R. Wang, L.-F. Wang, and Q.-A. Huang, Nonlinear exceptional points boost phase-shift sensing in silicon micromechanical resonators, ''Phys. Rev. Applied'' '''25''', 034039 (2026).</ref> Hybrid quantum systems with nonlinear bistability show a related route, where nonlinear transition dynamics can enhance sensing signal-to-noise ratio.<ref name="Wang2026">H. Wang ''et al.'', Exceptional sensitivity near the bistable transition point of a hybrid quantum system, ''Nature Physics'' '''22''', 577-583 (2026).</ref>

These examples support the same mechanism rather than unrelated applications. At the same time, enhanced response alone is not enough to prove better sensing. Noise can shift or weaken the apparent nonlinear exceptional point and can limit the signal-to-noise improvement.<ref name="Zheng2025">X. Zheng and Y. D. Chong, Noise constraints for nonlinear exceptional point sensing, ''Phys. Rev. Lett.'' '''134''', 133801 (2025).</ref> This is why our project makes a careful claim: we demonstrate the frequency-response mechanism on a breadboard, but we do not claim a complete metrological noise advantage.

== Report objectives and scope ==
The purpose of this work is to study the circuit principle behind '''nonlinear-exceptional-point-enhanced inductive sensing''' in a coupled electronic resonator platform. The scope is intentionally narrower than the full experiment reported in Ref. [1]. We are not attempting to redesign the active branch, derive all nonlinear bifurcation theory from first principles, reconstruct the auxiliary steady-state branches, or characterize every parasitic element. Instead, we adopt the framework of Ref. [1] and SM-4 and ask whether the breadboard circuit can reproduce the experimentally accessible frequency response.

The main target of our experiment is the <math>f-L_A</math> curve corresponding to Fig. S11(a) of SM-4. In the reference experiment, the steady-state frequency, voltage ratio <math>|V_A/V_B|</math>, and relative phase <math>\theta_A-\theta_B</math> are all measured as functions of <math>L_A</math>.<ref name="Bai2024SM" /> In our implementation, the frequency-versus-inductance curve is the primary observable. We tune <math>L_A</math> by connecting small inductors in series with approximately <math>2~\mu\mathrm{H}</math> resolution. We also use a homebrew coil in a separate demonstration where mechanical compression or insertion of an iron rod changes the effective inductance. The experimental goal is therefore to verify that the TL071 breadboard circuit settles into a stable oscillating state and that the measured frequency changes with <math>L_A</math> in the same qualitative manner as Fig. S11(a).

From the sensing perspective, the most relevant figure of merit is the local frequency responsivity. If

<math display="block">
|f-f_{\mathrm{NEP}}| = A |\delta L_A|^{1/3},
</math>

then differentiation gives

<math display="block">
S_f \equiv \left|\frac{df}{dL_A}\right| \propto |\delta L_A|^{-2/3}.
</math>

Thus, as the operating point approaches <math>L_{A,\mathrm{NEP}}</math>, the local spectral responsivity increases rapidly. The main practical constraint is that the system must remain on the intended stable branch while measurement noise and slow parameter drift are kept below the scale of the induced frequency shift.

== Setups ==
[[File:Experimental_setup.png|750px|thumb|center|Experimental setup adapted from Ref. [1].]]

=== Circuit topology and component implementation ===
The reference platform consists of two coupled RLC resonators. Both resonators contain the same nominal shunt capacitance <math>C_0</math>, while the inductances <math>L_A</math> and <math>L_B</math> set their bare resonance frequencies. The two resonators are coupled through a parallel resistor-capacitor channel characterized by <math>R_c</math> and <math>C_c</math>. Resonator <math>B</math> contains an explicit positive resistance <math>R_B</math> and therefore plays the role of the lossy resonator. Resonator <math>A</math>, by contrast, contains an active branch whose effective resistance is negative at small amplitude and becomes less negative as the oscillation amplitude increases, thereby providing saturable gain.<ref name="Bai2024" /><ref name="Bai2024SM" />

Our implementation follows this topology but is built on a breadboard rather than a custom PCB. The active branch is implemented with an op-amp negative-resistance circuit using a '''TL071''' operational amplifier. This is an important practical difference from the supplementary experiment, where the active components explicitly listed are a TI '''LM7171''' operational amplifier and an '''Onsemi BAV99L''' diode.<ref name="Bai2024SM" /> The TL071 substitution and the breadboard layout introduce different bandwidth limits, wiring parasitics, and contact resistances, so the numerical operating point is not expected to match the PCB experiment. For this reason, our comparison emphasizes the observable trend of <math>f</math> versus <math>L_A</math>.

The supplementary material reports the following representative component values for the PCB experiment:

<math display="block">
C_0 = 18.5~\mathrm{nF}, \qquad
C_c = 3.9~\mathrm{nF}, \qquad
R_c = 760~\Omega,
</math>

<math display="block">
L_B = 197.7~\mu\mathrm{H}, \qquad
R_B = 1314~\Omega.
</math>

A representative operating point shown in the main figure uses

<math display="block">
L_A = 245.4~\mu\mathrm{H}.
</math>

Our breadboard circuit follows the same functional topology, but its component values and parasitic elements are not identical to the PCB values above. The horizontal axis of our main <math>f-L_A</math> plot is therefore the series-combined <math>L_A</math> selected from our available small inductors, while the measured frequency <math>f</math> is the robust experimental readout.

=== Inductance tuning methods ===
We use two complementary methods to change the inductance seen by resonator <math>A</math>.

The first method is used for the main <math>f-L_A</math> sweep. Small inductors are connected in series so that the total inductance changes in discrete steps of approximately <math>2~\mu\mathrm{H}</math>. If the inductors are sufficiently separated so that their mutual magnetic coupling is small, the total series inductance is approximately the sum of the individual inductances,

<math display="block">
L_A \approx L_{\mathrm{base}}+\sum_i L_i .
</math>

This method is simple and reproducible. Adding one small inductor increases <math>L_A</math>; removing it decreases <math>L_A</math>. The circuit frequency is then measured for each chosen value. This produces a discrete version of the <math>f-L_A</math> curve in Fig. S11(a) of SM-4. The step size is not as fine as a commercial variable inductor, but <math>2~\mu\mathrm{H}</math> spacing is small enough to observe the frequency trend over the accessible range.

The second method is a homebrew compressible variable inductor. This coil is useful because it makes the sensing mechanism visible: a mechanical displacement changes the coil geometry, which changes the inductance, which then changes the oscillation frequency. The basic idea can be understood from the approximate solenoid formula

<math display="block">
L \approx \mu_\mathrm{eff}\frac{N^2 A}{\ell},
</math>

where <math>N</math> is the number of turns, <math>A</math> is the cross-sectional area, <math>\ell</math> is the coil length, and <math>\mu_\mathrm{eff}</math> describes the effective magnetic permeability of the core and surrounding region.<ref name="Chen2014">S.-C. Chen, D.-K. Le, and V.-S. Nguyen, Inductive displacement sensors with a notch filter for an active magnetic bearing system, ''Sensors'' '''14''', 12640-12657 (2014).</ref> During compression, the number of turns <math>N</math> is unchanged and the cross-sectional area <math>A</math> remains approximately unchanged, but the length <math>\ell</math> decreases. Equivalently, the turn density

<math display="block">
n=\frac{N}{\ell}
</math>

increases. The magnetic field produced by a solenoid is proportional to turn density. Therefore, compressing the coil increases the magnetic flux linkage through the turns and increases the inductance. Stretching the coil has the opposite effect.

This mechanical tuning is not identical to the precision tunable inductor used in the reference experiment. A real compressed coil is not an ideal long solenoid: the spacing between turns is finite, the end effects change as the coil length changes, nearby wires and the breadboard add parasitic capacitance, and the leads contribute additional series inductance and resistance. For this reason, the homebrew inductor is best treated as an experimentally characterized tuning element rather than an ideal component whose value is known exactly from geometry. In the present project, it provides a direct mechanical demonstration of how displacement can perturb <math>L_A</math>.

Experimentally, we measured the effective inductance of this homebrew variable inductor at different compression states. The compression process changes the coil length and wire density while leaving the coil area approximately fixed. Therefore, the tuning mechanism is physically the same as changing <math>\ell</math> and <math>n=N/\ell</math> in the solenoid model, rather than changing the number of turns or the coil diameter.

=== Kirchhoff model of the nonlinear circuit ===
A convenient state-space description uses the node voltages <math>V_A(t)</math> and <math>V_B(t)</math> together with the inductor currents <math>I_A(t)</math> and <math>I_B(t)</math>. A node voltage is the voltage measured at one resonator node relative to ground. Kirchhoff's current law says that the algebraic sum of all currents flowing away from a node must be zero. This gives a direct way to translate the circuit diagram into differential equations.

Denoting the amplitude-dependent conductance of the active branch by <math>G_A(|V_A|)</math>, where <math>G_A<0</math> in the small-signal regime, the nodal equations can be written as

<math display="block">
\begin{aligned}
C_0 \frac{dV_A}{dt} + I_A + G_A(|V_A|)\,V_A + \frac{V_A-V_B}{R_c}
+ C_c\frac{d(V_A-V_B)}{dt} &= 0,\\
C_0 \frac{dV_B}{dt} + I_B + \frac{V_B}{R_B} + \frac{V_B-V_A}{R_c}
+ C_c\frac{d(V_B-V_A)}{dt} &= 0,\\
L_A \frac{dI_A}{dt} &= V_A,\\
L_B \frac{dI_B}{dt} &= V_B.
\end{aligned}
</math>

Each term has a simple circuit meaning. The terms <math>C_0 dV_A/dt</math> and <math>C_0 dV_B/dt</math> are capacitor currents. The terms <math>I_A</math> and <math>I_B</math> are currents through the inductors. The term <math>G_A(|V_A|)V_A</math> is the active branch current; because <math>G_A</math> can be negative, this branch can inject energy instead of dissipating it. The term <math>V_B/R_B</math> is the ordinary loss current through the resistor in resonator <math>B</math>. The terms involving <math>R_c</math> and <math>C_c</math> describe current flowing between resonators <math>A</math> and <math>B</math>, so they are the electrical coupling between the two resonators.

These equations show explicitly how the experiment differs from a linear EP circuit. If <math>G_A</math> were a constant negative conductance, the model would reduce to a linear non-Hermitian dimer. Here, however, <math>G_A</math> depends on the voltage amplitude, so the steady state must be determined self-consistently together with the oscillation amplitude itself.

=== Harmonic-balance formulation of the steady state ===
After the transient has died away, the oscilloscope waveform is approximately periodic. The simplest approximation is therefore to keep only the dominant sinusoidal component. This is called a harmonic-balance approximation. For a stable activated oscillating state, one writes

<math display="block">
V_j(t)=\mathrm{Re}\!\left[\tilde V_j e^{i\omega t}\right],
\qquad j=A,B.
</math>

Substituting this ansatz into the circuit equations yields a nonlinear algebraic problem of the form

<math display="block">
\mathbf{Y}(\omega,|\tilde V_A|)
\begin{pmatrix}
\tilde V_A\\
\tilde V_B
\end{pmatrix}
=
\begin{pmatrix}
Y_A(\omega,|\tilde V_A|)+Y_c(\omega) & -Y_c(\omega)\\
-Y_c(\omega) & Y_B(\omega)+Y_c(\omega)
\end{pmatrix}
\begin{pmatrix}
\tilde V_A\\
\tilde V_B
\end{pmatrix}
=0,
</math>

with

<math display="block">
Y_A(\omega,|\tilde V_A|)=G_A(|\tilde V_A|)+i\omega C_0+\frac{1}{i\omega L_A},
</math>

<math display="block">
Y_B(\omega)=\frac{1}{R_B}+i\omega C_0+\frac{1}{i\omega L_B},
\qquad
Y_c(\omega)=\frac{1}{R_c}+i\omega C_c.
</math>

A nontrivial solution requires

<math display="block">
\det \mathbf{Y}(\omega,|\tilde V_A|)=0.
</math>

This determinant condition is the circuit analogue of an eigenfrequency condition. In a passive LC resonator, the resonance condition gives a single frequency <math>\omega_0=1/\sqrt{LC}</math>. In the present two-resonator active circuit, the same idea becomes a matrix equation because the two resonators are coupled. The amplitudes <math>\tilde V_A</math> and <math>\tilde V_B</math> are not arbitrary: a nonzero oscillation exists only when the admittance matrix has a zero eigenvalue.

Because the active admittance depends on amplitude, this condition is nonlinear and supports multiple branches. The supplementary material shows that the steady-state frequency can be reduced to a '''real-coefficient cubic polynomial''' in <math>\omega</math>. The existence of up to three real steady-state branches is therefore not accidental; it is the algebraic origin of the experimentally relevant <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM" />

=== Effective nonlinear-Hamiltonian picture ===
The same physics is often rewritten in coupled-mode form. This notation is common in non-Hermitian physics because it makes the circuit look like two coupled modes. The variables <math>\psi_A</math> and <math>\psi_B</math> represent the complex oscillation amplitudes in resonators <math>A</math> and <math>B</math>. The model is

<math display="block">
i\frac{d}{dt}
\begin{pmatrix}
\psi_A\\
\psi_B
\end{pmatrix}
=
\begin{pmatrix}
\omega_A+i\,g(|\psi_A|) & \kappa\\
\kappa & \omega_B-i\ell
\end{pmatrix}
\begin{pmatrix}
\psi_A\\
\psi_B
\end{pmatrix},
</math>

where <math>\omega_A</math> and <math>\omega_B</math> are the effective modal frequencies, <math>\kappa</math> is the coupling strength, <math>\ell</math> is the loss of resonator <math>B</math>, and <math>g(|\psi_A|)</math> is the saturable gain of resonator <math>A</math>. The steady-state problem then becomes

<math display="block">
H_s(\Psi)\Psi = \Omega \Psi,
</math>

with a real oscillation eigenfrequency <math>\Omega=2\pi f</math> for the stable oscillating branch. The word "real" is important: it means the imaginary part has vanished, so the mode is neither being amplified nor decaying in the steady state. In the cited experiment, this physical stable branch and two auxiliary steady branches coalesce at the singular point, producing a third-order nonlinear exceptional point rather than a conventional second-order linear EP. The auxiliary branches help form the cubic-root singularity but are not themselves stable physical outputs of the circuit.<ref name="Bai2024" /><ref name="Bai2024SM" />

== Measurements ==

=== Acquisition protocol and steady-state selection ===
The measurement is performed in the time domain. In the reference experiment, an arbitrary waveform generator first applies an external start-up signal of 1 V at 70 kHz in order to bring the system into the basin of attraction of the stable oscillating state.<ref name="Bai2024" /> In our breadboard measurement, an external voltage is likewise required to activate the resonators. The measured frequency is extracted from the activated steady waveform recorded on the oscilloscope.

The physical meaning of this procedure is important. The oscillation is not assumed to appear without activation; a supplied voltage is needed to bring the resonators into the relevant dynamical state. The measured oscillation frequency is then interpreted as the frequency selected by the nonlinear steady-state condition.

=== Extraction of the <math>f-L_A</math> observable ===
Ref. [1] does not treat the oscilloscope traces merely as qualitative evidence of oscillation. The supplementary explicitly states that the full reference experiment reports the steady-state frequency, the voltage ratio <math>|V_A/V_B|</math>, and the relative phase <math>\theta_A-\theta_B</math>, all extracted from the dominant peaks of the Fourier spectra of <math>V_A</math> and <math>V_B</math>.<ref name="Bai2024SM" />

Our available observable is the steady-state frequency. For each chosen value of <math>L_A</math>, the resonators are voltage-activated, the circuit reaches a stable oscillating state, and the dominant spectral peak is recorded. If we denote the Fourier transforms by

<math display="block">
\tilde V_A(f)=\mathcal{F}\{V_A(t)\},
\qquad
\tilde V_B(f)=\mathcal{F}\{V_B(t)\},
</math>

then the dominant oscillation frequency is obtained from

<math display="block">
f_0=\arg\max_f |\tilde V_A(f)| \approx \arg\max_f |\tilde V_B(f)|.
</math>

The experimental data set is therefore

<math display="block">
\{L_A,\ f_0(L_A)\},
</math>

which is compared with the frequency branch plotted in Fig. S11(a) of SM-4. In the full reference measurement, the amplitude ratio

<math display="block">
\rho = \frac{|\tilde V_A(f_0)|}{|\tilde V_B(f_0)|}
</math>

and relative phase

<math display="block">
\Delta\phi = \arg \tilde V_A(f_0)-\arg \tilde V_B(f_0)
</math>

provide additional state verification. Those quantities are useful but are not the main basis of our breadboard validation.

In practice, the same frequency can also be estimated directly from the time-domain period <math>T</math> using <math>f_0=1/T</math> when the waveform is clean. Fourier analysis is preferred because it uses a longer section of waveform and identifies the dominant oscillation peak even when there is small amplitude noise or waveform distortion.

=== Practical inductance sweep ===
In SM-4, the inductive tuning is carried out by varying <math>L_A</math>. The supplementary notes that the tunable inductor exhibits less than about 0.4% inductance variation over the frequency range of interest, which is sufficiently small for controlled sweeping near the singular point. To further suppress voltage-dependent drift during the sweep, the reference experiment slightly adjusts the active-branch resistance so as to maintain approximately <math>V_B \approx 2~\mathrm{V}</math> throughout the measurement series.<ref name="Bai2024SM" />

Our sweep uses controlled series combinations of small inductors, giving an approximately <math>2~\mu\mathrm{H}</math> step in <math>L_A</math>. The homebrew compressible coil is then used to show that a mechanical change of the sensing element also changes the inductance. The conclusion drawn from our data is therefore not a precision determination of <math>L_{A,\mathrm{NEP}}</math>, but a validation that the breadboard circuit produces the same kind of frequency response as the stable branch in Fig. S11(a).

=== Long-window spectral analysis and noise inspection ===
The supplementary also discusses long-time measurements over a window of <math>0<t<5~\mathrm{s}</math>. The motivation is that the noise around the stable oscillation is too small to be judged reliably from raw scope traces alone, so it is more informative to inspect the Fourier spectra and the extracted spectral center frequency and linewidth.<ref name="Bai2024SM" />

This is a useful methodological point for a sensing-oriented reproduction. In an NEP-based sensor, one ultimately cares about how precisely the frequency peak can be resolved and tracked. A long-window Fourier analysis is therefore closer to the sensing task than a purely visual time-domain inspection.

== Results ==

=== Result display: steady oscillation ===
[[File:PC5271inductive1.png|750px|thumb|center|Breadboard implementation and representative steady oscillation waveform]]

This figure should show the breadboard circuit and a representative voltage waveform after the resonators are activated by the supplied voltage. The displayed waveform is the evidence used to identify a stable oscillation frequency.

=== Result display: <math>f-L_A</math> sweep ===
[[File:PC5271inductive3.jpeg|750px|thumb|center|Measured breadboard <math>f-L_A</math> curve.]]

This figure should show the measured oscillation frequency <math>f</math> as a function of <math>L_A</math>. The values of <math>L_A</math> in the main sweep are produced by connecting small inductors in series with approximately <math>2~\mu\mathrm{H}</math> step size. The reference curve for comparison is the stable frequency branch in Fig. S11(a) of SM-4.<ref name="Bai2024SM" />

=== Result display: iron-rod demonstration ===
[[File:PC5271inductive2.png|750px|thumb|center|Displacement measurement by compressing the solenoid and inserting iron rod into the homebrew coil]]

This figure should show the homebrew coil with an inserted iron rod and the corresponding change in the measured oscillation frequency. The experimental variable is the rod position or insertion state, and the measured output is the activated steady oscillation frequency.

== Evaluation ==

=== Stable activated oscillation ===
The first required experimental signature is that, after the resonators are activated by the supplied voltage, the circuit reaches a stable oscillating waveform instead of decaying immediately or being amplified without bound. A stable waveform means that the TL071 active branch supplies enough small-signal negative resistance to compensate loss, while gain saturation prevents unbounded amplification.

Physically, this behavior confirms the role of the nonlinear active branch. At small amplitude, the effective negative resistance in resonator <math>A</math> overcomes dissipation. At larger amplitude, the gain becomes weaker, so the oscillation amplitude is clamped. The system self-stabilizes because the physical steady branch is the one with zero imaginary part of the eigenfrequency. The auxiliary branches near the nonlinear exceptional point are not stable physical outputs; they are mathematical branches of the nonlinear steady-state equation. The experiment is therefore not a passive ring-down measurement; it is a nonlinear steady-state selection process.

=== Evaluation of the <math>f-L_A</math> trend ===
The main experimental comparison is the measured <math>f-L_A</math> trend against Fig. S11(a) of SM-4. The reference experiment reports the stable frequency branch of the nonlinear circuit as <math>L_A</math> is tuned.<ref name="Bai2024SM" /> This branch is the physical one because its eigenfrequency is real in the steady state. In our breadboard experiment, the same observable is measured using the series-inductor sweep.

It is useful to separate two statements. First, any LC resonator changes frequency when <math>L</math> changes, because <math>f_0=1/(2\pi\sqrt{LC})</math>. This alone would not demonstrate the nonlinear exceptional-point mechanism. Second, in the coupled active-lossy circuit, the frequency is not just the bare LC resonance frequency of one isolated inductor and capacitor. It is the frequency selected by the nonlinear steady-state balance of gain, loss, and coupling. The reason Fig. S11(a) matters is that it shows this selected stable branch as <math>L_A</math> is swept.

Our breadboard circuit uses different component values and a different op-amp from the PCB experiment, so exact numerical overlap is not expected. The meaningful evaluation is whether the measured frequency branch follows the same qualitative <math>f-L_A</math> behavior as the stable branch in SM-4. Observing that behavior is enough to show that the constructed circuit realizes the intended nonlinear voltage-activated oscillating mechanism.

=== Evaluation of the displacement demonstration ===
In the iron-rod demonstration, the homebrew coil becomes a displacement-dependent inductive element. As the rod enters the coil, the effective magnetic permeability inside the coil increases. In the solenoid expression

<math display="block">
L(x) \approx \mu_\mathrm{eff}(x)\frac{N^2 A}{\ell},
</math>

the insertion depth <math>x</math> changes <math>\mu_\mathrm{eff}</math>. The coil geometry <math>N</math>, <math>A</math>, and <math>\ell</math> are approximately fixed during this demonstration, so the dominant effect is the change in magnetic core material inside the coil. Inductive displacement sensors commonly use this principle: the position of a ferromagnetic object changes a coil inductance, and the electronics convert that inductance change into an electrical signal.<ref name="Chen2014" />

In our circuit, the electrical signal is the activated steady oscillation frequency rather than a bridge voltage. The sensing chain is

<math display="block">
\mathrm{rod\ displacement}\ x
\longrightarrow
\mathrm{effective\ inductance}\ L_A(x)
\longrightarrow
\mathrm{oscillation\ frequency}\ f(x).
</math>

This demonstration shows how the inductive readout can be turned into a displacement-sensing readout after calibration of the relation between rod position and frequency.

=== Cubic-root perturbation law and sensing meaning ===
Near the singular operating point, the eigenfrequency shift follows the hallmark cubic-root law,

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3}.
</math>

This law is the main sensing-related result of the reference experiment.<ref name="Bai2024" /><ref name="Bai2024SM" /> Compared with the linear response of an ordinary isolated resonator, the cubic-root dependence can produce a larger frequency displacement for the same small perturbation of inductance when the device is biased sufficiently close to <math>L_{A,\mathrm{NEP}}</math>.

A useful way to interpret the result is through the local derivative,

<math display="block">
\left|\frac{df}{dL_A}\right| \propto |\delta L_A|^{-2/3}.
</math>

This increasing local responsivity is why the circuit is attractive for sensing. Our experiment supports this sensing logic by showing that the breadboard circuit has a measurable stable frequency branch as <math>L_A</math> is changed. A full quantitative exponent extraction would require identifying <math>L_{A,\mathrm{NEP}}</math> and <math>f_{\mathrm{NEP}}</math>, then plotting <math>|f-f_{\mathrm{NEP}}|</math> against <math>|L_A-L_{A,\mathrm{NEP}}|</math> on logarithmic axes.

=== Basis completeness and noise robustness ===
The supplementary material evaluates the Petermann factor as a diagnostic of modal nonorthogonality and basis completeness. For a non-Hermitian eigenproblem, one may write

<math display="block">
\mathrm{PF}_n =
\frac{\langle \Phi_n^{L}|\Phi_n^{L}\rangle \langle \Phi_n^{R}|\Phi_n^{R}\rangle}
{\left|\langle \Phi_n^{L}|\Phi_n^{R}\rangle\right|^2},
</math>

where <math>|\Phi_n^{R}\rangle</math> and <math>|\Phi_n^{L}\rangle</math> are the right and left eigenvectors of the effective steady-state Hamiltonian. At a conventional linear EP, this quantity is associated with diverging excess-noise sensitivity. In Ref. [1], the cited analysis finds that the effective steady-state Hamiltonian remains non-defective and the Petermann factor is finite at the <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM" />

This point explains the significance of the reference experiment. The system does not merely realize a large mathematical responsivity; it realizes that responsivity in a way that avoids the most severe basis-collapse problem of a linear EP. Our breadboard experiment does not directly test basis completeness, but the result provides the theoretical reason why reproducing the stable <math>f-L_A</math> branch is physically meaningful.

=== Overall interpretation ===
In this platform, the measurand is encoded as a perturbation of <math>L_A</math>. Any physical mechanism that modifies the effective inductance of resonator <math>A</math>, for example displacement, magnetic coupling, or the presence of a nearby magnetic target, moves the system away from the operating point and shifts the oscillation frequency. The primary readout in our implementation is therefore spectral: we track <math>f</math> as <math>L_A</math> is changed.

Taken together, Ref. [1] establishes the complete sensing logic: a nonlinear active-lossy circuit is tuned to an <math>\mathrm{NEP}_3</math>; after voltage activation, the system settles onto a stable oscillating branch; the dominant spectral peak is tracked as the perturbation readout; and the enhanced cubic-root response is obtained without the defective-basis pathology of a conventional linear EP. Our breadboard result demonstrates the experimentally accessible core of this logic by reproducing the stable frequency response corresponding to Fig. S11(a) of SM-4.

== References ==
<references />

Inductive Sensors of Ultra-high Sensitivity Based on Nonlinear Exceptional Point

2026-04-23T06:06:37Z

Siyu: /* Result display: f-L_A sweep */

== Introduction ==
This project is a repetition of the experiment reported in ''Observation of Nonlinear Exceptional Points with a Complete Basis in Dynamics'', adapted to our available laboratory equipment.<ref name="Bai2024">K. Bai, T.-R. Liu, L. Fang, J.-Z. Li, C. Lin, D. Wan, and M. Xiao, Observation of nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''132''', 073802 (2024).</ref> The circuit contains two coupled resonators. One resonator has gain, meaning that it can supply energy to the oscillation, while the other has loss, meaning that it dissipates energy. A system with both gain and loss is called a '''non-Hermitian''' system. Such systems can show unusual degeneracies called '''exceptional points''' (EPs), where both the eigenfrequencies and the eigenmodes merge.

Exceptional points are interesting for sensing because the measured frequency can respond very strongly to a small perturbation. In an inductive sensor, the perturbation is a change in inductance. If a small change in inductance produces a large change in oscillation frequency, the circuit can in principle detect small physical changes. However, ordinary linear EPs also suffer from strong eigenmode nonorthogonality and a defective basis, which can increase noise sensitivity and complicate practical sensing.<ref name="Bai2024" />

The system studied in this project belongs to the recently developed class of '''nonlinear exceptional points''' (NEPs). The physical platform is a pair of coupled circuit resonators. Resonator <math>A</math> is made active by a saturable negative-resistance branch, while resonator <math>B</math> is intentionally lossy. Because the gain in resonator <math>A</math> depends on the oscillation amplitude, the steady-state problem is nonlinear and admits multiple self-consistent branches. At the singular operating point reported in Ref. [1], one stable steady branch and two auxiliary steady branches merge into a third-order nonlinear exceptional point, denoted <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM">Supplemental Material for K. Bai, T.-R. Liu, L. Fang, J.-Z. Li, C. Lin, D. Wan, and M. Xiao, Observation of nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''132''', 073802 (2024).</ref>

The experimentally important consequence is that the oscillation eigenfrequency obeys a cubic-root perturbation law near the singularity,

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3},
</math>

where <math>\delta L_A = L_A-L_{A,\mathrm{NEP}}</math> is the inductive detuning from the exceptional operating point. This response is steeper than the ordinary linear dependence of a conventional resonator and is therefore useful for ultra-sensitive inductive sensing.<ref name="Bai2024" /><ref name="Bai2024SM" />

A second and equally important point is that the cited work does not interpret the singularity as a conventional defective linear EP. Instead, the effective steady-state Hamiltonian remains non-defective in dynamics, and the associated Petermann factor stays finite at the operating point. In other words, the experiment realizes enhanced spectral responsivity without inheriting the strongest basis-collapse pathology of a standard linear EP.<ref name="Bai2024" /><ref name="Bai2024SM" />

In this project, our goal is to reproduce the experimental logic of Ref. [1] and its supplementary material under course-lab conditions. Unlike the PCB implementation in the paper, our circuit is assembled on a breadboard and the active branch uses a '''TL071''' operational amplifier. Our main observable is the steady oscillation frequency <math>f</math> as a function of the chosen inductance <math>L_A</math>. We tune <math>L_A</math> in two ways. For the main <math>f-L_A</math> sweep, we join small inductors in series so that the discrete inductance step is about <math>2~\mu\mathrm{H}</math>. In addition, we use a homebrew compressible coil to demonstrate the sensing mechanism directly. Compressing the coil changes its length and turn density while keeping the cross-sectional area approximately unchanged. We therefore use the <math>f-L_A</math> behavior in Fig. S11(a) of the supplementary material as the main experimental benchmark. Reproducing this trend is sufficient for demonstrating that our setup captures the intended nonlinear voltage-activated, self-stabilized oscillating response, although it is not a full reproduction of every diagnostic reported in Ref. [1].<ref name="Bai2024" /><ref name="Bai2024SM" />

== Team Members ==
Wang Peikun — E1538091@u.nus.edu 
Zhu Ziyang — E1583446@u.nus.edu 
Yuan Si-Yu — siyu_yuan@u.nus.edu 
Li Xunyu — xunyu@u.nus.edu

== Background Knowledge ==

=== Ordinary LC and RLC resonators ===
The basic building block of the experiment is an electrical resonator. An ideal LC resonator contains an inductor <math>L</math> and a capacitor <math>C</math>. Energy repeatedly moves between the magnetic field of the inductor and the electric field of the capacitor. The natural angular frequency is

<math display="block">
\omega_0=\frac{1}{\sqrt{LC}},
</math>

or, in ordinary frequency units,

<math display="block">
f_0=\frac{1}{2\pi\sqrt{LC}}.
</math>

This formula is important for the whole project because it explains why changing an inductance changes the measured frequency. A larger <math>L</math> gives a lower resonance frequency, while a smaller <math>L</math> gives a higher resonance frequency.

Real circuits are not ideal. Wires, inductors, capacitors, breadboard contacts, and measuring probes all dissipate energy. This loss is often represented by a resistor <math>R</math>, giving an RLC resonator. In a passive RLC circuit, the oscillation amplitude decays with time because energy is continuously lost as heat. If a circuit is to maintain a steady activated oscillation, the loss must be compensated by a gain element.

=== Gain, loss, and negative resistance ===
In electronics, gain can be represented as an effective negative resistance. A normal positive resistance removes energy from the resonator. A negative resistance does the opposite: it supplies energy to the resonator. In this experiment, resonator <math>A</math> contains an active op-amp branch that behaves like a negative resistance at small amplitude, while resonator <math>B</math> contains ordinary positive loss.

The gain cannot remain negative and constant forever. If it did, any activated oscillation would be amplified without bound until the op-amp clipped or the circuit failed. The important nonlinear ingredient is '''saturable gain''': the active branch gives strong gain at small amplitude but weaker gain at larger amplitude. The stable oscillation amplitude is selected when the supplied gain balances the total loss. This nonlinear steady state is why the experiment is a nonlinear dynamical problem rather than only a passive resonance measurement.

=== What "non-Hermitian" means in this circuit ===
In ordinary conservative physics, a Hermitian Hamiltonian describes a closed system whose total energy is conserved. Its eigenfrequencies are real, and its eigenmodes form a complete orthogonal basis. This is the familiar setting of many quantum-mechanics courses.

A circuit with gain and loss is an open system. Energy can enter through the op-amp branch and leave through resistive loss. The effective dynamical matrix is therefore generally '''non-Hermitian'''. In a non-Hermitian system, eigenfrequencies can be complex:

<math display="block">
\omega=\omega_\mathrm{r}+i\omega_\mathrm{i}.
</math>

The real part <math>\omega_\mathrm{r}</math> gives the oscillation frequency, while the imaginary part <math>\omega_\mathrm{i}</math> describes amplification or decay. A positive imaginary part means the mode is amplified, a negative imaginary part means it decays, and a stable steady oscillation corresponds to zero imaginary part, or equivalently zero net amplification. This is the sense in which the circuit is self-stabilized after voltage activation.

=== Exceptional points ===
In a usual Hermitian system, two modes can have the same eigenfrequency while still remaining two independent modes. This is an ordinary degeneracy. A non-Hermitian system can show a stronger kind of degeneracy called an '''exceptional point''' (EP). At an EP, not only the eigenvalues but also the eigenvectors coalesce. In simple language, two modes collapse into one mode.

This collapse makes EPs interesting for sensing. Near a second-order linear EP, a small perturbation can produce a square-root frequency splitting:

<math display="block">
\Delta f \propto \sqrt{\epsilon},
</math>

where <math>\epsilon</math> is the perturbation. This response is steeper than an ordinary linear response when <math>\epsilon</math> is very small. However, a conventional linear EP also has a serious drawback: the eigenbasis becomes defective, meaning that the system no longer has enough independent eigenvectors. This is associated with strong modal nonorthogonality and enhanced noise sensitivity. The Petermann factor is one common way to quantify this nonorthogonality.

=== Why the nonlinear exceptional point is different ===
The reference paper studies a nonlinear exceptional point. The word "nonlinear" is essential. The circuit does not have a fixed gain coefficient; instead, the gain depends on the oscillation amplitude. The steady state must therefore be solved self-consistently: the frequency, voltage amplitudes, and saturated gain must all agree with one another.

In the theory of Ref. [1], this nonlinear steady-state problem can reduce to a cubic equation for the oscillation frequency. A cubic equation can have a triple root. At the special operating point, one stable physical branch and two auxiliary branches merge, giving a third-order nonlinear exceptional point, written as <math>\mathrm{NEP}_3</math>. The stable physical branch is the one with real oscillation frequency, i.e. without an imaginary part. The auxiliary branches are needed to define the higher-order coalescence mathematically, but they are not physically stable operating states of the circuit. The corresponding frequency response has the form

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3}.
</math>

This cubic-root response is the key sensing signature. It is steeper than both an ordinary linear response and the square-root response of a second-order EP. The important conceptual point of Ref. [1] is that this enhanced response is obtained while the effective steady-state Hamiltonian can still retain a complete basis in dynamics.<ref name="Bai2024" /><ref name="Bai2024SM" />

=== Sensing mechanism ===
The circuit can be understood as a frequency-generating sensor. It converts an inductance change into a frequency change.

The first step is activation. A voltage is supplied to the resonators to excite the circuit into an oscillating state. The active branch in resonator <math>A</math> then supplies energy through an effective negative resistance and amplifies the selected oscillation mode.

The second step is self-stabilization. The gain is saturable, which means it becomes weaker when the voltage amplitude becomes larger. The activated oscillation therefore is not amplified forever. The circuit settles onto the physical steady branch where the imaginary part of the eigenfrequency is zero. At that point, the gain supplied by resonator <math>A</math> balances the loss in the circuit, especially the loss in resonator <math>B</math>.

The third step is frequency selection. Because resonators <math>A</math> and <math>B</math> are coupled, the final oscillation frequency is not simply the resonance frequency of either isolated resonator. It is selected by the whole coupled circuit: the inductors, capacitors, coupling elements, loss, and saturated gain all contribute.

The fourth step is sensing. The inductance <math>L_A</math> controls the bare frequency of resonator <math>A</math>. When <math>L_A</math> changes, the self-consistent frequency selected by the nonlinear circuit also changes. Therefore, tracking the spectral peak <math>f</math> tells us how the effective inductance has changed.

The fifth step is the nonlinear exceptional-point response. Near the nonlinear exceptional point discussed in Ref. [1], the physical steady branch and the two auxiliary branches coalesce in the nonlinear steady-state equation. The frequency shift of the stable branch can then scale as the cube root of the inductance perturbation. This means that a small inductance change can produce a comparatively large frequency change. Our breadboard experiment focuses on verifying the stable <math>f-L_A</math> branch corresponding to Fig. S11(a) of SM-4, which is the experimentally accessible part of this sensing mechanism.

=== Same mechanism in other platforms ===
Several references are included to show that this is a platform-independent sensing mechanism, not only a feature of our breadboard circuit. The common chain is

<math display="block">
\mathrm{physical\ perturbation}
\longrightarrow
\mathrm{parameter\ change}
\longrightarrow
\mathrm{eigenfrequency\ shift\ of\ a\ resonator/cavity\ with\ gain\ and\ loss}
\longrightarrow
\mathrm{higher\ order\ steady\ state\ response}
\longrightarrow
\mathrm{sensing}.
</math>

In our circuit, the physical perturbation is a change in the inductance <math>L_A</math>, and the measured output is the oscillation frequency <math>f</math>. In coupled optical, mechanical, electrical, or hybrid resonators, the perturbed parameter may instead be a cavity detuning, mechanical frequency, phase shift, magnetic-field-dependent transition, or another effective resonator parameter. The shared point is that gain, loss, coupling, and nonlinearity convert the parameter change into a higher-order steady-state spectral response.

The earlier theoretical work on nonlinear exceptional points showed that nonlinear coupled resonators can host higher-order NEPs while retaining a complete dynamical basis.<ref name="Bai2023">K. Bai, J.-Z. Li, T.-R. Liu, L. Fang, D. Wan, and M. Xiao, Nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''130''', 266901 (2023).</ref> The 2024 experiment implemented the same mechanism in coupled electrical resonators and measured the stable nonlinear frequency branch.<ref name="Bai2024" /><ref name="Bai2024SM" /> Silicon micromechanical resonators provide another realization: there, the measured output can be a phase difference, and the nonlinear exceptional-point response can enhance phase-shift sensing.<ref name="Xie2026">Y.-J. Xie, S.-Y. Zhang, M.-N. Zhang, R. Wang, L.-F. Wang, and Q.-A. Huang, Nonlinear exceptional points boost phase-shift sensing in silicon micromechanical resonators, ''Phys. Rev. Applied'' '''25''', 034039 (2026).</ref> Hybrid quantum systems with nonlinear bistability show a related route, where nonlinear transition dynamics can enhance sensing signal-to-noise ratio.<ref name="Wang2026">H. Wang ''et al.'', Exceptional sensitivity near the bistable transition point of a hybrid quantum system, ''Nature Physics'' '''22''', 577-583 (2026).</ref>

These examples support the same mechanism rather than unrelated applications. At the same time, enhanced response alone is not enough to prove better sensing. Noise can shift or weaken the apparent nonlinear exceptional point and can limit the signal-to-noise improvement.<ref name="Zheng2025">X. Zheng and Y. D. Chong, Noise constraints for nonlinear exceptional point sensing, ''Phys. Rev. Lett.'' '''134''', 133801 (2025).</ref> This is why our project makes a careful claim: we demonstrate the frequency-response mechanism on a breadboard, but we do not claim a complete metrological noise advantage.

== Report objectives and scope ==
The purpose of this work is to study the circuit principle behind '''nonlinear-exceptional-point-enhanced inductive sensing''' in a coupled electronic resonator platform. The scope is intentionally narrower than the full experiment reported in Ref. [1]. We are not attempting to redesign the active branch, derive all nonlinear bifurcation theory from first principles, reconstruct the auxiliary steady-state branches, or characterize every parasitic element. Instead, we adopt the framework of Ref. [1] and SM-4 and ask whether the breadboard circuit can reproduce the experimentally accessible frequency response.

The main target of our experiment is the <math>f-L_A</math> curve corresponding to Fig. S11(a) of SM-4. In the reference experiment, the steady-state frequency, voltage ratio <math>|V_A/V_B|</math>, and relative phase <math>\theta_A-\theta_B</math> are all measured as functions of <math>L_A</math>.<ref name="Bai2024SM" /> In our implementation, the frequency-versus-inductance curve is the primary observable. We tune <math>L_A</math> by connecting small inductors in series with approximately <math>2~\mu\mathrm{H}</math> resolution. We also use a homebrew coil in a separate demonstration where mechanical compression or insertion of an iron rod changes the effective inductance. The experimental goal is therefore to verify that the TL071 breadboard circuit settles into a stable oscillating state and that the measured frequency changes with <math>L_A</math> in the same qualitative manner as Fig. S11(a).

From the sensing perspective, the most relevant figure of merit is the local frequency responsivity. If

<math display="block">
|f-f_{\mathrm{NEP}}| = A |\delta L_A|^{1/3},
</math>

then differentiation gives

<math display="block">
S_f \equiv \left|\frac{df}{dL_A}\right| \propto |\delta L_A|^{-2/3}.
</math>

Thus, as the operating point approaches <math>L_{A,\mathrm{NEP}}</math>, the local spectral responsivity increases rapidly. The main practical constraint is that the system must remain on the intended stable branch while measurement noise and slow parameter drift are kept below the scale of the induced frequency shift.

== Setups ==
[[File:Experimental_setup.png|750px|thumb|center|Experimental setup adapted from Ref. [1].]]

=== Circuit topology and component implementation ===
The reference platform consists of two coupled RLC resonators. Both resonators contain the same nominal shunt capacitance <math>C_0</math>, while the inductances <math>L_A</math> and <math>L_B</math> set their bare resonance frequencies. The two resonators are coupled through a parallel resistor-capacitor channel characterized by <math>R_c</math> and <math>C_c</math>. Resonator <math>B</math> contains an explicit positive resistance <math>R_B</math> and therefore plays the role of the lossy resonator. Resonator <math>A</math>, by contrast, contains an active branch whose effective resistance is negative at small amplitude and becomes less negative as the oscillation amplitude increases, thereby providing saturable gain.<ref name="Bai2024" /><ref name="Bai2024SM" />

Our implementation follows this topology but is built on a breadboard rather than a custom PCB. The active branch is implemented with an op-amp negative-resistance circuit using a '''TL071''' operational amplifier. This is an important practical difference from the supplementary experiment, where the active components explicitly listed are a TI '''LM7171''' operational amplifier and an '''Onsemi BAV99L''' diode.<ref name="Bai2024SM" /> The TL071 substitution and the breadboard layout introduce different bandwidth limits, wiring parasitics, and contact resistances, so the numerical operating point is not expected to match the PCB experiment. For this reason, our comparison emphasizes the observable trend of <math>f</math> versus <math>L_A</math>.

The supplementary material reports the following representative component values for the PCB experiment:

<math display="block">
C_0 = 18.5~\mathrm{nF}, \qquad
C_c = 3.9~\mathrm{nF}, \qquad
R_c = 760~\Omega,
</math>

<math display="block">
L_B = 197.7~\mu\mathrm{H}, \qquad
R_B = 1314~\Omega.
</math>

A representative operating point shown in the main figure uses

<math display="block">
L_A = 245.4~\mu\mathrm{H}.
</math>

Our breadboard circuit follows the same functional topology, but its component values and parasitic elements are not identical to the PCB values above. The horizontal axis of our main <math>f-L_A</math> plot is therefore the series-combined <math>L_A</math> selected from our available small inductors, while the measured frequency <math>f</math> is the robust experimental readout.

=== Inductance tuning methods ===
We use two complementary methods to change the inductance seen by resonator <math>A</math>.

The first method is used for the main <math>f-L_A</math> sweep. Small inductors are connected in series so that the total inductance changes in discrete steps of approximately <math>2~\mu\mathrm{H}</math>. If the inductors are sufficiently separated so that their mutual magnetic coupling is small, the total series inductance is approximately the sum of the individual inductances,

<math display="block">
L_A \approx L_{\mathrm{base}}+\sum_i L_i .
</math>

This method is simple and reproducible. Adding one small inductor increases <math>L_A</math>; removing it decreases <math>L_A</math>. The circuit frequency is then measured for each chosen value. This produces a discrete version of the <math>f-L_A</math> curve in Fig. S11(a) of SM-4. The step size is not as fine as a commercial variable inductor, but <math>2~\mu\mathrm{H}</math> spacing is small enough to observe the frequency trend over the accessible range.

The second method is a homebrew compressible variable inductor. This coil is useful because it makes the sensing mechanism visible: a mechanical displacement changes the coil geometry, which changes the inductance, which then changes the oscillation frequency. The basic idea can be understood from the approximate solenoid formula

<math display="block">
L \approx \mu_\mathrm{eff}\frac{N^2 A}{\ell},
</math>

where <math>N</math> is the number of turns, <math>A</math> is the cross-sectional area, <math>\ell</math> is the coil length, and <math>\mu_\mathrm{eff}</math> describes the effective magnetic permeability of the core and surrounding region.<ref name="Chen2014">S.-C. Chen, D.-K. Le, and V.-S. Nguyen, Inductive displacement sensors with a notch filter for an active magnetic bearing system, ''Sensors'' '''14''', 12640-12657 (2014).</ref> During compression, the number of turns <math>N</math> is unchanged and the cross-sectional area <math>A</math> remains approximately unchanged, but the length <math>\ell</math> decreases. Equivalently, the turn density

<math display="block">
n=\frac{N}{\ell}
</math>

increases. The magnetic field produced by a solenoid is proportional to turn density. Therefore, compressing the coil increases the magnetic flux linkage through the turns and increases the inductance. Stretching the coil has the opposite effect.

This mechanical tuning is not identical to the precision tunable inductor used in the reference experiment. A real compressed coil is not an ideal long solenoid: the spacing between turns is finite, the end effects change as the coil length changes, nearby wires and the breadboard add parasitic capacitance, and the leads contribute additional series inductance and resistance. For this reason, the homebrew inductor is best treated as an experimentally characterized tuning element rather than an ideal component whose value is known exactly from geometry. In the present project, it provides a direct mechanical demonstration of how displacement can perturb <math>L_A</math>.

Experimentally, we measured the effective inductance of this homebrew variable inductor at different compression states. The compression process changes the coil length and wire density while leaving the coil area approximately fixed. Therefore, the tuning mechanism is physically the same as changing <math>\ell</math> and <math>n=N/\ell</math> in the solenoid model, rather than changing the number of turns or the coil diameter.

=== Kirchhoff model of the nonlinear circuit ===
A convenient state-space description uses the node voltages <math>V_A(t)</math> and <math>V_B(t)</math> together with the inductor currents <math>I_A(t)</math> and <math>I_B(t)</math>. A node voltage is the voltage measured at one resonator node relative to ground. Kirchhoff's current law says that the algebraic sum of all currents flowing away from a node must be zero. This gives a direct way to translate the circuit diagram into differential equations.

Denoting the amplitude-dependent conductance of the active branch by <math>G_A(|V_A|)</math>, where <math>G_A<0</math> in the small-signal regime, the nodal equations can be written as

<math display="block">
\begin{aligned}
C_0 \frac{dV_A}{dt} + I_A + G_A(|V_A|)\,V_A + \frac{V_A-V_B}{R_c}
+ C_c\frac{d(V_A-V_B)}{dt} &= 0,\\
C_0 \frac{dV_B}{dt} + I_B + \frac{V_B}{R_B} + \frac{V_B-V_A}{R_c}
+ C_c\frac{d(V_B-V_A)}{dt} &= 0,\\
L_A \frac{dI_A}{dt} &= V_A,\\
L_B \frac{dI_B}{dt} &= V_B.
\end{aligned}
</math>

Each term has a simple circuit meaning. The terms <math>C_0 dV_A/dt</math> and <math>C_0 dV_B/dt</math> are capacitor currents. The terms <math>I_A</math> and <math>I_B</math> are currents through the inductors. The term <math>G_A(|V_A|)V_A</math> is the active branch current; because <math>G_A</math> can be negative, this branch can inject energy instead of dissipating it. The term <math>V_B/R_B</math> is the ordinary loss current through the resistor in resonator <math>B</math>. The terms involving <math>R_c</math> and <math>C_c</math> describe current flowing between resonators <math>A</math> and <math>B</math>, so they are the electrical coupling between the two resonators.

These equations show explicitly how the experiment differs from a linear EP circuit. If <math>G_A</math> were a constant negative conductance, the model would reduce to a linear non-Hermitian dimer. Here, however, <math>G_A</math> depends on the voltage amplitude, so the steady state must be determined self-consistently together with the oscillation amplitude itself.

=== Harmonic-balance formulation of the steady state ===
After the transient has died away, the oscilloscope waveform is approximately periodic. The simplest approximation is therefore to keep only the dominant sinusoidal component. This is called a harmonic-balance approximation. For a stable activated oscillating state, one writes

<math display="block">
V_j(t)=\mathrm{Re}\!\left[\tilde V_j e^{i\omega t}\right],
\qquad j=A,B.
</math>

Substituting this ansatz into the circuit equations yields a nonlinear algebraic problem of the form

<math display="block">
\mathbf{Y}(\omega,|\tilde V_A|)
\begin{pmatrix}
\tilde V_A\\
\tilde V_B
\end{pmatrix}
=
\begin{pmatrix}
Y_A(\omega,|\tilde V_A|)+Y_c(\omega) & -Y_c(\omega)\\
-Y_c(\omega) & Y_B(\omega)+Y_c(\omega)
\end{pmatrix}
\begin{pmatrix}
\tilde V_A\\
\tilde V_B
\end{pmatrix}
=0,
</math>

with

<math display="block">
Y_A(\omega,|\tilde V_A|)=G_A(|\tilde V_A|)+i\omega C_0+\frac{1}{i\omega L_A},
</math>

<math display="block">
Y_B(\omega)=\frac{1}{R_B}+i\omega C_0+\frac{1}{i\omega L_B},
\qquad
Y_c(\omega)=\frac{1}{R_c}+i\omega C_c.
</math>

A nontrivial solution requires

<math display="block">
\det \mathbf{Y}(\omega,|\tilde V_A|)=0.
</math>

This determinant condition is the circuit analogue of an eigenfrequency condition. In a passive LC resonator, the resonance condition gives a single frequency <math>\omega_0=1/\sqrt{LC}</math>. In the present two-resonator active circuit, the same idea becomes a matrix equation because the two resonators are coupled. The amplitudes <math>\tilde V_A</math> and <math>\tilde V_B</math> are not arbitrary: a nonzero oscillation exists only when the admittance matrix has a zero eigenvalue.

Because the active admittance depends on amplitude, this condition is nonlinear and supports multiple branches. The supplementary material shows that the steady-state frequency can be reduced to a '''real-coefficient cubic polynomial''' in <math>\omega</math>. The existence of up to three real steady-state branches is therefore not accidental; it is the algebraic origin of the experimentally relevant <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM" />

=== Effective nonlinear-Hamiltonian picture ===
The same physics is often rewritten in coupled-mode form. This notation is common in non-Hermitian physics because it makes the circuit look like two coupled modes. The variables <math>\psi_A</math> and <math>\psi_B</math> represent the complex oscillation amplitudes in resonators <math>A</math> and <math>B</math>. The model is

<math display="block">
i\frac{d}{dt}
\begin{pmatrix}
\psi_A\\
\psi_B
\end{pmatrix}
=
\begin{pmatrix}
\omega_A+i\,g(|\psi_A|) & \kappa\\
\kappa & \omega_B-i\ell
\end{pmatrix}
\begin{pmatrix}
\psi_A\\
\psi_B
\end{pmatrix},
</math>

where <math>\omega_A</math> and <math>\omega_B</math> are the effective modal frequencies, <math>\kappa</math> is the coupling strength, <math>\ell</math> is the loss of resonator <math>B</math>, and <math>g(|\psi_A|)</math> is the saturable gain of resonator <math>A</math>. The steady-state problem then becomes

<math display="block">
H_s(\Psi)\Psi = \Omega \Psi,
</math>

with a real oscillation eigenfrequency <math>\Omega=2\pi f</math> for the stable oscillating branch. The word "real" is important: it means the imaginary part has vanished, so the mode is neither being amplified nor decaying in the steady state. In the cited experiment, this physical stable branch and two auxiliary steady branches coalesce at the singular point, producing a third-order nonlinear exceptional point rather than a conventional second-order linear EP. The auxiliary branches help form the cubic-root singularity but are not themselves stable physical outputs of the circuit.<ref name="Bai2024" /><ref name="Bai2024SM" />

== Measurements ==

=== Acquisition protocol and steady-state selection ===
The measurement is performed in the time domain. In the reference experiment, an arbitrary waveform generator first applies an external start-up signal of 1 V at 70 kHz in order to bring the system into the basin of attraction of the stable oscillating state.<ref name="Bai2024" /> In our breadboard measurement, an external voltage is likewise required to activate the resonators. The measured frequency is extracted from the activated steady waveform recorded on the oscilloscope.

The physical meaning of this procedure is important. The oscillation is not assumed to appear without activation; a supplied voltage is needed to bring the resonators into the relevant dynamical state. The measured oscillation frequency is then interpreted as the frequency selected by the nonlinear steady-state condition.

=== Extraction of the <math>f-L_A</math> observable ===
Ref. [1] does not treat the oscilloscope traces merely as qualitative evidence of oscillation. The supplementary explicitly states that the full reference experiment reports the steady-state frequency, the voltage ratio <math>|V_A/V_B|</math>, and the relative phase <math>\theta_A-\theta_B</math>, all extracted from the dominant peaks of the Fourier spectra of <math>V_A</math> and <math>V_B</math>.<ref name="Bai2024SM" />

Our available observable is the steady-state frequency. For each chosen value of <math>L_A</math>, the resonators are voltage-activated, the circuit reaches a stable oscillating state, and the dominant spectral peak is recorded. If we denote the Fourier transforms by

<math display="block">
\tilde V_A(f)=\mathcal{F}\{V_A(t)\},
\qquad
\tilde V_B(f)=\mathcal{F}\{V_B(t)\},
</math>

then the dominant oscillation frequency is obtained from

<math display="block">
f_0=\arg\max_f |\tilde V_A(f)| \approx \arg\max_f |\tilde V_B(f)|.
</math>

The experimental data set is therefore

<math display="block">
\{L_A,\ f_0(L_A)\},
</math>

which is compared with the frequency branch plotted in Fig. S11(a) of SM-4. In the full reference measurement, the amplitude ratio

<math display="block">
\rho = \frac{|\tilde V_A(f_0)|}{|\tilde V_B(f_0)|}
</math>

and relative phase

<math display="block">
\Delta\phi = \arg \tilde V_A(f_0)-\arg \tilde V_B(f_0)
</math>

provide additional state verification. Those quantities are useful but are not the main basis of our breadboard validation.

In practice, the same frequency can also be estimated directly from the time-domain period <math>T</math> using <math>f_0=1/T</math> when the waveform is clean. Fourier analysis is preferred because it uses a longer section of waveform and identifies the dominant oscillation peak even when there is small amplitude noise or waveform distortion.

=== Practical inductance sweep ===
In SM-4, the inductive tuning is carried out by varying <math>L_A</math>. The supplementary notes that the tunable inductor exhibits less than about 0.4% inductance variation over the frequency range of interest, which is sufficiently small for controlled sweeping near the singular point. To further suppress voltage-dependent drift during the sweep, the reference experiment slightly adjusts the active-branch resistance so as to maintain approximately <math>V_B \approx 2~\mathrm{V}</math> throughout the measurement series.<ref name="Bai2024SM" />

Our sweep uses controlled series combinations of small inductors, giving an approximately <math>2~\mu\mathrm{H}</math> step in <math>L_A</math>. The homebrew compressible coil is then used to show that a mechanical change of the sensing element also changes the inductance. The conclusion drawn from our data is therefore not a precision determination of <math>L_{A,\mathrm{NEP}}</math>, but a validation that the breadboard circuit produces the same kind of frequency response as the stable branch in Fig. S11(a).

=== Long-window spectral analysis and noise inspection ===
The supplementary also discusses long-time measurements over a window of <math>0<t<5~\mathrm{s}</math>. The motivation is that the noise around the stable oscillation is too small to be judged reliably from raw scope traces alone, so it is more informative to inspect the Fourier spectra and the extracted spectral center frequency and linewidth.<ref name="Bai2024SM" />

This is a useful methodological point for a sensing-oriented reproduction. In an NEP-based sensor, one ultimately cares about how precisely the frequency peak can be resolved and tracked. A long-window Fourier analysis is therefore closer to the sensing task than a purely visual time-domain inspection.

== Results ==

=== Result display: steady oscillation ===
[[File:PC5271inductive1.png|750px|thumb|center|Breadboard implementation and representative steady oscillation waveform]]

This figure should show the breadboard circuit and a representative voltage waveform after the resonators are activated by the supplied voltage. The displayed waveform is the evidence used to identify a stable oscillation frequency.

=== Result display: <math>f-L_A</math> sweep ===
[[File:PC5271inductive3.png|750px|thumb|center|Measured breadboard <math>f-L_A</math> curve.]]

This figure should show the measured oscillation frequency <math>f</math> as a function of <math>L_A</math>. The values of <math>L_A</math> in the main sweep are produced by connecting small inductors in series with approximately <math>2~\mu\mathrm{H}</math> step size. The reference curve for comparison is the stable frequency branch in Fig. S11(a) of SM-4.<ref name="Bai2024SM" />

=== Result display: iron-rod demonstration ===
[[File:PC5271inductive2.png|750px|thumb|center|Displacement measurement by compressing the solenoid and inserting iron rod into the homebrew coil]]

This figure should show the homebrew coil with an inserted iron rod and the corresponding change in the measured oscillation frequency. The experimental variable is the rod position or insertion state, and the measured output is the activated steady oscillation frequency.

== Evaluation ==

=== Stable activated oscillation ===
The first required experimental signature is that, after the resonators are activated by the supplied voltage, the circuit reaches a stable oscillating waveform instead of decaying immediately or being amplified without bound. A stable waveform means that the TL071 active branch supplies enough small-signal negative resistance to compensate loss, while gain saturation prevents unbounded amplification.

Physically, this behavior confirms the role of the nonlinear active branch. At small amplitude, the effective negative resistance in resonator <math>A</math> overcomes dissipation. At larger amplitude, the gain becomes weaker, so the oscillation amplitude is clamped. The system self-stabilizes because the physical steady branch is the one with zero imaginary part of the eigenfrequency. The auxiliary branches near the nonlinear exceptional point are not stable physical outputs; they are mathematical branches of the nonlinear steady-state equation. The experiment is therefore not a passive ring-down measurement; it is a nonlinear steady-state selection process.

=== Evaluation of the <math>f-L_A</math> trend ===
The main experimental comparison is the measured <math>f-L_A</math> trend against Fig. S11(a) of SM-4. The reference experiment reports the stable frequency branch of the nonlinear circuit as <math>L_A</math> is tuned.<ref name="Bai2024SM" /> This branch is the physical one because its eigenfrequency is real in the steady state. In our breadboard experiment, the same observable is measured using the series-inductor sweep.

It is useful to separate two statements. First, any LC resonator changes frequency when <math>L</math> changes, because <math>f_0=1/(2\pi\sqrt{LC})</math>. This alone would not demonstrate the nonlinear exceptional-point mechanism. Second, in the coupled active-lossy circuit, the frequency is not just the bare LC resonance frequency of one isolated inductor and capacitor. It is the frequency selected by the nonlinear steady-state balance of gain, loss, and coupling. The reason Fig. S11(a) matters is that it shows this selected stable branch as <math>L_A</math> is swept.

Our breadboard circuit uses different component values and a different op-amp from the PCB experiment, so exact numerical overlap is not expected. The meaningful evaluation is whether the measured frequency branch follows the same qualitative <math>f-L_A</math> behavior as the stable branch in SM-4. Observing that behavior is enough to show that the constructed circuit realizes the intended nonlinear voltage-activated oscillating mechanism.

=== Evaluation of the displacement demonstration ===
In the iron-rod demonstration, the homebrew coil becomes a displacement-dependent inductive element. As the rod enters the coil, the effective magnetic permeability inside the coil increases. In the solenoid expression

<math display="block">
L(x) \approx \mu_\mathrm{eff}(x)\frac{N^2 A}{\ell},
</math>

the insertion depth <math>x</math> changes <math>\mu_\mathrm{eff}</math>. The coil geometry <math>N</math>, <math>A</math>, and <math>\ell</math> are approximately fixed during this demonstration, so the dominant effect is the change in magnetic core material inside the coil. Inductive displacement sensors commonly use this principle: the position of a ferromagnetic object changes a coil inductance, and the electronics convert that inductance change into an electrical signal.<ref name="Chen2014" />

In our circuit, the electrical signal is the activated steady oscillation frequency rather than a bridge voltage. The sensing chain is

<math display="block">
\mathrm{rod\ displacement}\ x
\longrightarrow
\mathrm{effective\ inductance}\ L_A(x)
\longrightarrow
\mathrm{oscillation\ frequency}\ f(x).
</math>

This demonstration shows how the inductive readout can be turned into a displacement-sensing readout after calibration of the relation between rod position and frequency.

=== Cubic-root perturbation law and sensing meaning ===
Near the singular operating point, the eigenfrequency shift follows the hallmark cubic-root law,

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3}.
</math>

This law is the main sensing-related result of the reference experiment.<ref name="Bai2024" /><ref name="Bai2024SM" /> Compared with the linear response of an ordinary isolated resonator, the cubic-root dependence can produce a larger frequency displacement for the same small perturbation of inductance when the device is biased sufficiently close to <math>L_{A,\mathrm{NEP}}</math>.

A useful way to interpret the result is through the local derivative,

<math display="block">
\left|\frac{df}{dL_A}\right| \propto |\delta L_A|^{-2/3}.
</math>

This increasing local responsivity is why the circuit is attractive for sensing. Our experiment supports this sensing logic by showing that the breadboard circuit has a measurable stable frequency branch as <math>L_A</math> is changed. A full quantitative exponent extraction would require identifying <math>L_{A,\mathrm{NEP}}</math> and <math>f_{\mathrm{NEP}}</math>, then plotting <math>|f-f_{\mathrm{NEP}}|</math> against <math>|L_A-L_{A,\mathrm{NEP}}|</math> on logarithmic axes.

=== Basis completeness and noise robustness ===
The supplementary material evaluates the Petermann factor as a diagnostic of modal nonorthogonality and basis completeness. For a non-Hermitian eigenproblem, one may write

<math display="block">
\mathrm{PF}_n =
\frac{\langle \Phi_n^{L}|\Phi_n^{L}\rangle \langle \Phi_n^{R}|\Phi_n^{R}\rangle}
{\left|\langle \Phi_n^{L}|\Phi_n^{R}\rangle\right|^2},
</math>

where <math>|\Phi_n^{R}\rangle</math> and <math>|\Phi_n^{L}\rangle</math> are the right and left eigenvectors of the effective steady-state Hamiltonian. At a conventional linear EP, this quantity is associated with diverging excess-noise sensitivity. In Ref. [1], the cited analysis finds that the effective steady-state Hamiltonian remains non-defective and the Petermann factor is finite at the <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM" />

This point explains the significance of the reference experiment. The system does not merely realize a large mathematical responsivity; it realizes that responsivity in a way that avoids the most severe basis-collapse problem of a linear EP. Our breadboard experiment does not directly test basis completeness, but the result provides the theoretical reason why reproducing the stable <math>f-L_A</math> branch is physically meaningful.

=== Overall interpretation ===
In this platform, the measurand is encoded as a perturbation of <math>L_A</math>. Any physical mechanism that modifies the effective inductance of resonator <math>A</math>, for example displacement, magnetic coupling, or the presence of a nearby magnetic target, moves the system away from the operating point and shifts the oscillation frequency. The primary readout in our implementation is therefore spectral: we track <math>f</math> as <math>L_A</math> is changed.

Taken together, Ref. [1] establishes the complete sensing logic: a nonlinear active-lossy circuit is tuned to an <math>\mathrm{NEP}_3</math>; after voltage activation, the system settles onto a stable oscillating branch; the dominant spectral peak is tracked as the perturbation readout; and the enhanced cubic-root response is obtained without the defective-basis pathology of a conventional linear EP. Our breadboard result demonstrates the experimentally accessible core of this logic by reproducing the stable frequency response corresponding to Fig. S11(a) of SM-4.

== References ==
<references />

File:PC5271inductive3.jpeg

2026-04-23T06:05:40Z

Siyu:

Inductive Sensors of Ultra-high Sensitivity Based on Nonlinear Exceptional Point

2026-04-23T06:01:44Z

Siyu: /* Same mechanism in other platforms */

== Introduction ==
This project is a repetition of the experiment reported in ''Observation of Nonlinear Exceptional Points with a Complete Basis in Dynamics'', adapted to our available laboratory equipment.<ref name="Bai2024">K. Bai, T.-R. Liu, L. Fang, J.-Z. Li, C. Lin, D. Wan, and M. Xiao, Observation of nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''132''', 073802 (2024).</ref> The circuit contains two coupled resonators. One resonator has gain, meaning that it can supply energy to the oscillation, while the other has loss, meaning that it dissipates energy. A system with both gain and loss is called a '''non-Hermitian''' system. Such systems can show unusual degeneracies called '''exceptional points''' (EPs), where both the eigenfrequencies and the eigenmodes merge.

Exceptional points are interesting for sensing because the measured frequency can respond very strongly to a small perturbation. In an inductive sensor, the perturbation is a change in inductance. If a small change in inductance produces a large change in oscillation frequency, the circuit can in principle detect small physical changes. However, ordinary linear EPs also suffer from strong eigenmode nonorthogonality and a defective basis, which can increase noise sensitivity and complicate practical sensing.<ref name="Bai2024" />

The system studied in this project belongs to the recently developed class of '''nonlinear exceptional points''' (NEPs). The physical platform is a pair of coupled circuit resonators. Resonator <math>A</math> is made active by a saturable negative-resistance branch, while resonator <math>B</math> is intentionally lossy. Because the gain in resonator <math>A</math> depends on the oscillation amplitude, the steady-state problem is nonlinear and admits multiple self-consistent branches. At the singular operating point reported in Ref. [1], one stable steady branch and two auxiliary steady branches merge into a third-order nonlinear exceptional point, denoted <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM">Supplemental Material for K. Bai, T.-R. Liu, L. Fang, J.-Z. Li, C. Lin, D. Wan, and M. Xiao, Observation of nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''132''', 073802 (2024).</ref>

The experimentally important consequence is that the oscillation eigenfrequency obeys a cubic-root perturbation law near the singularity,

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3},
</math>

where <math>\delta L_A = L_A-L_{A,\mathrm{NEP}}</math> is the inductive detuning from the exceptional operating point. This response is steeper than the ordinary linear dependence of a conventional resonator and is therefore useful for ultra-sensitive inductive sensing.<ref name="Bai2024" /><ref name="Bai2024SM" />

A second and equally important point is that the cited work does not interpret the singularity as a conventional defective linear EP. Instead, the effective steady-state Hamiltonian remains non-defective in dynamics, and the associated Petermann factor stays finite at the operating point. In other words, the experiment realizes enhanced spectral responsivity without inheriting the strongest basis-collapse pathology of a standard linear EP.<ref name="Bai2024" /><ref name="Bai2024SM" />

In this project, our goal is to reproduce the experimental logic of Ref. [1] and its supplementary material under course-lab conditions. Unlike the PCB implementation in the paper, our circuit is assembled on a breadboard and the active branch uses a '''TL071''' operational amplifier. Our main observable is the steady oscillation frequency <math>f</math> as a function of the chosen inductance <math>L_A</math>. We tune <math>L_A</math> in two ways. For the main <math>f-L_A</math> sweep, we join small inductors in series so that the discrete inductance step is about <math>2~\mu\mathrm{H}</math>. In addition, we use a homebrew compressible coil to demonstrate the sensing mechanism directly. Compressing the coil changes its length and turn density while keeping the cross-sectional area approximately unchanged. We therefore use the <math>f-L_A</math> behavior in Fig. S11(a) of the supplementary material as the main experimental benchmark. Reproducing this trend is sufficient for demonstrating that our setup captures the intended nonlinear voltage-activated, self-stabilized oscillating response, although it is not a full reproduction of every diagnostic reported in Ref. [1].<ref name="Bai2024" /><ref name="Bai2024SM" />

== Team Members ==
Wang Peikun — E1538091@u.nus.edu 
Zhu Ziyang — E1583446@u.nus.edu 
Yuan Si-Yu — siyu_yuan@u.nus.edu 
Li Xunyu — xunyu@u.nus.edu

== Background Knowledge ==

=== Ordinary LC and RLC resonators ===
The basic building block of the experiment is an electrical resonator. An ideal LC resonator contains an inductor <math>L</math> and a capacitor <math>C</math>. Energy repeatedly moves between the magnetic field of the inductor and the electric field of the capacitor. The natural angular frequency is

<math display="block">
\omega_0=\frac{1}{\sqrt{LC}},
</math>

or, in ordinary frequency units,

<math display="block">
f_0=\frac{1}{2\pi\sqrt{LC}}.
</math>

This formula is important for the whole project because it explains why changing an inductance changes the measured frequency. A larger <math>L</math> gives a lower resonance frequency, while a smaller <math>L</math> gives a higher resonance frequency.

Real circuits are not ideal. Wires, inductors, capacitors, breadboard contacts, and measuring probes all dissipate energy. This loss is often represented by a resistor <math>R</math>, giving an RLC resonator. In a passive RLC circuit, the oscillation amplitude decays with time because energy is continuously lost as heat. If a circuit is to maintain a steady activated oscillation, the loss must be compensated by a gain element.

=== Gain, loss, and negative resistance ===
In electronics, gain can be represented as an effective negative resistance. A normal positive resistance removes energy from the resonator. A negative resistance does the opposite: it supplies energy to the resonator. In this experiment, resonator <math>A</math> contains an active op-amp branch that behaves like a negative resistance at small amplitude, while resonator <math>B</math> contains ordinary positive loss.

The gain cannot remain negative and constant forever. If it did, any activated oscillation would be amplified without bound until the op-amp clipped or the circuit failed. The important nonlinear ingredient is '''saturable gain''': the active branch gives strong gain at small amplitude but weaker gain at larger amplitude. The stable oscillation amplitude is selected when the supplied gain balances the total loss. This nonlinear steady state is why the experiment is a nonlinear dynamical problem rather than only a passive resonance measurement.

=== What "non-Hermitian" means in this circuit ===
In ordinary conservative physics, a Hermitian Hamiltonian describes a closed system whose total energy is conserved. Its eigenfrequencies are real, and its eigenmodes form a complete orthogonal basis. This is the familiar setting of many quantum-mechanics courses.

A circuit with gain and loss is an open system. Energy can enter through the op-amp branch and leave through resistive loss. The effective dynamical matrix is therefore generally '''non-Hermitian'''. In a non-Hermitian system, eigenfrequencies can be complex:

<math display="block">
\omega=\omega_\mathrm{r}+i\omega_\mathrm{i}.
</math>

The real part <math>\omega_\mathrm{r}</math> gives the oscillation frequency, while the imaginary part <math>\omega_\mathrm{i}</math> describes amplification or decay. A positive imaginary part means the mode is amplified, a negative imaginary part means it decays, and a stable steady oscillation corresponds to zero imaginary part, or equivalently zero net amplification. This is the sense in which the circuit is self-stabilized after voltage activation.

=== Exceptional points ===
In a usual Hermitian system, two modes can have the same eigenfrequency while still remaining two independent modes. This is an ordinary degeneracy. A non-Hermitian system can show a stronger kind of degeneracy called an '''exceptional point''' (EP). At an EP, not only the eigenvalues but also the eigenvectors coalesce. In simple language, two modes collapse into one mode.

This collapse makes EPs interesting for sensing. Near a second-order linear EP, a small perturbation can produce a square-root frequency splitting:

<math display="block">
\Delta f \propto \sqrt{\epsilon},
</math>

where <math>\epsilon</math> is the perturbation. This response is steeper than an ordinary linear response when <math>\epsilon</math> is very small. However, a conventional linear EP also has a serious drawback: the eigenbasis becomes defective, meaning that the system no longer has enough independent eigenvectors. This is associated with strong modal nonorthogonality and enhanced noise sensitivity. The Petermann factor is one common way to quantify this nonorthogonality.

=== Why the nonlinear exceptional point is different ===
The reference paper studies a nonlinear exceptional point. The word "nonlinear" is essential. The circuit does not have a fixed gain coefficient; instead, the gain depends on the oscillation amplitude. The steady state must therefore be solved self-consistently: the frequency, voltage amplitudes, and saturated gain must all agree with one another.

In the theory of Ref. [1], this nonlinear steady-state problem can reduce to a cubic equation for the oscillation frequency. A cubic equation can have a triple root. At the special operating point, one stable physical branch and two auxiliary branches merge, giving a third-order nonlinear exceptional point, written as <math>\mathrm{NEP}_3</math>. The stable physical branch is the one with real oscillation frequency, i.e. without an imaginary part. The auxiliary branches are needed to define the higher-order coalescence mathematically, but they are not physically stable operating states of the circuit. The corresponding frequency response has the form

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3}.
</math>

This cubic-root response is the key sensing signature. It is steeper than both an ordinary linear response and the square-root response of a second-order EP. The important conceptual point of Ref. [1] is that this enhanced response is obtained while the effective steady-state Hamiltonian can still retain a complete basis in dynamics.<ref name="Bai2024" /><ref name="Bai2024SM" />

=== Sensing mechanism ===
The circuit can be understood as a frequency-generating sensor. It converts an inductance change into a frequency change.

The first step is activation. A voltage is supplied to the resonators to excite the circuit into an oscillating state. The active branch in resonator <math>A</math> then supplies energy through an effective negative resistance and amplifies the selected oscillation mode.

The second step is self-stabilization. The gain is saturable, which means it becomes weaker when the voltage amplitude becomes larger. The activated oscillation therefore is not amplified forever. The circuit settles onto the physical steady branch where the imaginary part of the eigenfrequency is zero. At that point, the gain supplied by resonator <math>A</math> balances the loss in the circuit, especially the loss in resonator <math>B</math>.

The third step is frequency selection. Because resonators <math>A</math> and <math>B</math> are coupled, the final oscillation frequency is not simply the resonance frequency of either isolated resonator. It is selected by the whole coupled circuit: the inductors, capacitors, coupling elements, loss, and saturated gain all contribute.

The fourth step is sensing. The inductance <math>L_A</math> controls the bare frequency of resonator <math>A</math>. When <math>L_A</math> changes, the self-consistent frequency selected by the nonlinear circuit also changes. Therefore, tracking the spectral peak <math>f</math> tells us how the effective inductance has changed.

The fifth step is the nonlinear exceptional-point response. Near the nonlinear exceptional point discussed in Ref. [1], the physical steady branch and the two auxiliary branches coalesce in the nonlinear steady-state equation. The frequency shift of the stable branch can then scale as the cube root of the inductance perturbation. This means that a small inductance change can produce a comparatively large frequency change. Our breadboard experiment focuses on verifying the stable <math>f-L_A</math> branch corresponding to Fig. S11(a) of SM-4, which is the experimentally accessible part of this sensing mechanism.

=== Same mechanism in other platforms ===
Several references are included to show that this is a platform-independent sensing mechanism, not only a feature of our breadboard circuit. The common chain is

<math display="block">
\mathrm{physical\ perturbation}
\longrightarrow
\mathrm{parameter\ change}
\longrightarrow
\mathrm{eigenfrequency\ shift\ of\ a\ resonator/cavity\ with\ gain\ and\ loss}
\longrightarrow
\mathrm{higher\ order\ steady\ state\ response}
\longrightarrow
\mathrm{sensing}.
</math>

In our circuit, the physical perturbation is a change in the inductance <math>L_A</math>, and the measured output is the oscillation frequency <math>f</math>. In coupled optical, mechanical, electrical, or hybrid resonators, the perturbed parameter may instead be a cavity detuning, mechanical frequency, phase shift, magnetic-field-dependent transition, or another effective resonator parameter. The shared point is that gain, loss, coupling, and nonlinearity convert the parameter change into a higher-order steady-state spectral response.

The earlier theoretical work on nonlinear exceptional points showed that nonlinear coupled resonators can host higher-order NEPs while retaining a complete dynamical basis.<ref name="Bai2023">K. Bai, J.-Z. Li, T.-R. Liu, L. Fang, D. Wan, and M. Xiao, Nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''130''', 266901 (2023).</ref> The 2024 experiment implemented the same mechanism in coupled electrical resonators and measured the stable nonlinear frequency branch.<ref name="Bai2024" /><ref name="Bai2024SM" /> Silicon micromechanical resonators provide another realization: there, the measured output can be a phase difference, and the nonlinear exceptional-point response can enhance phase-shift sensing.<ref name="Xie2026">Y.-J. Xie, S.-Y. Zhang, M.-N. Zhang, R. Wang, L.-F. Wang, and Q.-A. Huang, Nonlinear exceptional points boost phase-shift sensing in silicon micromechanical resonators, ''Phys. Rev. Applied'' '''25''', 034039 (2026).</ref> Hybrid quantum systems with nonlinear bistability show a related route, where nonlinear transition dynamics can enhance sensing signal-to-noise ratio.<ref name="Wang2026">H. Wang ''et al.'', Exceptional sensitivity near the bistable transition point of a hybrid quantum system, ''Nature Physics'' '''22''', 577-583 (2026).</ref>

These examples support the same mechanism rather than unrelated applications. At the same time, enhanced response alone is not enough to prove better sensing. Noise can shift or weaken the apparent nonlinear exceptional point and can limit the signal-to-noise improvement.<ref name="Zheng2025">X. Zheng and Y. D. Chong, Noise constraints for nonlinear exceptional point sensing, ''Phys. Rev. Lett.'' '''134''', 133801 (2025).</ref> This is why our project makes a careful claim: we demonstrate the frequency-response mechanism on a breadboard, but we do not claim a complete metrological noise advantage.

== Report objectives and scope ==
The purpose of this work is to study the circuit principle behind '''nonlinear-exceptional-point-enhanced inductive sensing''' in a coupled electronic resonator platform. The scope is intentionally narrower than the full experiment reported in Ref. [1]. We are not attempting to redesign the active branch, derive all nonlinear bifurcation theory from first principles, reconstruct the auxiliary steady-state branches, or characterize every parasitic element. Instead, we adopt the framework of Ref. [1] and SM-4 and ask whether the breadboard circuit can reproduce the experimentally accessible frequency response.

The main target of our experiment is the <math>f-L_A</math> curve corresponding to Fig. S11(a) of SM-4. In the reference experiment, the steady-state frequency, voltage ratio <math>|V_A/V_B|</math>, and relative phase <math>\theta_A-\theta_B</math> are all measured as functions of <math>L_A</math>.<ref name="Bai2024SM" /> In our implementation, the frequency-versus-inductance curve is the primary observable. We tune <math>L_A</math> by connecting small inductors in series with approximately <math>2~\mu\mathrm{H}</math> resolution. We also use a homebrew coil in a separate demonstration where mechanical compression or insertion of an iron rod changes the effective inductance. The experimental goal is therefore to verify that the TL071 breadboard circuit settles into a stable oscillating state and that the measured frequency changes with <math>L_A</math> in the same qualitative manner as Fig. S11(a).

From the sensing perspective, the most relevant figure of merit is the local frequency responsivity. If

<math display="block">
|f-f_{\mathrm{NEP}}| = A |\delta L_A|^{1/3},
</math>

then differentiation gives

<math display="block">
S_f \equiv \left|\frac{df}{dL_A}\right| \propto |\delta L_A|^{-2/3}.
</math>

Thus, as the operating point approaches <math>L_{A,\mathrm{NEP}}</math>, the local spectral responsivity increases rapidly. The main practical constraint is that the system must remain on the intended stable branch while measurement noise and slow parameter drift are kept below the scale of the induced frequency shift.

== Setups ==
[[File:Experimental_setup.png|750px|thumb|center|Experimental setup adapted from Ref. [1].]]

=== Circuit topology and component implementation ===
The reference platform consists of two coupled RLC resonators. Both resonators contain the same nominal shunt capacitance <math>C_0</math>, while the inductances <math>L_A</math> and <math>L_B</math> set their bare resonance frequencies. The two resonators are coupled through a parallel resistor-capacitor channel characterized by <math>R_c</math> and <math>C_c</math>. Resonator <math>B</math> contains an explicit positive resistance <math>R_B</math> and therefore plays the role of the lossy resonator. Resonator <math>A</math>, by contrast, contains an active branch whose effective resistance is negative at small amplitude and becomes less negative as the oscillation amplitude increases, thereby providing saturable gain.<ref name="Bai2024" /><ref name="Bai2024SM" />

Our implementation follows this topology but is built on a breadboard rather than a custom PCB. The active branch is implemented with an op-amp negative-resistance circuit using a '''TL071''' operational amplifier. This is an important practical difference from the supplementary experiment, where the active components explicitly listed are a TI '''LM7171''' operational amplifier and an '''Onsemi BAV99L''' diode.<ref name="Bai2024SM" /> The TL071 substitution and the breadboard layout introduce different bandwidth limits, wiring parasitics, and contact resistances, so the numerical operating point is not expected to match the PCB experiment. For this reason, our comparison emphasizes the observable trend of <math>f</math> versus <math>L_A</math>.

The supplementary material reports the following representative component values for the PCB experiment:

<math display="block">
C_0 = 18.5~\mathrm{nF}, \qquad
C_c = 3.9~\mathrm{nF}, \qquad
R_c = 760~\Omega,
</math>

<math display="block">
L_B = 197.7~\mu\mathrm{H}, \qquad
R_B = 1314~\Omega.
</math>

A representative operating point shown in the main figure uses

<math display="block">
L_A = 245.4~\mu\mathrm{H}.
</math>

Our breadboard circuit follows the same functional topology, but its component values and parasitic elements are not identical to the PCB values above. The horizontal axis of our main <math>f-L_A</math> plot is therefore the series-combined <math>L_A</math> selected from our available small inductors, while the measured frequency <math>f</math> is the robust experimental readout.

=== Inductance tuning methods ===
We use two complementary methods to change the inductance seen by resonator <math>A</math>.

The first method is used for the main <math>f-L_A</math> sweep. Small inductors are connected in series so that the total inductance changes in discrete steps of approximately <math>2~\mu\mathrm{H}</math>. If the inductors are sufficiently separated so that their mutual magnetic coupling is small, the total series inductance is approximately the sum of the individual inductances,

<math display="block">
L_A \approx L_{\mathrm{base}}+\sum_i L_i .
</math>

This method is simple and reproducible. Adding one small inductor increases <math>L_A</math>; removing it decreases <math>L_A</math>. The circuit frequency is then measured for each chosen value. This produces a discrete version of the <math>f-L_A</math> curve in Fig. S11(a) of SM-4. The step size is not as fine as a commercial variable inductor, but <math>2~\mu\mathrm{H}</math> spacing is small enough to observe the frequency trend over the accessible range.

The second method is a homebrew compressible variable inductor. This coil is useful because it makes the sensing mechanism visible: a mechanical displacement changes the coil geometry, which changes the inductance, which then changes the oscillation frequency. The basic idea can be understood from the approximate solenoid formula

<math display="block">
L \approx \mu_\mathrm{eff}\frac{N^2 A}{\ell},
</math>

where <math>N</math> is the number of turns, <math>A</math> is the cross-sectional area, <math>\ell</math> is the coil length, and <math>\mu_\mathrm{eff}</math> describes the effective magnetic permeability of the core and surrounding region.<ref name="Chen2014">S.-C. Chen, D.-K. Le, and V.-S. Nguyen, Inductive displacement sensors with a notch filter for an active magnetic bearing system, ''Sensors'' '''14''', 12640-12657 (2014).</ref> During compression, the number of turns <math>N</math> is unchanged and the cross-sectional area <math>A</math> remains approximately unchanged, but the length <math>\ell</math> decreases. Equivalently, the turn density

<math display="block">
n=\frac{N}{\ell}
</math>

increases. The magnetic field produced by a solenoid is proportional to turn density. Therefore, compressing the coil increases the magnetic flux linkage through the turns and increases the inductance. Stretching the coil has the opposite effect.

This mechanical tuning is not identical to the precision tunable inductor used in the reference experiment. A real compressed coil is not an ideal long solenoid: the spacing between turns is finite, the end effects change as the coil length changes, nearby wires and the breadboard add parasitic capacitance, and the leads contribute additional series inductance and resistance. For this reason, the homebrew inductor is best treated as an experimentally characterized tuning element rather than an ideal component whose value is known exactly from geometry. In the present project, it provides a direct mechanical demonstration of how displacement can perturb <math>L_A</math>.

Experimentally, we measured the effective inductance of this homebrew variable inductor at different compression states. The compression process changes the coil length and wire density while leaving the coil area approximately fixed. Therefore, the tuning mechanism is physically the same as changing <math>\ell</math> and <math>n=N/\ell</math> in the solenoid model, rather than changing the number of turns or the coil diameter.

=== Kirchhoff model of the nonlinear circuit ===
A convenient state-space description uses the node voltages <math>V_A(t)</math> and <math>V_B(t)</math> together with the inductor currents <math>I_A(t)</math> and <math>I_B(t)</math>. A node voltage is the voltage measured at one resonator node relative to ground. Kirchhoff's current law says that the algebraic sum of all currents flowing away from a node must be zero. This gives a direct way to translate the circuit diagram into differential equations.

Denoting the amplitude-dependent conductance of the active branch by <math>G_A(|V_A|)</math>, where <math>G_A<0</math> in the small-signal regime, the nodal equations can be written as

<math display="block">
\begin{aligned}
C_0 \frac{dV_A}{dt} + I_A + G_A(|V_A|)\,V_A + \frac{V_A-V_B}{R_c}
+ C_c\frac{d(V_A-V_B)}{dt} &= 0,\\
C_0 \frac{dV_B}{dt} + I_B + \frac{V_B}{R_B} + \frac{V_B-V_A}{R_c}
+ C_c\frac{d(V_B-V_A)}{dt} &= 0,\\
L_A \frac{dI_A}{dt} &= V_A,\\
L_B \frac{dI_B}{dt} &= V_B.
\end{aligned}
</math>

Each term has a simple circuit meaning. The terms <math>C_0 dV_A/dt</math> and <math>C_0 dV_B/dt</math> are capacitor currents. The terms <math>I_A</math> and <math>I_B</math> are currents through the inductors. The term <math>G_A(|V_A|)V_A</math> is the active branch current; because <math>G_A</math> can be negative, this branch can inject energy instead of dissipating it. The term <math>V_B/R_B</math> is the ordinary loss current through the resistor in resonator <math>B</math>. The terms involving <math>R_c</math> and <math>C_c</math> describe current flowing between resonators <math>A</math> and <math>B</math>, so they are the electrical coupling between the two resonators.

These equations show explicitly how the experiment differs from a linear EP circuit. If <math>G_A</math> were a constant negative conductance, the model would reduce to a linear non-Hermitian dimer. Here, however, <math>G_A</math> depends on the voltage amplitude, so the steady state must be determined self-consistently together with the oscillation amplitude itself.

=== Harmonic-balance formulation of the steady state ===
After the transient has died away, the oscilloscope waveform is approximately periodic. The simplest approximation is therefore to keep only the dominant sinusoidal component. This is called a harmonic-balance approximation. For a stable activated oscillating state, one writes

<math display="block">
V_j(t)=\mathrm{Re}\!\left[\tilde V_j e^{i\omega t}\right],
\qquad j=A,B.
</math>

Substituting this ansatz into the circuit equations yields a nonlinear algebraic problem of the form

<math display="block">
\mathbf{Y}(\omega,|\tilde V_A|)
\begin{pmatrix}
\tilde V_A\\
\tilde V_B
\end{pmatrix}
=
\begin{pmatrix}
Y_A(\omega,|\tilde V_A|)+Y_c(\omega) & -Y_c(\omega)\\
-Y_c(\omega) & Y_B(\omega)+Y_c(\omega)
\end{pmatrix}
\begin{pmatrix}
\tilde V_A\\
\tilde V_B
\end{pmatrix}
=0,
</math>

with

<math display="block">
Y_A(\omega,|\tilde V_A|)=G_A(|\tilde V_A|)+i\omega C_0+\frac{1}{i\omega L_A},
</math>

<math display="block">
Y_B(\omega)=\frac{1}{R_B}+i\omega C_0+\frac{1}{i\omega L_B},
\qquad
Y_c(\omega)=\frac{1}{R_c}+i\omega C_c.
</math>

A nontrivial solution requires

<math display="block">
\det \mathbf{Y}(\omega,|\tilde V_A|)=0.
</math>

This determinant condition is the circuit analogue of an eigenfrequency condition. In a passive LC resonator, the resonance condition gives a single frequency <math>\omega_0=1/\sqrt{LC}</math>. In the present two-resonator active circuit, the same idea becomes a matrix equation because the two resonators are coupled. The amplitudes <math>\tilde V_A</math> and <math>\tilde V_B</math> are not arbitrary: a nonzero oscillation exists only when the admittance matrix has a zero eigenvalue.

Because the active admittance depends on amplitude, this condition is nonlinear and supports multiple branches. The supplementary material shows that the steady-state frequency can be reduced to a '''real-coefficient cubic polynomial''' in <math>\omega</math>. The existence of up to three real steady-state branches is therefore not accidental; it is the algebraic origin of the experimentally relevant <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM" />

=== Effective nonlinear-Hamiltonian picture ===
The same physics is often rewritten in coupled-mode form. This notation is common in non-Hermitian physics because it makes the circuit look like two coupled modes. The variables <math>\psi_A</math> and <math>\psi_B</math> represent the complex oscillation amplitudes in resonators <math>A</math> and <math>B</math>. The model is

<math display="block">
i\frac{d}{dt}
\begin{pmatrix}
\psi_A\\
\psi_B
\end{pmatrix}
=
\begin{pmatrix}
\omega_A+i\,g(|\psi_A|) & \kappa\\
\kappa & \omega_B-i\ell
\end{pmatrix}
\begin{pmatrix}
\psi_A\\
\psi_B
\end{pmatrix},
</math>

where <math>\omega_A</math> and <math>\omega_B</math> are the effective modal frequencies, <math>\kappa</math> is the coupling strength, <math>\ell</math> is the loss of resonator <math>B</math>, and <math>g(|\psi_A|)</math> is the saturable gain of resonator <math>A</math>. The steady-state problem then becomes

<math display="block">
H_s(\Psi)\Psi = \Omega \Psi,
</math>

with a real oscillation eigenfrequency <math>\Omega=2\pi f</math> for the stable oscillating branch. The word "real" is important: it means the imaginary part has vanished, so the mode is neither being amplified nor decaying in the steady state. In the cited experiment, this physical stable branch and two auxiliary steady branches coalesce at the singular point, producing a third-order nonlinear exceptional point rather than a conventional second-order linear EP. The auxiliary branches help form the cubic-root singularity but are not themselves stable physical outputs of the circuit.<ref name="Bai2024" /><ref name="Bai2024SM" />

== Measurements ==

=== Acquisition protocol and steady-state selection ===
The measurement is performed in the time domain. In the reference experiment, an arbitrary waveform generator first applies an external start-up signal of 1 V at 70 kHz in order to bring the system into the basin of attraction of the stable oscillating state.<ref name="Bai2024" /> In our breadboard measurement, an external voltage is likewise required to activate the resonators. The measured frequency is extracted from the activated steady waveform recorded on the oscilloscope.

The physical meaning of this procedure is important. The oscillation is not assumed to appear without activation; a supplied voltage is needed to bring the resonators into the relevant dynamical state. The measured oscillation frequency is then interpreted as the frequency selected by the nonlinear steady-state condition.

=== Extraction of the <math>f-L_A</math> observable ===
Ref. [1] does not treat the oscilloscope traces merely as qualitative evidence of oscillation. The supplementary explicitly states that the full reference experiment reports the steady-state frequency, the voltage ratio <math>|V_A/V_B|</math>, and the relative phase <math>\theta_A-\theta_B</math>, all extracted from the dominant peaks of the Fourier spectra of <math>V_A</math> and <math>V_B</math>.<ref name="Bai2024SM" />

Our available observable is the steady-state frequency. For each chosen value of <math>L_A</math>, the resonators are voltage-activated, the circuit reaches a stable oscillating state, and the dominant spectral peak is recorded. If we denote the Fourier transforms by

<math display="block">
\tilde V_A(f)=\mathcal{F}\{V_A(t)\},
\qquad
\tilde V_B(f)=\mathcal{F}\{V_B(t)\},
</math>

then the dominant oscillation frequency is obtained from

<math display="block">
f_0=\arg\max_f |\tilde V_A(f)| \approx \arg\max_f |\tilde V_B(f)|.
</math>

The experimental data set is therefore

<math display="block">
\{L_A,\ f_0(L_A)\},
</math>

which is compared with the frequency branch plotted in Fig. S11(a) of SM-4. In the full reference measurement, the amplitude ratio

<math display="block">
\rho = \frac{|\tilde V_A(f_0)|}{|\tilde V_B(f_0)|}
</math>

and relative phase

<math display="block">
\Delta\phi = \arg \tilde V_A(f_0)-\arg \tilde V_B(f_0)
</math>

provide additional state verification. Those quantities are useful but are not the main basis of our breadboard validation.

In practice, the same frequency can also be estimated directly from the time-domain period <math>T</math> using <math>f_0=1/T</math> when the waveform is clean. Fourier analysis is preferred because it uses a longer section of waveform and identifies the dominant oscillation peak even when there is small amplitude noise or waveform distortion.

=== Practical inductance sweep ===
In SM-4, the inductive tuning is carried out by varying <math>L_A</math>. The supplementary notes that the tunable inductor exhibits less than about 0.4% inductance variation over the frequency range of interest, which is sufficiently small for controlled sweeping near the singular point. To further suppress voltage-dependent drift during the sweep, the reference experiment slightly adjusts the active-branch resistance so as to maintain approximately <math>V_B \approx 2~\mathrm{V}</math> throughout the measurement series.<ref name="Bai2024SM" />

Our sweep uses controlled series combinations of small inductors, giving an approximately <math>2~\mu\mathrm{H}</math> step in <math>L_A</math>. The homebrew compressible coil is then used to show that a mechanical change of the sensing element also changes the inductance. The conclusion drawn from our data is therefore not a precision determination of <math>L_{A,\mathrm{NEP}}</math>, but a validation that the breadboard circuit produces the same kind of frequency response as the stable branch in Fig. S11(a).

=== Long-window spectral analysis and noise inspection ===
The supplementary also discusses long-time measurements over a window of <math>0<t<5~\mathrm{s}</math>. The motivation is that the noise around the stable oscillation is too small to be judged reliably from raw scope traces alone, so it is more informative to inspect the Fourier spectra and the extracted spectral center frequency and linewidth.<ref name="Bai2024SM" />

This is a useful methodological point for a sensing-oriented reproduction. In an NEP-based sensor, one ultimately cares about how precisely the frequency peak can be resolved and tracked. A long-window Fourier analysis is therefore closer to the sensing task than a purely visual time-domain inspection.

== Results ==

=== Result display: steady oscillation ===
[[File:PC5271inductive1.png|750px|thumb|center|Breadboard implementation and representative steady oscillation waveform]]

This figure should show the breadboard circuit and a representative voltage waveform after the resonators are activated by the supplied voltage. The displayed waveform is the evidence used to identify a stable oscillation frequency.

=== Result display: <math>f-L_A</math> sweep ===
'''[INSERT FIGURE: measured breadboard <math>f-L_A</math> curve compared with Fig. S11(a) of SM-4]'''

This figure should show the measured oscillation frequency <math>f</math> as a function of <math>L_A</math>. The values of <math>L_A</math> in the main sweep are produced by connecting small inductors in series with approximately <math>2~\mu\mathrm{H}</math> step size. The reference curve for comparison is the stable frequency branch in Fig. S11(a) of SM-4.<ref name="Bai2024SM" />

=== Result display: iron-rod demonstration ===
[[File:PC5271inductive2.png|750px|thumb|center|Displacement measurement by compressing the solenoid and inserting iron rod into the homebrew coil]]

This figure should show the homebrew coil with an inserted iron rod and the corresponding change in the measured oscillation frequency. The experimental variable is the rod position or insertion state, and the measured output is the activated steady oscillation frequency.

== Evaluation ==

=== Stable activated oscillation ===
The first required experimental signature is that, after the resonators are activated by the supplied voltage, the circuit reaches a stable oscillating waveform instead of decaying immediately or being amplified without bound. A stable waveform means that the TL071 active branch supplies enough small-signal negative resistance to compensate loss, while gain saturation prevents unbounded amplification.

Physically, this behavior confirms the role of the nonlinear active branch. At small amplitude, the effective negative resistance in resonator <math>A</math> overcomes dissipation. At larger amplitude, the gain becomes weaker, so the oscillation amplitude is clamped. The system self-stabilizes because the physical steady branch is the one with zero imaginary part of the eigenfrequency. The auxiliary branches near the nonlinear exceptional point are not stable physical outputs; they are mathematical branches of the nonlinear steady-state equation. The experiment is therefore not a passive ring-down measurement; it is a nonlinear steady-state selection process.

=== Evaluation of the <math>f-L_A</math> trend ===
The main experimental comparison is the measured <math>f-L_A</math> trend against Fig. S11(a) of SM-4. The reference experiment reports the stable frequency branch of the nonlinear circuit as <math>L_A</math> is tuned.<ref name="Bai2024SM" /> This branch is the physical one because its eigenfrequency is real in the steady state. In our breadboard experiment, the same observable is measured using the series-inductor sweep.

It is useful to separate two statements. First, any LC resonator changes frequency when <math>L</math> changes, because <math>f_0=1/(2\pi\sqrt{LC})</math>. This alone would not demonstrate the nonlinear exceptional-point mechanism. Second, in the coupled active-lossy circuit, the frequency is not just the bare LC resonance frequency of one isolated inductor and capacitor. It is the frequency selected by the nonlinear steady-state balance of gain, loss, and coupling. The reason Fig. S11(a) matters is that it shows this selected stable branch as <math>L_A</math> is swept.

Our breadboard circuit uses different component values and a different op-amp from the PCB experiment, so exact numerical overlap is not expected. The meaningful evaluation is whether the measured frequency branch follows the same qualitative <math>f-L_A</math> behavior as the stable branch in SM-4. Observing that behavior is enough to show that the constructed circuit realizes the intended nonlinear voltage-activated oscillating mechanism.

=== Evaluation of the displacement demonstration ===
In the iron-rod demonstration, the homebrew coil becomes a displacement-dependent inductive element. As the rod enters the coil, the effective magnetic permeability inside the coil increases. In the solenoid expression

<math display="block">
L(x) \approx \mu_\mathrm{eff}(x)\frac{N^2 A}{\ell},
</math>

the insertion depth <math>x</math> changes <math>\mu_\mathrm{eff}</math>. The coil geometry <math>N</math>, <math>A</math>, and <math>\ell</math> are approximately fixed during this demonstration, so the dominant effect is the change in magnetic core material inside the coil. Inductive displacement sensors commonly use this principle: the position of a ferromagnetic object changes a coil inductance, and the electronics convert that inductance change into an electrical signal.<ref name="Chen2014" />

In our circuit, the electrical signal is the activated steady oscillation frequency rather than a bridge voltage. The sensing chain is

<math display="block">
\mathrm{rod\ displacement}\ x
\longrightarrow
\mathrm{effective\ inductance}\ L_A(x)
\longrightarrow
\mathrm{oscillation\ frequency}\ f(x).
</math>

This demonstration shows how the inductive readout can be turned into a displacement-sensing readout after calibration of the relation between rod position and frequency.

=== Cubic-root perturbation law and sensing meaning ===
Near the singular operating point, the eigenfrequency shift follows the hallmark cubic-root law,

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3}.
</math>

This law is the main sensing-related result of the reference experiment.<ref name="Bai2024" /><ref name="Bai2024SM" /> Compared with the linear response of an ordinary isolated resonator, the cubic-root dependence can produce a larger frequency displacement for the same small perturbation of inductance when the device is biased sufficiently close to <math>L_{A,\mathrm{NEP}}</math>.

A useful way to interpret the result is through the local derivative,

<math display="block">
\left|\frac{df}{dL_A}\right| \propto |\delta L_A|^{-2/3}.
</math>

This increasing local responsivity is why the circuit is attractive for sensing. Our experiment supports this sensing logic by showing that the breadboard circuit has a measurable stable frequency branch as <math>L_A</math> is changed. A full quantitative exponent extraction would require identifying <math>L_{A,\mathrm{NEP}}</math> and <math>f_{\mathrm{NEP}}</math>, then plotting <math>|f-f_{\mathrm{NEP}}|</math> against <math>|L_A-L_{A,\mathrm{NEP}}|</math> on logarithmic axes.

=== Basis completeness and noise robustness ===
The supplementary material evaluates the Petermann factor as a diagnostic of modal nonorthogonality and basis completeness. For a non-Hermitian eigenproblem, one may write

<math display="block">
\mathrm{PF}_n =
\frac{\langle \Phi_n^{L}|\Phi_n^{L}\rangle \langle \Phi_n^{R}|\Phi_n^{R}\rangle}
{\left|\langle \Phi_n^{L}|\Phi_n^{R}\rangle\right|^2},
</math>

where <math>|\Phi_n^{R}\rangle</math> and <math>|\Phi_n^{L}\rangle</math> are the right and left eigenvectors of the effective steady-state Hamiltonian. At a conventional linear EP, this quantity is associated with diverging excess-noise sensitivity. In Ref. [1], the cited analysis finds that the effective steady-state Hamiltonian remains non-defective and the Petermann factor is finite at the <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM" />

This point explains the significance of the reference experiment. The system does not merely realize a large mathematical responsivity; it realizes that responsivity in a way that avoids the most severe basis-collapse problem of a linear EP. Our breadboard experiment does not directly test basis completeness, but the result provides the theoretical reason why reproducing the stable <math>f-L_A</math> branch is physically meaningful.

=== Overall interpretation ===
In this platform, the measurand is encoded as a perturbation of <math>L_A</math>. Any physical mechanism that modifies the effective inductance of resonator <math>A</math>, for example displacement, magnetic coupling, or the presence of a nearby magnetic target, moves the system away from the operating point and shifts the oscillation frequency. The primary readout in our implementation is therefore spectral: we track <math>f</math> as <math>L_A</math> is changed.

Taken together, Ref. [1] establishes the complete sensing logic: a nonlinear active-lossy circuit is tuned to an <math>\mathrm{NEP}_3</math>; after voltage activation, the system settles onto a stable oscillating branch; the dominant spectral peak is tracked as the perturbation readout; and the enhanced cubic-root response is obtained without the defective-basis pathology of a conventional linear EP. Our breadboard result demonstrates the experimentally accessible core of this logic by reproducing the stable frequency response corresponding to Fig. S11(a) of SM-4.

== References ==
<references />

Inductive Sensors of Ultra-high Sensitivity Based on Nonlinear Exceptional Point

2026-04-23T05:56:01Z

Siyu: /* Background */

== Introduction ==
This project is a repetition of the experiment reported in ''Observation of Nonlinear Exceptional Points with a Complete Basis in Dynamics'', adapted to our available laboratory equipment.<ref name="Bai2024">K. Bai, T.-R. Liu, L. Fang, J.-Z. Li, C. Lin, D. Wan, and M. Xiao, Observation of nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''132''', 073802 (2024).</ref> The circuit contains two coupled resonators. One resonator has gain, meaning that it can supply energy to the oscillation, while the other has loss, meaning that it dissipates energy. A system with both gain and loss is called a '''non-Hermitian''' system. Such systems can show unusual degeneracies called '''exceptional points''' (EPs), where both the eigenfrequencies and the eigenmodes merge.

Exceptional points are interesting for sensing because the measured frequency can respond very strongly to a small perturbation. In an inductive sensor, the perturbation is a change in inductance. If a small change in inductance produces a large change in oscillation frequency, the circuit can in principle detect small physical changes. However, ordinary linear EPs also suffer from strong eigenmode nonorthogonality and a defective basis, which can increase noise sensitivity and complicate practical sensing.<ref name="Bai2024" />

The system studied in this project belongs to the recently developed class of '''nonlinear exceptional points''' (NEPs). The physical platform is a pair of coupled circuit resonators. Resonator <math>A</math> is made active by a saturable negative-resistance branch, while resonator <math>B</math> is intentionally lossy. Because the gain in resonator <math>A</math> depends on the oscillation amplitude, the steady-state problem is nonlinear and admits multiple self-consistent branches. At the singular operating point reported in Ref. [1], one stable steady branch and two auxiliary steady branches merge into a third-order nonlinear exceptional point, denoted <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM">Supplemental Material for K. Bai, T.-R. Liu, L. Fang, J.-Z. Li, C. Lin, D. Wan, and M. Xiao, Observation of nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''132''', 073802 (2024).</ref>

The experimentally important consequence is that the oscillation eigenfrequency obeys a cubic-root perturbation law near the singularity,

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3},
</math>

where <math>\delta L_A = L_A-L_{A,\mathrm{NEP}}</math> is the inductive detuning from the exceptional operating point. This response is steeper than the ordinary linear dependence of a conventional resonator and is therefore useful for ultra-sensitive inductive sensing.<ref name="Bai2024" /><ref name="Bai2024SM" />

A second and equally important point is that the cited work does not interpret the singularity as a conventional defective linear EP. Instead, the effective steady-state Hamiltonian remains non-defective in dynamics, and the associated Petermann factor stays finite at the operating point. In other words, the experiment realizes enhanced spectral responsivity without inheriting the strongest basis-collapse pathology of a standard linear EP.<ref name="Bai2024" /><ref name="Bai2024SM" />

In this project, our goal is to reproduce the experimental logic of Ref. [1] and its supplementary material under course-lab conditions. Unlike the PCB implementation in the paper, our circuit is assembled on a breadboard and the active branch uses a '''TL071''' operational amplifier. Our main observable is the steady oscillation frequency <math>f</math> as a function of the chosen inductance <math>L_A</math>. We tune <math>L_A</math> in two ways. For the main <math>f-L_A</math> sweep, we join small inductors in series so that the discrete inductance step is about <math>2~\mu\mathrm{H}</math>. In addition, we use a homebrew compressible coil to demonstrate the sensing mechanism directly. Compressing the coil changes its length and turn density while keeping the cross-sectional area approximately unchanged. We therefore use the <math>f-L_A</math> behavior in Fig. S11(a) of the supplementary material as the main experimental benchmark. Reproducing this trend is sufficient for demonstrating that our setup captures the intended nonlinear voltage-activated, self-stabilized oscillating response, although it is not a full reproduction of every diagnostic reported in Ref. [1].<ref name="Bai2024" /><ref name="Bai2024SM" />

== Team Members ==
Wang Peikun — E1538091@u.nus.edu 
Zhu Ziyang — E1583446@u.nus.edu 
Yuan Si-Yu — siyu_yuan@u.nus.edu 
Li Xunyu — xunyu@u.nus.edu

== Background Knowledge ==

=== Ordinary LC and RLC resonators ===
The basic building block of the experiment is an electrical resonator. An ideal LC resonator contains an inductor <math>L</math> and a capacitor <math>C</math>. Energy repeatedly moves between the magnetic field of the inductor and the electric field of the capacitor. The natural angular frequency is

<math display="block">
\omega_0=\frac{1}{\sqrt{LC}},
</math>

or, in ordinary frequency units,

<math display="block">
f_0=\frac{1}{2\pi\sqrt{LC}}.
</math>

This formula is important for the whole project because it explains why changing an inductance changes the measured frequency. A larger <math>L</math> gives a lower resonance frequency, while a smaller <math>L</math> gives a higher resonance frequency.

Real circuits are not ideal. Wires, inductors, capacitors, breadboard contacts, and measuring probes all dissipate energy. This loss is often represented by a resistor <math>R</math>, giving an RLC resonator. In a passive RLC circuit, the oscillation amplitude decays with time because energy is continuously lost as heat. If a circuit is to maintain a steady activated oscillation, the loss must be compensated by a gain element.

=== Gain, loss, and negative resistance ===
In electronics, gain can be represented as an effective negative resistance. A normal positive resistance removes energy from the resonator. A negative resistance does the opposite: it supplies energy to the resonator. In this experiment, resonator <math>A</math> contains an active op-amp branch that behaves like a negative resistance at small amplitude, while resonator <math>B</math> contains ordinary positive loss.

The gain cannot remain negative and constant forever. If it did, any activated oscillation would be amplified without bound until the op-amp clipped or the circuit failed. The important nonlinear ingredient is '''saturable gain''': the active branch gives strong gain at small amplitude but weaker gain at larger amplitude. The stable oscillation amplitude is selected when the supplied gain balances the total loss. This nonlinear steady state is why the experiment is a nonlinear dynamical problem rather than only a passive resonance measurement.

=== What "non-Hermitian" means in this circuit ===
In ordinary conservative physics, a Hermitian Hamiltonian describes a closed system whose total energy is conserved. Its eigenfrequencies are real, and its eigenmodes form a complete orthogonal basis. This is the familiar setting of many quantum-mechanics courses.

A circuit with gain and loss is an open system. Energy can enter through the op-amp branch and leave through resistive loss. The effective dynamical matrix is therefore generally '''non-Hermitian'''. In a non-Hermitian system, eigenfrequencies can be complex:

<math display="block">
\omega=\omega_\mathrm{r}+i\omega_\mathrm{i}.
</math>

The real part <math>\omega_\mathrm{r}</math> gives the oscillation frequency, while the imaginary part <math>\omega_\mathrm{i}</math> describes amplification or decay. A positive imaginary part means the mode is amplified, a negative imaginary part means it decays, and a stable steady oscillation corresponds to zero imaginary part, or equivalently zero net amplification. This is the sense in which the circuit is self-stabilized after voltage activation.

=== Exceptional points ===
In a usual Hermitian system, two modes can have the same eigenfrequency while still remaining two independent modes. This is an ordinary degeneracy. A non-Hermitian system can show a stronger kind of degeneracy called an '''exceptional point''' (EP). At an EP, not only the eigenvalues but also the eigenvectors coalesce. In simple language, two modes collapse into one mode.

This collapse makes EPs interesting for sensing. Near a second-order linear EP, a small perturbation can produce a square-root frequency splitting:

<math display="block">
\Delta f \propto \sqrt{\epsilon},
</math>

where <math>\epsilon</math> is the perturbation. This response is steeper than an ordinary linear response when <math>\epsilon</math> is very small. However, a conventional linear EP also has a serious drawback: the eigenbasis becomes defective, meaning that the system no longer has enough independent eigenvectors. This is associated with strong modal nonorthogonality and enhanced noise sensitivity. The Petermann factor is one common way to quantify this nonorthogonality.

=== Why the nonlinear exceptional point is different ===
The reference paper studies a nonlinear exceptional point. The word "nonlinear" is essential. The circuit does not have a fixed gain coefficient; instead, the gain depends on the oscillation amplitude. The steady state must therefore be solved self-consistently: the frequency, voltage amplitudes, and saturated gain must all agree with one another.

In the theory of Ref. [1], this nonlinear steady-state problem can reduce to a cubic equation for the oscillation frequency. A cubic equation can have a triple root. At the special operating point, one stable physical branch and two auxiliary branches merge, giving a third-order nonlinear exceptional point, written as <math>\mathrm{NEP}_3</math>. The stable physical branch is the one with real oscillation frequency, i.e. without an imaginary part. The auxiliary branches are needed to define the higher-order coalescence mathematically, but they are not physically stable operating states of the circuit. The corresponding frequency response has the form

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3}.
</math>

This cubic-root response is the key sensing signature. It is steeper than both an ordinary linear response and the square-root response of a second-order EP. The important conceptual point of Ref. [1] is that this enhanced response is obtained while the effective steady-state Hamiltonian can still retain a complete basis in dynamics.<ref name="Bai2024" /><ref name="Bai2024SM" />

=== Sensing mechanism ===
The circuit can be understood as a frequency-generating sensor. It converts an inductance change into a frequency change.

The first step is activation. A voltage is supplied to the resonators to excite the circuit into an oscillating state. The active branch in resonator <math>A</math> then supplies energy through an effective negative resistance and amplifies the selected oscillation mode.

The second step is self-stabilization. The gain is saturable, which means it becomes weaker when the voltage amplitude becomes larger. The activated oscillation therefore is not amplified forever. The circuit settles onto the physical steady branch where the imaginary part of the eigenfrequency is zero. At that point, the gain supplied by resonator <math>A</math> balances the loss in the circuit, especially the loss in resonator <math>B</math>.

The third step is frequency selection. Because resonators <math>A</math> and <math>B</math> are coupled, the final oscillation frequency is not simply the resonance frequency of either isolated resonator. It is selected by the whole coupled circuit: the inductors, capacitors, coupling elements, loss, and saturated gain all contribute.

The fourth step is sensing. The inductance <math>L_A</math> controls the bare frequency of resonator <math>A</math>. When <math>L_A</math> changes, the self-consistent frequency selected by the nonlinear circuit also changes. Therefore, tracking the spectral peak <math>f</math> tells us how the effective inductance has changed.

The fifth step is the nonlinear exceptional-point response. Near the nonlinear exceptional point discussed in Ref. [1], the physical steady branch and the two auxiliary branches coalesce in the nonlinear steady-state equation. The frequency shift of the stable branch can then scale as the cube root of the inductance perturbation. This means that a small inductance change can produce a comparatively large frequency change. Our breadboard experiment focuses on verifying the stable <math>f-L_A</math> branch corresponding to Fig. S11(a) of SM-4, which is the experimentally accessible part of this sensing mechanism.

=== Same mechanism in other platforms ===
Several references are included to show that this is a platform-independent sensing mechanism, not only a feature of our breadboard circuit. The common chain is

<math display="block">
\mathrm{physical\ perturbation}
\longrightarrow
\mathrm{parameter\ change}
\longrightarrow
\mathrm{eigenfrequency\ shift\ of\ a\ resonator/cavity\ with\ gain\ and\ loss}
\longrightarrow
\mathrm{higher\ order\ steady\ state\ response}
\longrightarrow
\mathrm{sensing}.
</math>

In our circuit, the physical perturbation is a change in the inductance <math>L_A</math>, and the measured output is the oscillation frequency <math>f</math>. In coupled optical, mechanical, electrical, or hybrid resonators, the perturbed parameter may instead be a cavity detuning, mechanical frequency, phase shift, magnetic-field-dependent transition, or another effective resonator parameter. The shared point is that gain, loss, coupling, and nonlinearity convert the parameter change into a higher-order steady-state spectral response.

The earlier theoretical work on nonlinear exceptional points showed that nonlinear coupled resonators can host higher-order NEPs while retaining a complete dynamical basis.<ref name="Bai2023">K. Bai, J.-Z. Li, T.-R. Liu, L. Fang, D. Wan, and M. Xiao, Nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''130''', 266901 (2023).</ref> The 2024 experiment implemented the same mechanism in coupled electrical resonators and measured the stable nonlinear frequency branch.<ref name="Bai2024" /><ref name="Bai2024SM" /> Silicon micromechanical resonators provide another realization: there, the measured output can be a phase difference, and the nonlinear exceptional-point response can enhance phase-shift sensing.<ref name="Xie2026">Y.-J. Xie, S.-Y. Zhang, M.-N. Zhang, R. Wang, L.-F. Wang, and Q.-A. Huang, Nonlinear exceptional points boost phase-shift sensing in silicon micromechanical resonators, ''Phys. Rev. Applied'' '''25''', 034039 (2026).</ref> Hybrid quantum systems with nonlinear bistability show a related route, where nonlinear transition dynamics can enhance sensing signal-to-noise ratio.<ref name="Wang2026">H. Wang ''et al.'', Exceptional sensitivity near the bistable transition point of a hybrid quantum system, ''Nature Physics'' '''22''', 577-583 (2026).</ref>

These examples support the same mechanism rather than unrelated applications. At the same time, enhanced response alone is not enough to prove better sensing. Noise can shift or weaken the apparent nonlinear exceptional point and can limit the signal-to-noise improvement.<ref name="Zheng2025">X. Zheng and Y. D. Chong, Noise constraints for nonlinear exceptional point sensing, arXiv:2410.08502v2 (2025).</ref> This is why our project makes a careful claim: we demonstrate the frequency-response mechanism on a breadboard, but we do not claim a complete metrological noise advantage.

== Report objectives and scope ==
The purpose of this work is to study the circuit principle behind '''nonlinear-exceptional-point-enhanced inductive sensing''' in a coupled electronic resonator platform. The scope is intentionally narrower than the full experiment reported in Ref. [1]. We are not attempting to redesign the active branch, derive all nonlinear bifurcation theory from first principles, reconstruct the auxiliary steady-state branches, or characterize every parasitic element. Instead, we adopt the framework of Ref. [1] and SM-4 and ask whether the breadboard circuit can reproduce the experimentally accessible frequency response.

The main target of our experiment is the <math>f-L_A</math> curve corresponding to Fig. S11(a) of SM-4. In the reference experiment, the steady-state frequency, voltage ratio <math>|V_A/V_B|</math>, and relative phase <math>\theta_A-\theta_B</math> are all measured as functions of <math>L_A</math>.<ref name="Bai2024SM" /> In our implementation, the frequency-versus-inductance curve is the primary observable. We tune <math>L_A</math> by connecting small inductors in series with approximately <math>2~\mu\mathrm{H}</math> resolution. We also use a homebrew coil in a separate demonstration where mechanical compression or insertion of an iron rod changes the effective inductance. The experimental goal is therefore to verify that the TL071 breadboard circuit settles into a stable oscillating state and that the measured frequency changes with <math>L_A</math> in the same qualitative manner as Fig. S11(a).

From the sensing perspective, the most relevant figure of merit is the local frequency responsivity. If

<math display="block">
|f-f_{\mathrm{NEP}}| = A |\delta L_A|^{1/3},
</math>

then differentiation gives

<math display="block">
S_f \equiv \left|\frac{df}{dL_A}\right| \propto |\delta L_A|^{-2/3}.
</math>

Thus, as the operating point approaches <math>L_{A,\mathrm{NEP}}</math>, the local spectral responsivity increases rapidly. The main practical constraint is that the system must remain on the intended stable branch while measurement noise and slow parameter drift are kept below the scale of the induced frequency shift.

== Setups ==
[[File:Experimental_setup.png|750px|thumb|center|Experimental setup adapted from Ref. [1].]]

=== Circuit topology and component implementation ===
The reference platform consists of two coupled RLC resonators. Both resonators contain the same nominal shunt capacitance <math>C_0</math>, while the inductances <math>L_A</math> and <math>L_B</math> set their bare resonance frequencies. The two resonators are coupled through a parallel resistor-capacitor channel characterized by <math>R_c</math> and <math>C_c</math>. Resonator <math>B</math> contains an explicit positive resistance <math>R_B</math> and therefore plays the role of the lossy resonator. Resonator <math>A</math>, by contrast, contains an active branch whose effective resistance is negative at small amplitude and becomes less negative as the oscillation amplitude increases, thereby providing saturable gain.<ref name="Bai2024" /><ref name="Bai2024SM" />

Our implementation follows this topology but is built on a breadboard rather than a custom PCB. The active branch is implemented with an op-amp negative-resistance circuit using a '''TL071''' operational amplifier. This is an important practical difference from the supplementary experiment, where the active components explicitly listed are a TI '''LM7171''' operational amplifier and an '''Onsemi BAV99L''' diode.<ref name="Bai2024SM" /> The TL071 substitution and the breadboard layout introduce different bandwidth limits, wiring parasitics, and contact resistances, so the numerical operating point is not expected to match the PCB experiment. For this reason, our comparison emphasizes the observable trend of <math>f</math> versus <math>L_A</math>.

The supplementary material reports the following representative component values for the PCB experiment:

<math display="block">
C_0 = 18.5~\mathrm{nF}, \qquad
C_c = 3.9~\mathrm{nF}, \qquad
R_c = 760~\Omega,
</math>

<math display="block">
L_B = 197.7~\mu\mathrm{H}, \qquad
R_B = 1314~\Omega.
</math>

A representative operating point shown in the main figure uses

<math display="block">
L_A = 245.4~\mu\mathrm{H}.
</math>

Our breadboard circuit follows the same functional topology, but its component values and parasitic elements are not identical to the PCB values above. The horizontal axis of our main <math>f-L_A</math> plot is therefore the series-combined <math>L_A</math> selected from our available small inductors, while the measured frequency <math>f</math> is the robust experimental readout.

=== Inductance tuning methods ===
We use two complementary methods to change the inductance seen by resonator <math>A</math>.

The first method is used for the main <math>f-L_A</math> sweep. Small inductors are connected in series so that the total inductance changes in discrete steps of approximately <math>2~\mu\mathrm{H}</math>. If the inductors are sufficiently separated so that their mutual magnetic coupling is small, the total series inductance is approximately the sum of the individual inductances,

<math display="block">
L_A \approx L_{\mathrm{base}}+\sum_i L_i .
</math>

This method is simple and reproducible. Adding one small inductor increases <math>L_A</math>; removing it decreases <math>L_A</math>. The circuit frequency is then measured for each chosen value. This produces a discrete version of the <math>f-L_A</math> curve in Fig. S11(a) of SM-4. The step size is not as fine as a commercial variable inductor, but <math>2~\mu\mathrm{H}</math> spacing is small enough to observe the frequency trend over the accessible range.

The second method is a homebrew compressible variable inductor. This coil is useful because it makes the sensing mechanism visible: a mechanical displacement changes the coil geometry, which changes the inductance, which then changes the oscillation frequency. The basic idea can be understood from the approximate solenoid formula

<math display="block">
L \approx \mu_\mathrm{eff}\frac{N^2 A}{\ell},
</math>

where <math>N</math> is the number of turns, <math>A</math> is the cross-sectional area, <math>\ell</math> is the coil length, and <math>\mu_\mathrm{eff}</math> describes the effective magnetic permeability of the core and surrounding region.<ref name="Chen2014">S.-C. Chen, D.-K. Le, and V.-S. Nguyen, Inductive displacement sensors with a notch filter for an active magnetic bearing system, ''Sensors'' '''14''', 12640-12657 (2014).</ref> During compression, the number of turns <math>N</math> is unchanged and the cross-sectional area <math>A</math> remains approximately unchanged, but the length <math>\ell</math> decreases. Equivalently, the turn density

<math display="block">
n=\frac{N}{\ell}
</math>

increases. The magnetic field produced by a solenoid is proportional to turn density. Therefore, compressing the coil increases the magnetic flux linkage through the turns and increases the inductance. Stretching the coil has the opposite effect.

This mechanical tuning is not identical to the precision tunable inductor used in the reference experiment. A real compressed coil is not an ideal long solenoid: the spacing between turns is finite, the end effects change as the coil length changes, nearby wires and the breadboard add parasitic capacitance, and the leads contribute additional series inductance and resistance. For this reason, the homebrew inductor is best treated as an experimentally characterized tuning element rather than an ideal component whose value is known exactly from geometry. In the present project, it provides a direct mechanical demonstration of how displacement can perturb <math>L_A</math>.

Experimentally, we measured the effective inductance of this homebrew variable inductor at different compression states. The compression process changes the coil length and wire density while leaving the coil area approximately fixed. Therefore, the tuning mechanism is physically the same as changing <math>\ell</math> and <math>n=N/\ell</math> in the solenoid model, rather than changing the number of turns or the coil diameter.

=== Kirchhoff model of the nonlinear circuit ===
A convenient state-space description uses the node voltages <math>V_A(t)</math> and <math>V_B(t)</math> together with the inductor currents <math>I_A(t)</math> and <math>I_B(t)</math>. A node voltage is the voltage measured at one resonator node relative to ground. Kirchhoff's current law says that the algebraic sum of all currents flowing away from a node must be zero. This gives a direct way to translate the circuit diagram into differential equations.

Denoting the amplitude-dependent conductance of the active branch by <math>G_A(|V_A|)</math>, where <math>G_A<0</math> in the small-signal regime, the nodal equations can be written as

<math display="block">
\begin{aligned}
C_0 \frac{dV_A}{dt} + I_A + G_A(|V_A|)\,V_A + \frac{V_A-V_B}{R_c}
+ C_c\frac{d(V_A-V_B)}{dt} &= 0,\\
C_0 \frac{dV_B}{dt} + I_B + \frac{V_B}{R_B} + \frac{V_B-V_A}{R_c}
+ C_c\frac{d(V_B-V_A)}{dt} &= 0,\\
L_A \frac{dI_A}{dt} &= V_A,\\
L_B \frac{dI_B}{dt} &= V_B.
\end{aligned}
</math>

Each term has a simple circuit meaning. The terms <math>C_0 dV_A/dt</math> and <math>C_0 dV_B/dt</math> are capacitor currents. The terms <math>I_A</math> and <math>I_B</math> are currents through the inductors. The term <math>G_A(|V_A|)V_A</math> is the active branch current; because <math>G_A</math> can be negative, this branch can inject energy instead of dissipating it. The term <math>V_B/R_B</math> is the ordinary loss current through the resistor in resonator <math>B</math>. The terms involving <math>R_c</math> and <math>C_c</math> describe current flowing between resonators <math>A</math> and <math>B</math>, so they are the electrical coupling between the two resonators.

These equations show explicitly how the experiment differs from a linear EP circuit. If <math>G_A</math> were a constant negative conductance, the model would reduce to a linear non-Hermitian dimer. Here, however, <math>G_A</math> depends on the voltage amplitude, so the steady state must be determined self-consistently together with the oscillation amplitude itself.

=== Harmonic-balance formulation of the steady state ===
After the transient has died away, the oscilloscope waveform is approximately periodic. The simplest approximation is therefore to keep only the dominant sinusoidal component. This is called a harmonic-balance approximation. For a stable activated oscillating state, one writes

<math display="block">
V_j(t)=\mathrm{Re}\!\left[\tilde V_j e^{i\omega t}\right],
\qquad j=A,B.
</math>

Substituting this ansatz into the circuit equations yields a nonlinear algebraic problem of the form

<math display="block">
\mathbf{Y}(\omega,|\tilde V_A|)
\begin{pmatrix}
\tilde V_A\\
\tilde V_B
\end{pmatrix}
=
\begin{pmatrix}
Y_A(\omega,|\tilde V_A|)+Y_c(\omega) & -Y_c(\omega)\\
-Y_c(\omega) & Y_B(\omega)+Y_c(\omega)
\end{pmatrix}
\begin{pmatrix}
\tilde V_A\\
\tilde V_B
\end{pmatrix}
=0,
</math>

with

<math display="block">
Y_A(\omega,|\tilde V_A|)=G_A(|\tilde V_A|)+i\omega C_0+\frac{1}{i\omega L_A},
</math>

<math display="block">
Y_B(\omega)=\frac{1}{R_B}+i\omega C_0+\frac{1}{i\omega L_B},
\qquad
Y_c(\omega)=\frac{1}{R_c}+i\omega C_c.
</math>

A nontrivial solution requires

<math display="block">
\det \mathbf{Y}(\omega,|\tilde V_A|)=0.
</math>

This determinant condition is the circuit analogue of an eigenfrequency condition. In a passive LC resonator, the resonance condition gives a single frequency <math>\omega_0=1/\sqrt{LC}</math>. In the present two-resonator active circuit, the same idea becomes a matrix equation because the two resonators are coupled. The amplitudes <math>\tilde V_A</math> and <math>\tilde V_B</math> are not arbitrary: a nonzero oscillation exists only when the admittance matrix has a zero eigenvalue.

Because the active admittance depends on amplitude, this condition is nonlinear and supports multiple branches. The supplementary material shows that the steady-state frequency can be reduced to a '''real-coefficient cubic polynomial''' in <math>\omega</math>. The existence of up to three real steady-state branches is therefore not accidental; it is the algebraic origin of the experimentally relevant <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM" />

=== Effective nonlinear-Hamiltonian picture ===
The same physics is often rewritten in coupled-mode form. This notation is common in non-Hermitian physics because it makes the circuit look like two coupled modes. The variables <math>\psi_A</math> and <math>\psi_B</math> represent the complex oscillation amplitudes in resonators <math>A</math> and <math>B</math>. The model is

<math display="block">
i\frac{d}{dt}
\begin{pmatrix}
\psi_A\\
\psi_B
\end{pmatrix}
=
\begin{pmatrix}
\omega_A+i\,g(|\psi_A|) & \kappa\\
\kappa & \omega_B-i\ell
\end{pmatrix}
\begin{pmatrix}
\psi_A\\
\psi_B
\end{pmatrix},
</math>

where <math>\omega_A</math> and <math>\omega_B</math> are the effective modal frequencies, <math>\kappa</math> is the coupling strength, <math>\ell</math> is the loss of resonator <math>B</math>, and <math>g(|\psi_A|)</math> is the saturable gain of resonator <math>A</math>. The steady-state problem then becomes

<math display="block">
H_s(\Psi)\Psi = \Omega \Psi,
</math>

with a real oscillation eigenfrequency <math>\Omega=2\pi f</math> for the stable oscillating branch. The word "real" is important: it means the imaginary part has vanished, so the mode is neither being amplified nor decaying in the steady state. In the cited experiment, this physical stable branch and two auxiliary steady branches coalesce at the singular point, producing a third-order nonlinear exceptional point rather than a conventional second-order linear EP. The auxiliary branches help form the cubic-root singularity but are not themselves stable physical outputs of the circuit.<ref name="Bai2024" /><ref name="Bai2024SM" />

== Measurements ==

=== Acquisition protocol and steady-state selection ===
The measurement is performed in the time domain. In the reference experiment, an arbitrary waveform generator first applies an external start-up signal of 1 V at 70 kHz in order to bring the system into the basin of attraction of the stable oscillating state.<ref name="Bai2024" /> In our breadboard measurement, an external voltage is likewise required to activate the resonators. The measured frequency is extracted from the activated steady waveform recorded on the oscilloscope.

The physical meaning of this procedure is important. The oscillation is not assumed to appear without activation; a supplied voltage is needed to bring the resonators into the relevant dynamical state. The measured oscillation frequency is then interpreted as the frequency selected by the nonlinear steady-state condition.

=== Extraction of the <math>f-L_A</math> observable ===
Ref. [1] does not treat the oscilloscope traces merely as qualitative evidence of oscillation. The supplementary explicitly states that the full reference experiment reports the steady-state frequency, the voltage ratio <math>|V_A/V_B|</math>, and the relative phase <math>\theta_A-\theta_B</math>, all extracted from the dominant peaks of the Fourier spectra of <math>V_A</math> and <math>V_B</math>.<ref name="Bai2024SM" />

Our available observable is the steady-state frequency. For each chosen value of <math>L_A</math>, the resonators are voltage-activated, the circuit reaches a stable oscillating state, and the dominant spectral peak is recorded. If we denote the Fourier transforms by

<math display="block">
\tilde V_A(f)=\mathcal{F}\{V_A(t)\},
\qquad
\tilde V_B(f)=\mathcal{F}\{V_B(t)\},
</math>

then the dominant oscillation frequency is obtained from

<math display="block">
f_0=\arg\max_f |\tilde V_A(f)| \approx \arg\max_f |\tilde V_B(f)|.
</math>

The experimental data set is therefore

<math display="block">
\{L_A,\ f_0(L_A)\},
</math>

which is compared with the frequency branch plotted in Fig. S11(a) of SM-4. In the full reference measurement, the amplitude ratio

<math display="block">
\rho = \frac{|\tilde V_A(f_0)|}{|\tilde V_B(f_0)|}
</math>

and relative phase

<math display="block">
\Delta\phi = \arg \tilde V_A(f_0)-\arg \tilde V_B(f_0)
</math>

provide additional state verification. Those quantities are useful but are not the main basis of our breadboard validation.

In practice, the same frequency can also be estimated directly from the time-domain period <math>T</math> using <math>f_0=1/T</math> when the waveform is clean. Fourier analysis is preferred because it uses a longer section of waveform and identifies the dominant oscillation peak even when there is small amplitude noise or waveform distortion.

=== Practical inductance sweep ===
In SM-4, the inductive tuning is carried out by varying <math>L_A</math>. The supplementary notes that the tunable inductor exhibits less than about 0.4% inductance variation over the frequency range of interest, which is sufficiently small for controlled sweeping near the singular point. To further suppress voltage-dependent drift during the sweep, the reference experiment slightly adjusts the active-branch resistance so as to maintain approximately <math>V_B \approx 2~\mathrm{V}</math> throughout the measurement series.<ref name="Bai2024SM" />

Our sweep uses controlled series combinations of small inductors, giving an approximately <math>2~\mu\mathrm{H}</math> step in <math>L_A</math>. The homebrew compressible coil is then used to show that a mechanical change of the sensing element also changes the inductance. The conclusion drawn from our data is therefore not a precision determination of <math>L_{A,\mathrm{NEP}}</math>, but a validation that the breadboard circuit produces the same kind of frequency response as the stable branch in Fig. S11(a).

=== Long-window spectral analysis and noise inspection ===
The supplementary also discusses long-time measurements over a window of <math>0<t<5~\mathrm{s}</math>. The motivation is that the noise around the stable oscillation is too small to be judged reliably from raw scope traces alone, so it is more informative to inspect the Fourier spectra and the extracted spectral center frequency and linewidth.<ref name="Bai2024SM" />

This is a useful methodological point for a sensing-oriented reproduction. In an NEP-based sensor, one ultimately cares about how precisely the frequency peak can be resolved and tracked. A long-window Fourier analysis is therefore closer to the sensing task than a purely visual time-domain inspection.

== Results ==

=== Result display: steady oscillation ===
[[File:PC5271inductive1.png|750px|thumb|center|Breadboard implementation and representative steady oscillation waveform]]

This figure should show the breadboard circuit and a representative voltage waveform after the resonators are activated by the supplied voltage. The displayed waveform is the evidence used to identify a stable oscillation frequency.

=== Result display: <math>f-L_A</math> sweep ===
'''[INSERT FIGURE: measured breadboard <math>f-L_A</math> curve compared with Fig. S11(a) of SM-4]'''

This figure should show the measured oscillation frequency <math>f</math> as a function of <math>L_A</math>. The values of <math>L_A</math> in the main sweep are produced by connecting small inductors in series with approximately <math>2~\mu\mathrm{H}</math> step size. The reference curve for comparison is the stable frequency branch in Fig. S11(a) of SM-4.<ref name="Bai2024SM" />

=== Result display: iron-rod demonstration ===
[[File:PC5271inductive2.png|750px|thumb|center|Displacement measurement by compressing the solenoid and inserting iron rod into the homebrew coil]]

This figure should show the homebrew coil with an inserted iron rod and the corresponding change in the measured oscillation frequency. The experimental variable is the rod position or insertion state, and the measured output is the activated steady oscillation frequency.

== Evaluation ==

=== Stable activated oscillation ===
The first required experimental signature is that, after the resonators are activated by the supplied voltage, the circuit reaches a stable oscillating waveform instead of decaying immediately or being amplified without bound. A stable waveform means that the TL071 active branch supplies enough small-signal negative resistance to compensate loss, while gain saturation prevents unbounded amplification.

Physically, this behavior confirms the role of the nonlinear active branch. At small amplitude, the effective negative resistance in resonator <math>A</math> overcomes dissipation. At larger amplitude, the gain becomes weaker, so the oscillation amplitude is clamped. The system self-stabilizes because the physical steady branch is the one with zero imaginary part of the eigenfrequency. The auxiliary branches near the nonlinear exceptional point are not stable physical outputs; they are mathematical branches of the nonlinear steady-state equation. The experiment is therefore not a passive ring-down measurement; it is a nonlinear steady-state selection process.

=== Evaluation of the <math>f-L_A</math> trend ===
The main experimental comparison is the measured <math>f-L_A</math> trend against Fig. S11(a) of SM-4. The reference experiment reports the stable frequency branch of the nonlinear circuit as <math>L_A</math> is tuned.<ref name="Bai2024SM" /> This branch is the physical one because its eigenfrequency is real in the steady state. In our breadboard experiment, the same observable is measured using the series-inductor sweep.

It is useful to separate two statements. First, any LC resonator changes frequency when <math>L</math> changes, because <math>f_0=1/(2\pi\sqrt{LC})</math>. This alone would not demonstrate the nonlinear exceptional-point mechanism. Second, in the coupled active-lossy circuit, the frequency is not just the bare LC resonance frequency of one isolated inductor and capacitor. It is the frequency selected by the nonlinear steady-state balance of gain, loss, and coupling. The reason Fig. S11(a) matters is that it shows this selected stable branch as <math>L_A</math> is swept.

Our breadboard circuit uses different component values and a different op-amp from the PCB experiment, so exact numerical overlap is not expected. The meaningful evaluation is whether the measured frequency branch follows the same qualitative <math>f-L_A</math> behavior as the stable branch in SM-4. Observing that behavior is enough to show that the constructed circuit realizes the intended nonlinear voltage-activated oscillating mechanism.

=== Evaluation of the displacement demonstration ===
In the iron-rod demonstration, the homebrew coil becomes a displacement-dependent inductive element. As the rod enters the coil, the effective magnetic permeability inside the coil increases. In the solenoid expression

<math display="block">
L(x) \approx \mu_\mathrm{eff}(x)\frac{N^2 A}{\ell},
</math>

the insertion depth <math>x</math> changes <math>\mu_\mathrm{eff}</math>. The coil geometry <math>N</math>, <math>A</math>, and <math>\ell</math> are approximately fixed during this demonstration, so the dominant effect is the change in magnetic core material inside the coil. Inductive displacement sensors commonly use this principle: the position of a ferromagnetic object changes a coil inductance, and the electronics convert that inductance change into an electrical signal.<ref name="Chen2014" />

In our circuit, the electrical signal is the activated steady oscillation frequency rather than a bridge voltage. The sensing chain is

<math display="block">
\mathrm{rod\ displacement}\ x
\longrightarrow
\mathrm{effective\ inductance}\ L_A(x)
\longrightarrow
\mathrm{oscillation\ frequency}\ f(x).
</math>

This demonstration shows how the inductive readout can be turned into a displacement-sensing readout after calibration of the relation between rod position and frequency.

=== Cubic-root perturbation law and sensing meaning ===
Near the singular operating point, the eigenfrequency shift follows the hallmark cubic-root law,

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3}.
</math>

This law is the main sensing-related result of the reference experiment.<ref name="Bai2024" /><ref name="Bai2024SM" /> Compared with the linear response of an ordinary isolated resonator, the cubic-root dependence can produce a larger frequency displacement for the same small perturbation of inductance when the device is biased sufficiently close to <math>L_{A,\mathrm{NEP}}</math>.

A useful way to interpret the result is through the local derivative,

<math display="block">
\left|\frac{df}{dL_A}\right| \propto |\delta L_A|^{-2/3}.
</math>

This increasing local responsivity is why the circuit is attractive for sensing. Our experiment supports this sensing logic by showing that the breadboard circuit has a measurable stable frequency branch as <math>L_A</math> is changed. A full quantitative exponent extraction would require identifying <math>L_{A,\mathrm{NEP}}</math> and <math>f_{\mathrm{NEP}}</math>, then plotting <math>|f-f_{\mathrm{NEP}}|</math> against <math>|L_A-L_{A,\mathrm{NEP}}|</math> on logarithmic axes.

=== Basis completeness and noise robustness ===
The supplementary material evaluates the Petermann factor as a diagnostic of modal nonorthogonality and basis completeness. For a non-Hermitian eigenproblem, one may write

<math display="block">
\mathrm{PF}_n =
\frac{\langle \Phi_n^{L}|\Phi_n^{L}\rangle \langle \Phi_n^{R}|\Phi_n^{R}\rangle}
{\left|\langle \Phi_n^{L}|\Phi_n^{R}\rangle\right|^2},
</math>

where <math>|\Phi_n^{R}\rangle</math> and <math>|\Phi_n^{L}\rangle</math> are the right and left eigenvectors of the effective steady-state Hamiltonian. At a conventional linear EP, this quantity is associated with diverging excess-noise sensitivity. In Ref. [1], the cited analysis finds that the effective steady-state Hamiltonian remains non-defective and the Petermann factor is finite at the <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM" />

This point explains the significance of the reference experiment. The system does not merely realize a large mathematical responsivity; it realizes that responsivity in a way that avoids the most severe basis-collapse problem of a linear EP. Our breadboard experiment does not directly test basis completeness, but the result provides the theoretical reason why reproducing the stable <math>f-L_A</math> branch is physically meaningful.

=== Overall interpretation ===
In this platform, the measurand is encoded as a perturbation of <math>L_A</math>. Any physical mechanism that modifies the effective inductance of resonator <math>A</math>, for example displacement, magnetic coupling, or the presence of a nearby magnetic target, moves the system away from the operating point and shifts the oscillation frequency. The primary readout in our implementation is therefore spectral: we track <math>f</math> as <math>L_A</math> is changed.

Taken together, Ref. [1] establishes the complete sensing logic: a nonlinear active-lossy circuit is tuned to an <math>\mathrm{NEP}_3</math>; after voltage activation, the system settles onto a stable oscillating branch; the dominant spectral peak is tracked as the perturbation readout; and the enhanced cubic-root response is obtained without the defective-basis pathology of a conventional linear EP. Our breadboard result demonstrates the experimentally accessible core of this logic by reproducing the stable frequency response corresponding to Fig. S11(a) of SM-4.

== References ==
<references />

Inductive Sensors of Ultra-high Sensitivity Based on Nonlinear Exceptional Point

2026-04-23T05:52:30Z

Siyu: /* Result display: iron-rod demonstration */

== Introduction ==
This project is a repetition of the experiment reported in ''Observation of Nonlinear Exceptional Points with a Complete Basis in Dynamics'', adapted to our available laboratory equipment.<ref name="Bai2024">K. Bai, T.-R. Liu, L. Fang, J.-Z. Li, C. Lin, D. Wan, and M. Xiao, Observation of nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''132''', 073802 (2024).</ref> The circuit contains two coupled resonators. One resonator has gain, meaning that it can supply energy to the oscillation, while the other has loss, meaning that it dissipates energy. A system with both gain and loss is called a '''non-Hermitian''' system. Such systems can show unusual degeneracies called '''exceptional points''' (EPs), where both the eigenfrequencies and the eigenmodes merge.

Exceptional points are interesting for sensing because the measured frequency can respond very strongly to a small perturbation. In an inductive sensor, the perturbation is a change in inductance. If a small change in inductance produces a large change in oscillation frequency, the circuit can in principle detect small physical changes. However, ordinary linear EPs also suffer from strong eigenmode nonorthogonality and a defective basis, which can increase noise sensitivity and complicate practical sensing.<ref name="Bai2024" />

The system studied in this project belongs to the recently developed class of '''nonlinear exceptional points''' (NEPs). The physical platform is a pair of coupled circuit resonators. Resonator <math>A</math> is made active by a saturable negative-resistance branch, while resonator <math>B</math> is intentionally lossy. Because the gain in resonator <math>A</math> depends on the oscillation amplitude, the steady-state problem is nonlinear and admits multiple self-consistent branches. At the singular operating point reported in Ref. [1], one stable steady branch and two auxiliary steady branches merge into a third-order nonlinear exceptional point, denoted <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM">Supplemental Material for K. Bai, T.-R. Liu, L. Fang, J.-Z. Li, C. Lin, D. Wan, and M. Xiao, Observation of nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''132''', 073802 (2024).</ref>

The experimentally important consequence is that the oscillation eigenfrequency obeys a cubic-root perturbation law near the singularity,

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3},
</math>

where <math>\delta L_A = L_A-L_{A,\mathrm{NEP}}</math> is the inductive detuning from the exceptional operating point. This response is steeper than the ordinary linear dependence of a conventional resonator and is therefore useful for ultra-sensitive inductive sensing.<ref name="Bai2024" /><ref name="Bai2024SM" />

A second and equally important point is that the cited work does not interpret the singularity as a conventional defective linear EP. Instead, the effective steady-state Hamiltonian remains non-defective in dynamics, and the associated Petermann factor stays finite at the operating point. In other words, the experiment realizes enhanced spectral responsivity without inheriting the strongest basis-collapse pathology of a standard linear EP.<ref name="Bai2024" /><ref name="Bai2024SM" />

In this project, our goal is to reproduce the experimental logic of Ref. [1] and its supplementary material under course-lab conditions. Unlike the PCB implementation in the paper, our circuit is assembled on a breadboard and the active branch uses a '''TL071''' operational amplifier. Our main observable is the steady oscillation frequency <math>f</math> as a function of the chosen inductance <math>L_A</math>. We tune <math>L_A</math> in two ways. For the main <math>f-L_A</math> sweep, we join small inductors in series so that the discrete inductance step is about <math>2~\mu\mathrm{H}</math>. In addition, we use a homebrew compressible coil to demonstrate the sensing mechanism directly. Compressing the coil changes its length and turn density while keeping the cross-sectional area approximately unchanged. We therefore use the <math>f-L_A</math> behavior in Fig. S11(a) of the supplementary material as the main experimental benchmark. Reproducing this trend is sufficient for demonstrating that our setup captures the intended nonlinear voltage-activated, self-stabilized oscillating response, although it is not a full reproduction of every diagnostic reported in Ref. [1].<ref name="Bai2024" /><ref name="Bai2024SM" />

== Team Members ==
Wang Peikun — E1538091@u.nus.edu 
Zhu Ziyang — E1583446@u.nus.edu 
Yuan Si-Yu — siyu_yuan@u.nus.edu 
Li Xunyu — xunyu@u.nus.edu

== Background ==

=== Ordinary LC and RLC resonators ===
The basic building block of the experiment is an electrical resonator. An ideal LC resonator contains an inductor <math>L</math> and a capacitor <math>C</math>. Energy repeatedly moves between the magnetic field of the inductor and the electric field of the capacitor. The natural angular frequency is

<math display="block">
\omega_0=\frac{1}{\sqrt{LC}},
</math>

or, in ordinary frequency units,

<math display="block">
f_0=\frac{1}{2\pi\sqrt{LC}}.
</math>

This formula is important for the whole project because it explains why changing an inductance changes the measured frequency. A larger <math>L</math> gives a lower resonance frequency, while a smaller <math>L</math> gives a higher resonance frequency.

Real circuits are not ideal. Wires, inductors, capacitors, breadboard contacts, and measuring probes all dissipate energy. This loss is often represented by a resistor <math>R</math>, giving an RLC resonator. In a passive RLC circuit, the oscillation amplitude decays with time because energy is continuously lost as heat. If a circuit is to maintain a steady activated oscillation, the loss must be compensated by a gain element.

=== Gain, loss, and negative resistance ===
In electronics, gain can be represented as an effective negative resistance. A normal positive resistance removes energy from the resonator. A negative resistance does the opposite: it supplies energy to the resonator. In this experiment, resonator <math>A</math> contains an active op-amp branch that behaves like a negative resistance at small amplitude, while resonator <math>B</math> contains ordinary positive loss.

The gain cannot remain negative and constant forever. If it did, any activated oscillation would be amplified without bound until the op-amp clipped or the circuit failed. The important nonlinear ingredient is '''saturable gain''': the active branch gives strong gain at small amplitude but weaker gain at larger amplitude. The stable oscillation amplitude is selected when the supplied gain balances the total loss. This nonlinear steady state is why the experiment is a nonlinear dynamical problem rather than only a passive resonance measurement.

=== What "non-Hermitian" means in this circuit ===
In ordinary conservative physics, a Hermitian Hamiltonian describes a closed system whose total energy is conserved. Its eigenfrequencies are real, and its eigenmodes form a complete orthogonal basis. This is the familiar setting of many quantum-mechanics courses.

A circuit with gain and loss is an open system. Energy can enter through the op-amp branch and leave through resistive loss. The effective dynamical matrix is therefore generally '''non-Hermitian'''. In a non-Hermitian system, eigenfrequencies can be complex:

<math display="block">
\omega=\omega_\mathrm{r}+i\omega_\mathrm{i}.
</math>

The real part <math>\omega_\mathrm{r}</math> gives the oscillation frequency, while the imaginary part <math>\omega_\mathrm{i}</math> describes amplification or decay. A positive imaginary part means the mode is amplified, a negative imaginary part means it decays, and a stable steady oscillation corresponds to zero imaginary part, or equivalently zero net amplification. This is the sense in which the circuit is self-stabilized after voltage activation.

=== Exceptional points ===
In a usual Hermitian system, two modes can have the same eigenfrequency while still remaining two independent modes. This is an ordinary degeneracy. A non-Hermitian system can show a stronger kind of degeneracy called an '''exceptional point''' (EP). At an EP, not only the eigenvalues but also the eigenvectors coalesce. In simple language, two modes collapse into one mode.

This collapse makes EPs interesting for sensing. Near a second-order linear EP, a small perturbation can produce a square-root frequency splitting:

<math display="block">
\Delta f \propto \sqrt{\epsilon},
</math>

where <math>\epsilon</math> is the perturbation. This response is steeper than an ordinary linear response when <math>\epsilon</math> is very small. However, a conventional linear EP also has a serious drawback: the eigenbasis becomes defective, meaning that the system no longer has enough independent eigenvectors. This is associated with strong modal nonorthogonality and enhanced noise sensitivity. The Petermann factor is one common way to quantify this nonorthogonality.

=== Why the nonlinear exceptional point is different ===
The reference paper studies a nonlinear exceptional point. The word "nonlinear" is essential. The circuit does not have a fixed gain coefficient; instead, the gain depends on the oscillation amplitude. The steady state must therefore be solved self-consistently: the frequency, voltage amplitudes, and saturated gain must all agree with one another.

In the theory of Ref. [1], this nonlinear steady-state problem can reduce to a cubic equation for the oscillation frequency. A cubic equation can have a triple root. At the special operating point, one stable physical branch and two auxiliary branches merge, giving a third-order nonlinear exceptional point, written as <math>\mathrm{NEP}_3</math>. The stable physical branch is the one with real oscillation frequency, i.e. without an imaginary part. The auxiliary branches are needed to define the higher-order coalescence mathematically, but they are not physically stable operating states of the circuit. The corresponding frequency response has the form

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3}.
</math>

This cubic-root response is the key sensing signature. It is steeper than both an ordinary linear response and the square-root response of a second-order EP. The important conceptual point of Ref. [1] is that this enhanced response is obtained while the effective steady-state Hamiltonian can still retain a complete basis in dynamics.<ref name="Bai2024" /><ref name="Bai2024SM" />

=== Sensing mechanism ===
The circuit can be understood as a frequency-generating sensor. It converts an inductance change into a frequency change.

The first step is activation. A voltage is supplied to the resonators to excite the circuit into an oscillating state. The active branch in resonator <math>A</math> then supplies energy through an effective negative resistance and amplifies the selected oscillation mode.

The second step is self-stabilization. The gain is saturable, which means it becomes weaker when the voltage amplitude becomes larger. The activated oscillation therefore is not amplified forever. The circuit settles onto the physical steady branch where the imaginary part of the eigenfrequency is zero. At that point, the gain supplied by resonator <math>A</math> balances the loss in the circuit, especially the loss in resonator <math>B</math>.

The third step is frequency selection. Because resonators <math>A</math> and <math>B</math> are coupled, the final oscillation frequency is not simply the resonance frequency of either isolated resonator. It is selected by the whole coupled circuit: the inductors, capacitors, coupling elements, loss, and saturated gain all contribute.

The fourth step is sensing. The inductance <math>L_A</math> controls the bare frequency of resonator <math>A</math>. When <math>L_A</math> changes, the self-consistent frequency selected by the nonlinear circuit also changes. Therefore, tracking the spectral peak <math>f</math> tells us how the effective inductance has changed.

The fifth step is the nonlinear exceptional-point response. Near the nonlinear exceptional point discussed in Ref. [1], the physical steady branch and the two auxiliary branches coalesce in the nonlinear steady-state equation. The frequency shift of the stable branch can then scale as the cube root of the inductance perturbation. This means that a small inductance change can produce a comparatively large frequency change. Our breadboard experiment focuses on verifying the stable <math>f-L_A</math> branch corresponding to Fig. S11(a) of SM-4, which is the experimentally accessible part of this sensing mechanism.

=== Same mechanism in other platforms ===
Several references are included to show that this is a platform-independent sensing mechanism, not only a feature of our breadboard circuit. The common chain is

<math display="block">
\mathrm{physical\ perturbation}
\longrightarrow
\mathrm{parameter\ change}
\longrightarrow
\mathrm{eigenfrequency\ shift\ of\ a\ resonator/cavity\ with\ gain\ and\ loss}
\longrightarrow
\mathrm{higher\ order\ steady\ state\ response}
\longrightarrow
\mathrm{sensing}.
</math>

In our circuit, the physical perturbation is a change in the inductance <math>L_A</math>, and the measured output is the oscillation frequency <math>f</math>. In coupled optical, mechanical, electrical, or hybrid resonators, the perturbed parameter may instead be a cavity detuning, mechanical frequency, phase shift, magnetic-field-dependent transition, or another effective resonator parameter. The shared point is that gain, loss, coupling, and nonlinearity convert the parameter change into a higher-order steady-state spectral response.

The earlier theoretical work on nonlinear exceptional points showed that nonlinear coupled resonators can host higher-order NEPs while retaining a complete dynamical basis.<ref name="Bai2023">K. Bai, J.-Z. Li, T.-R. Liu, L. Fang, D. Wan, and M. Xiao, Nonlinear exceptional points with a complete basis in dynamics, ''Phys. Rev. Lett.'' '''130''', 266901 (2023).</ref> The 2024 experiment implemented the same mechanism in coupled electrical resonators and measured the stable nonlinear frequency branch.<ref name="Bai2024" /><ref name="Bai2024SM" /> Silicon micromechanical resonators provide another realization: there, the measured output can be a phase difference, and the nonlinear exceptional-point response can enhance phase-shift sensing.<ref name="Xie2026">Y.-J. Xie, S.-Y. Zhang, M.-N. Zhang, R. Wang, L.-F. Wang, and Q.-A. Huang, Nonlinear exceptional points boost phase-shift sensing in silicon micromechanical resonators, ''Phys. Rev. Applied'' '''25''', 034039 (2026).</ref> Hybrid quantum systems with nonlinear bistability show a related route, where nonlinear transition dynamics can enhance sensing signal-to-noise ratio.<ref name="Wang2026">H. Wang ''et al.'', Exceptional sensitivity near the bistable transition point of a hybrid quantum system, ''Nature Physics'' '''22''', 577-583 (2026).</ref>

These examples support the same mechanism rather than unrelated applications. At the same time, enhanced response alone is not enough to prove better sensing. Noise can shift or weaken the apparent nonlinear exceptional point and can limit the signal-to-noise improvement.<ref name="Zheng2025">X. Zheng and Y. D. Chong, Noise constraints for nonlinear exceptional point sensing, arXiv:2410.08502v2 (2025).</ref> This is why our project makes a careful claim: we demonstrate the frequency-response mechanism on a breadboard, but we do not claim a complete metrological noise advantage.

== Report objectives and scope ==
The purpose of this work is to study the circuit principle behind '''nonlinear-exceptional-point-enhanced inductive sensing''' in a coupled electronic resonator platform. The scope is intentionally narrower than the full experiment reported in Ref. [1]. We are not attempting to redesign the active branch, derive all nonlinear bifurcation theory from first principles, reconstruct the auxiliary steady-state branches, or characterize every parasitic element. Instead, we adopt the framework of Ref. [1] and SM-4 and ask whether the breadboard circuit can reproduce the experimentally accessible frequency response.

The main target of our experiment is the <math>f-L_A</math> curve corresponding to Fig. S11(a) of SM-4. In the reference experiment, the steady-state frequency, voltage ratio <math>|V_A/V_B|</math>, and relative phase <math>\theta_A-\theta_B</math> are all measured as functions of <math>L_A</math>.<ref name="Bai2024SM" /> In our implementation, the frequency-versus-inductance curve is the primary observable. We tune <math>L_A</math> by connecting small inductors in series with approximately <math>2~\mu\mathrm{H}</math> resolution. We also use a homebrew coil in a separate demonstration where mechanical compression or insertion of an iron rod changes the effective inductance. The experimental goal is therefore to verify that the TL071 breadboard circuit settles into a stable oscillating state and that the measured frequency changes with <math>L_A</math> in the same qualitative manner as Fig. S11(a).

From the sensing perspective, the most relevant figure of merit is the local frequency responsivity. If

<math display="block">
|f-f_{\mathrm{NEP}}| = A |\delta L_A|^{1/3},
</math>

then differentiation gives

<math display="block">
S_f \equiv \left|\frac{df}{dL_A}\right| \propto |\delta L_A|^{-2/3}.
</math>

Thus, as the operating point approaches <math>L_{A,\mathrm{NEP}}</math>, the local spectral responsivity increases rapidly. The main practical constraint is that the system must remain on the intended stable branch while measurement noise and slow parameter drift are kept below the scale of the induced frequency shift.

== Setups ==
[[File:Experimental_setup.png|750px|thumb|center|Experimental setup adapted from Ref. [1].]]

=== Circuit topology and component implementation ===
The reference platform consists of two coupled RLC resonators. Both resonators contain the same nominal shunt capacitance <math>C_0</math>, while the inductances <math>L_A</math> and <math>L_B</math> set their bare resonance frequencies. The two resonators are coupled through a parallel resistor-capacitor channel characterized by <math>R_c</math> and <math>C_c</math>. Resonator <math>B</math> contains an explicit positive resistance <math>R_B</math> and therefore plays the role of the lossy resonator. Resonator <math>A</math>, by contrast, contains an active branch whose effective resistance is negative at small amplitude and becomes less negative as the oscillation amplitude increases, thereby providing saturable gain.<ref name="Bai2024" /><ref name="Bai2024SM" />

Our implementation follows this topology but is built on a breadboard rather than a custom PCB. The active branch is implemented with an op-amp negative-resistance circuit using a '''TL071''' operational amplifier. This is an important practical difference from the supplementary experiment, where the active components explicitly listed are a TI '''LM7171''' operational amplifier and an '''Onsemi BAV99L''' diode.<ref name="Bai2024SM" /> The TL071 substitution and the breadboard layout introduce different bandwidth limits, wiring parasitics, and contact resistances, so the numerical operating point is not expected to match the PCB experiment. For this reason, our comparison emphasizes the observable trend of <math>f</math> versus <math>L_A</math>.

The supplementary material reports the following representative component values for the PCB experiment:

<math display="block">
C_0 = 18.5~\mathrm{nF}, \qquad
C_c = 3.9~\mathrm{nF}, \qquad
R_c = 760~\Omega,
</math>

<math display="block">
L_B = 197.7~\mu\mathrm{H}, \qquad
R_B = 1314~\Omega.
</math>

A representative operating point shown in the main figure uses

<math display="block">
L_A = 245.4~\mu\mathrm{H}.
</math>

Our breadboard circuit follows the same functional topology, but its component values and parasitic elements are not identical to the PCB values above. The horizontal axis of our main <math>f-L_A</math> plot is therefore the series-combined <math>L_A</math> selected from our available small inductors, while the measured frequency <math>f</math> is the robust experimental readout.

=== Inductance tuning methods ===
We use two complementary methods to change the inductance seen by resonator <math>A</math>.

The first method is used for the main <math>f-L_A</math> sweep. Small inductors are connected in series so that the total inductance changes in discrete steps of approximately <math>2~\mu\mathrm{H}</math>. If the inductors are sufficiently separated so that their mutual magnetic coupling is small, the total series inductance is approximately the sum of the individual inductances,

<math display="block">
L_A \approx L_{\mathrm{base}}+\sum_i L_i .
</math>

This method is simple and reproducible. Adding one small inductor increases <math>L_A</math>; removing it decreases <math>L_A</math>. The circuit frequency is then measured for each chosen value. This produces a discrete version of the <math>f-L_A</math> curve in Fig. S11(a) of SM-4. The step size is not as fine as a commercial variable inductor, but <math>2~\mu\mathrm{H}</math> spacing is small enough to observe the frequency trend over the accessible range.

The second method is a homebrew compressible variable inductor. This coil is useful because it makes the sensing mechanism visible: a mechanical displacement changes the coil geometry, which changes the inductance, which then changes the oscillation frequency. The basic idea can be understood from the approximate solenoid formula

<math display="block">
L \approx \mu_\mathrm{eff}\frac{N^2 A}{\ell},
</math>

where <math>N</math> is the number of turns, <math>A</math> is the cross-sectional area, <math>\ell</math> is the coil length, and <math>\mu_\mathrm{eff}</math> describes the effective magnetic permeability of the core and surrounding region.<ref name="Chen2014">S.-C. Chen, D.-K. Le, and V.-S. Nguyen, Inductive displacement sensors with a notch filter for an active magnetic bearing system, ''Sensors'' '''14''', 12640-12657 (2014).</ref> During compression, the number of turns <math>N</math> is unchanged and the cross-sectional area <math>A</math> remains approximately unchanged, but the length <math>\ell</math> decreases. Equivalently, the turn density

<math display="block">
n=\frac{N}{\ell}
</math>

increases. The magnetic field produced by a solenoid is proportional to turn density. Therefore, compressing the coil increases the magnetic flux linkage through the turns and increases the inductance. Stretching the coil has the opposite effect.

This mechanical tuning is not identical to the precision tunable inductor used in the reference experiment. A real compressed coil is not an ideal long solenoid: the spacing between turns is finite, the end effects change as the coil length changes, nearby wires and the breadboard add parasitic capacitance, and the leads contribute additional series inductance and resistance. For this reason, the homebrew inductor is best treated as an experimentally characterized tuning element rather than an ideal component whose value is known exactly from geometry. In the present project, it provides a direct mechanical demonstration of how displacement can perturb <math>L_A</math>.

Experimentally, we measured the effective inductance of this homebrew variable inductor at different compression states. The compression process changes the coil length and wire density while leaving the coil area approximately fixed. Therefore, the tuning mechanism is physically the same as changing <math>\ell</math> and <math>n=N/\ell</math> in the solenoid model, rather than changing the number of turns or the coil diameter.

=== Kirchhoff model of the nonlinear circuit ===
A convenient state-space description uses the node voltages <math>V_A(t)</math> and <math>V_B(t)</math> together with the inductor currents <math>I_A(t)</math> and <math>I_B(t)</math>. A node voltage is the voltage measured at one resonator node relative to ground. Kirchhoff's current law says that the algebraic sum of all currents flowing away from a node must be zero. This gives a direct way to translate the circuit diagram into differential equations.

Denoting the amplitude-dependent conductance of the active branch by <math>G_A(|V_A|)</math>, where <math>G_A<0</math> in the small-signal regime, the nodal equations can be written as

<math display="block">
\begin{aligned}
C_0 \frac{dV_A}{dt} + I_A + G_A(|V_A|)\,V_A + \frac{V_A-V_B}{R_c}
+ C_c\frac{d(V_A-V_B)}{dt} &= 0,\\
C_0 \frac{dV_B}{dt} + I_B + \frac{V_B}{R_B} + \frac{V_B-V_A}{R_c}
+ C_c\frac{d(V_B-V_A)}{dt} &= 0,\\
L_A \frac{dI_A}{dt} &= V_A,\\
L_B \frac{dI_B}{dt} &= V_B.
\end{aligned}
</math>

Each term has a simple circuit meaning. The terms <math>C_0 dV_A/dt</math> and <math>C_0 dV_B/dt</math> are capacitor currents. The terms <math>I_A</math> and <math>I_B</math> are currents through the inductors. The term <math>G_A(|V_A|)V_A</math> is the active branch current; because <math>G_A</math> can be negative, this branch can inject energy instead of dissipating it. The term <math>V_B/R_B</math> is the ordinary loss current through the resistor in resonator <math>B</math>. The terms involving <math>R_c</math> and <math>C_c</math> describe current flowing between resonators <math>A</math> and <math>B</math>, so they are the electrical coupling between the two resonators.

These equations show explicitly how the experiment differs from a linear EP circuit. If <math>G_A</math> were a constant negative conductance, the model would reduce to a linear non-Hermitian dimer. Here, however, <math>G_A</math> depends on the voltage amplitude, so the steady state must be determined self-consistently together with the oscillation amplitude itself.

=== Harmonic-balance formulation of the steady state ===
After the transient has died away, the oscilloscope waveform is approximately periodic. The simplest approximation is therefore to keep only the dominant sinusoidal component. This is called a harmonic-balance approximation. For a stable activated oscillating state, one writes

<math display="block">
V_j(t)=\mathrm{Re}\!\left[\tilde V_j e^{i\omega t}\right],
\qquad j=A,B.
</math>

Substituting this ansatz into the circuit equations yields a nonlinear algebraic problem of the form

<math display="block">
\mathbf{Y}(\omega,|\tilde V_A|)
\begin{pmatrix}
\tilde V_A\\
\tilde V_B
\end{pmatrix}
=
\begin{pmatrix}
Y_A(\omega,|\tilde V_A|)+Y_c(\omega) & -Y_c(\omega)\\
-Y_c(\omega) & Y_B(\omega)+Y_c(\omega)
\end{pmatrix}
\begin{pmatrix}
\tilde V_A\\
\tilde V_B
\end{pmatrix}
=0,
</math>

with

<math display="block">
Y_A(\omega,|\tilde V_A|)=G_A(|\tilde V_A|)+i\omega C_0+\frac{1}{i\omega L_A},
</math>

<math display="block">
Y_B(\omega)=\frac{1}{R_B}+i\omega C_0+\frac{1}{i\omega L_B},
\qquad
Y_c(\omega)=\frac{1}{R_c}+i\omega C_c.
</math>

A nontrivial solution requires

<math display="block">
\det \mathbf{Y}(\omega,|\tilde V_A|)=0.
</math>

This determinant condition is the circuit analogue of an eigenfrequency condition. In a passive LC resonator, the resonance condition gives a single frequency <math>\omega_0=1/\sqrt{LC}</math>. In the present two-resonator active circuit, the same idea becomes a matrix equation because the two resonators are coupled. The amplitudes <math>\tilde V_A</math> and <math>\tilde V_B</math> are not arbitrary: a nonzero oscillation exists only when the admittance matrix has a zero eigenvalue.

Because the active admittance depends on amplitude, this condition is nonlinear and supports multiple branches. The supplementary material shows that the steady-state frequency can be reduced to a '''real-coefficient cubic polynomial''' in <math>\omega</math>. The existence of up to three real steady-state branches is therefore not accidental; it is the algebraic origin of the experimentally relevant <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM" />

=== Effective nonlinear-Hamiltonian picture ===
The same physics is often rewritten in coupled-mode form. This notation is common in non-Hermitian physics because it makes the circuit look like two coupled modes. The variables <math>\psi_A</math> and <math>\psi_B</math> represent the complex oscillation amplitudes in resonators <math>A</math> and <math>B</math>. The model is

<math display="block">
i\frac{d}{dt}
\begin{pmatrix}
\psi_A\\
\psi_B
\end{pmatrix}
=
\begin{pmatrix}
\omega_A+i\,g(|\psi_A|) & \kappa\\
\kappa & \omega_B-i\ell
\end{pmatrix}
\begin{pmatrix}
\psi_A\\
\psi_B
\end{pmatrix},
</math>

where <math>\omega_A</math> and <math>\omega_B</math> are the effective modal frequencies, <math>\kappa</math> is the coupling strength, <math>\ell</math> is the loss of resonator <math>B</math>, and <math>g(|\psi_A|)</math> is the saturable gain of resonator <math>A</math>. The steady-state problem then becomes

<math display="block">
H_s(\Psi)\Psi = \Omega \Psi,
</math>

with a real oscillation eigenfrequency <math>\Omega=2\pi f</math> for the stable oscillating branch. The word "real" is important: it means the imaginary part has vanished, so the mode is neither being amplified nor decaying in the steady state. In the cited experiment, this physical stable branch and two auxiliary steady branches coalesce at the singular point, producing a third-order nonlinear exceptional point rather than a conventional second-order linear EP. The auxiliary branches help form the cubic-root singularity but are not themselves stable physical outputs of the circuit.<ref name="Bai2024" /><ref name="Bai2024SM" />

== Measurements ==

=== Acquisition protocol and steady-state selection ===
The measurement is performed in the time domain. In the reference experiment, an arbitrary waveform generator first applies an external start-up signal of 1 V at 70 kHz in order to bring the system into the basin of attraction of the stable oscillating state.<ref name="Bai2024" /> In our breadboard measurement, an external voltage is likewise required to activate the resonators. The measured frequency is extracted from the activated steady waveform recorded on the oscilloscope.

The physical meaning of this procedure is important. The oscillation is not assumed to appear without activation; a supplied voltage is needed to bring the resonators into the relevant dynamical state. The measured oscillation frequency is then interpreted as the frequency selected by the nonlinear steady-state condition.

=== Extraction of the <math>f-L_A</math> observable ===
Ref. [1] does not treat the oscilloscope traces merely as qualitative evidence of oscillation. The supplementary explicitly states that the full reference experiment reports the steady-state frequency, the voltage ratio <math>|V_A/V_B|</math>, and the relative phase <math>\theta_A-\theta_B</math>, all extracted from the dominant peaks of the Fourier spectra of <math>V_A</math> and <math>V_B</math>.<ref name="Bai2024SM" />

Our available observable is the steady-state frequency. For each chosen value of <math>L_A</math>, the resonators are voltage-activated, the circuit reaches a stable oscillating state, and the dominant spectral peak is recorded. If we denote the Fourier transforms by

<math display="block">
\tilde V_A(f)=\mathcal{F}\{V_A(t)\},
\qquad
\tilde V_B(f)=\mathcal{F}\{V_B(t)\},
</math>

then the dominant oscillation frequency is obtained from

<math display="block">
f_0=\arg\max_f |\tilde V_A(f)| \approx \arg\max_f |\tilde V_B(f)|.
</math>

The experimental data set is therefore

<math display="block">
\{L_A,\ f_0(L_A)\},
</math>

which is compared with the frequency branch plotted in Fig. S11(a) of SM-4. In the full reference measurement, the amplitude ratio

<math display="block">
\rho = \frac{|\tilde V_A(f_0)|}{|\tilde V_B(f_0)|}
</math>

and relative phase

<math display="block">
\Delta\phi = \arg \tilde V_A(f_0)-\arg \tilde V_B(f_0)
</math>

provide additional state verification. Those quantities are useful but are not the main basis of our breadboard validation.

In practice, the same frequency can also be estimated directly from the time-domain period <math>T</math> using <math>f_0=1/T</math> when the waveform is clean. Fourier analysis is preferred because it uses a longer section of waveform and identifies the dominant oscillation peak even when there is small amplitude noise or waveform distortion.

=== Practical inductance sweep ===
In SM-4, the inductive tuning is carried out by varying <math>L_A</math>. The supplementary notes that the tunable inductor exhibits less than about 0.4% inductance variation over the frequency range of interest, which is sufficiently small for controlled sweeping near the singular point. To further suppress voltage-dependent drift during the sweep, the reference experiment slightly adjusts the active-branch resistance so as to maintain approximately <math>V_B \approx 2~\mathrm{V}</math> throughout the measurement series.<ref name="Bai2024SM" />

Our sweep uses controlled series combinations of small inductors, giving an approximately <math>2~\mu\mathrm{H}</math> step in <math>L_A</math>. The homebrew compressible coil is then used to show that a mechanical change of the sensing element also changes the inductance. The conclusion drawn from our data is therefore not a precision determination of <math>L_{A,\mathrm{NEP}}</math>, but a validation that the breadboard circuit produces the same kind of frequency response as the stable branch in Fig. S11(a).

=== Long-window spectral analysis and noise inspection ===
The supplementary also discusses long-time measurements over a window of <math>0<t<5~\mathrm{s}</math>. The motivation is that the noise around the stable oscillation is too small to be judged reliably from raw scope traces alone, so it is more informative to inspect the Fourier spectra and the extracted spectral center frequency and linewidth.<ref name="Bai2024SM" />

This is a useful methodological point for a sensing-oriented reproduction. In an NEP-based sensor, one ultimately cares about how precisely the frequency peak can be resolved and tracked. A long-window Fourier analysis is therefore closer to the sensing task than a purely visual time-domain inspection.

== Results ==

=== Result display: steady oscillation ===
[[File:PC5271inductive1.png|750px|thumb|center|Breadboard implementation and representative steady oscillation waveform]]

This figure should show the breadboard circuit and a representative voltage waveform after the resonators are activated by the supplied voltage. The displayed waveform is the evidence used to identify a stable oscillation frequency.

=== Result display: <math>f-L_A</math> sweep ===
'''[INSERT FIGURE: measured breadboard <math>f-L_A</math> curve compared with Fig. S11(a) of SM-4]'''

This figure should show the measured oscillation frequency <math>f</math> as a function of <math>L_A</math>. The values of <math>L_A</math> in the main sweep are produced by connecting small inductors in series with approximately <math>2~\mu\mathrm{H}</math> step size. The reference curve for comparison is the stable frequency branch in Fig. S11(a) of SM-4.<ref name="Bai2024SM" />

=== Result display: iron-rod demonstration ===
[[File:PC5271inductive2.png|750px|thumb|center|Displacement measurement by compressing the solenoid and inserting iron rod into the homebrew coil]]

This figure should show the homebrew coil with an inserted iron rod and the corresponding change in the measured oscillation frequency. The experimental variable is the rod position or insertion state, and the measured output is the activated steady oscillation frequency.

== Evaluation ==

=== Stable activated oscillation ===
The first required experimental signature is that, after the resonators are activated by the supplied voltage, the circuit reaches a stable oscillating waveform instead of decaying immediately or being amplified without bound. A stable waveform means that the TL071 active branch supplies enough small-signal negative resistance to compensate loss, while gain saturation prevents unbounded amplification.

Physically, this behavior confirms the role of the nonlinear active branch. At small amplitude, the effective negative resistance in resonator <math>A</math> overcomes dissipation. At larger amplitude, the gain becomes weaker, so the oscillation amplitude is clamped. The system self-stabilizes because the physical steady branch is the one with zero imaginary part of the eigenfrequency. The auxiliary branches near the nonlinear exceptional point are not stable physical outputs; they are mathematical branches of the nonlinear steady-state equation. The experiment is therefore not a passive ring-down measurement; it is a nonlinear steady-state selection process.

=== Evaluation of the <math>f-L_A</math> trend ===
The main experimental comparison is the measured <math>f-L_A</math> trend against Fig. S11(a) of SM-4. The reference experiment reports the stable frequency branch of the nonlinear circuit as <math>L_A</math> is tuned.<ref name="Bai2024SM" /> This branch is the physical one because its eigenfrequency is real in the steady state. In our breadboard experiment, the same observable is measured using the series-inductor sweep.

It is useful to separate two statements. First, any LC resonator changes frequency when <math>L</math> changes, because <math>f_0=1/(2\pi\sqrt{LC})</math>. This alone would not demonstrate the nonlinear exceptional-point mechanism. Second, in the coupled active-lossy circuit, the frequency is not just the bare LC resonance frequency of one isolated inductor and capacitor. It is the frequency selected by the nonlinear steady-state balance of gain, loss, and coupling. The reason Fig. S11(a) matters is that it shows this selected stable branch as <math>L_A</math> is swept.

Our breadboard circuit uses different component values and a different op-amp from the PCB experiment, so exact numerical overlap is not expected. The meaningful evaluation is whether the measured frequency branch follows the same qualitative <math>f-L_A</math> behavior as the stable branch in SM-4. Observing that behavior is enough to show that the constructed circuit realizes the intended nonlinear voltage-activated oscillating mechanism.

=== Evaluation of the displacement demonstration ===
In the iron-rod demonstration, the homebrew coil becomes a displacement-dependent inductive element. As the rod enters the coil, the effective magnetic permeability inside the coil increases. In the solenoid expression

<math display="block">
L(x) \approx \mu_\mathrm{eff}(x)\frac{N^2 A}{\ell},
</math>

the insertion depth <math>x</math> changes <math>\mu_\mathrm{eff}</math>. The coil geometry <math>N</math>, <math>A</math>, and <math>\ell</math> are approximately fixed during this demonstration, so the dominant effect is the change in magnetic core material inside the coil. Inductive displacement sensors commonly use this principle: the position of a ferromagnetic object changes a coil inductance, and the electronics convert that inductance change into an electrical signal.<ref name="Chen2014" />

In our circuit, the electrical signal is the activated steady oscillation frequency rather than a bridge voltage. The sensing chain is

<math display="block">
\mathrm{rod\ displacement}\ x
\longrightarrow
\mathrm{effective\ inductance}\ L_A(x)
\longrightarrow
\mathrm{oscillation\ frequency}\ f(x).
</math>

This demonstration shows how the inductive readout can be turned into a displacement-sensing readout after calibration of the relation between rod position and frequency.

=== Cubic-root perturbation law and sensing meaning ===
Near the singular operating point, the eigenfrequency shift follows the hallmark cubic-root law,

<math display="block">
|f-f_{\mathrm{NEP}}| \propto |\delta L_A|^{1/3}.
</math>

This law is the main sensing-related result of the reference experiment.<ref name="Bai2024" /><ref name="Bai2024SM" /> Compared with the linear response of an ordinary isolated resonator, the cubic-root dependence can produce a larger frequency displacement for the same small perturbation of inductance when the device is biased sufficiently close to <math>L_{A,\mathrm{NEP}}</math>.

A useful way to interpret the result is through the local derivative,

<math display="block">
\left|\frac{df}{dL_A}\right| \propto |\delta L_A|^{-2/3}.
</math>

This increasing local responsivity is why the circuit is attractive for sensing. Our experiment supports this sensing logic by showing that the breadboard circuit has a measurable stable frequency branch as <math>L_A</math> is changed. A full quantitative exponent extraction would require identifying <math>L_{A,\mathrm{NEP}}</math> and <math>f_{\mathrm{NEP}}</math>, then plotting <math>|f-f_{\mathrm{NEP}}|</math> against <math>|L_A-L_{A,\mathrm{NEP}}|</math> on logarithmic axes.

=== Basis completeness and noise robustness ===
The supplementary material evaluates the Petermann factor as a diagnostic of modal nonorthogonality and basis completeness. For a non-Hermitian eigenproblem, one may write

<math display="block">
\mathrm{PF}_n =
\frac{\langle \Phi_n^{L}|\Phi_n^{L}\rangle \langle \Phi_n^{R}|\Phi_n^{R}\rangle}
{\left|\langle \Phi_n^{L}|\Phi_n^{R}\rangle\right|^2},
</math>

where <math>|\Phi_n^{R}\rangle</math> and <math>|\Phi_n^{L}\rangle</math> are the right and left eigenvectors of the effective steady-state Hamiltonian. At a conventional linear EP, this quantity is associated with diverging excess-noise sensitivity. In Ref. [1], the cited analysis finds that the effective steady-state Hamiltonian remains non-defective and the Petermann factor is finite at the <math>\mathrm{NEP}_3</math>.<ref name="Bai2024SM" />

This point explains the significance of the reference experiment. The system does not merely realize a large mathematical responsivity; it realizes that responsivity in a way that avoids the most severe basis-collapse problem of a linear EP. Our breadboard experiment does not directly test basis completeness, but the result provides the theoretical reason why reproducing the stable <math>f-L_A</math> branch is physically meaningful.

=== Overall interpretation ===
In this platform, the measurand is encoded as a perturbation of <math>L_A</math>. Any physical mechanism that modifies the effective inductance of resonator <math>A</math>, for example displacement, magnetic coupling, or the presence of a nearby magnetic target, moves the system away from the operating point and shifts the oscillation frequency. The primary readout in our implementation is therefore spectral: we track <math>f</math> as <math>L_A</math> is changed.

Taken together, Ref. [1] establishes the complete sensing logic: a nonlinear active-lossy circuit is tuned to an <math>\mathrm{NEP}_3</math>; after voltage activation, the system settles onto a stable oscillating branch; the dominant spectral peak is tracked as the perturbation readout; and the enhanced cubic-root response is obtained without the defective-basis pathology of a conventional linear EP. Our breadboard result demonstrates the experimentally accessible core of this logic by reproducing the stable frequency response corresponding to Fig. S11(a) of SM-4.

== References ==
<references />

File:PC5271inductive2.png

2026-04-23T05:50:10Z

Siyu:

2026-01-30T09:14:20Z

Siyu: /* Inductor Sensors of Ultra-high Sensitivity Based on Nonlinear Exceptional Point */

'''Welcome to the wiki page for the course PC5271: Physics of Sensors "(in AY25/26 Sem 2)!'''

This is the repository where projects are documented. You will need to create an account for editing/creating pages. If you need an account, please contact Christian.

'''Logistics''':
Our '''location is S11-02-04''', time slots for "classes" are '''Tue and Fri 10:00am-12:00noon'''.

==Projects==
===[[Project 1 (Example)]]===
Keep a very brief description of a project or even a suggestion here, and perhaps the names of the team members, or who to contact if there is interest to join. Once the project has stabilized, keep stuff in the project page linked by the headline.

===[[Inductor Sensors of Ultra-high Sensitivity Based on Nonlinear Exceptional Point]]===
Team members: Yuan Siyu; Zhu Ziyang; Wang Peikun; Li Xunyu

We are building two coupled oscillating circuits: one that naturally loses energy (lossy) and one that gains energy (active) using a specific amplifier that saturates at high amplitudes. When tuning these two circuits to a nonlinear Exceptional Point (NEP), the system becomes extremely sensitive to small perturbations in inductance, following a steep cubic-root response curve, while remaining resistant to noise.

'''CK:''' We likely have all the parts for this, but let us know the frequency so we can find the proper amplifier and circuit board.

‘’’SY:’’’ Thanks for your confirmation. The operating frequency is roughly 70-80 kHz.

===[[EA Spectroscopy as a series of sensors: Investigating the Impact of Film-Processing Temperature on Mobility in Organic Diodes]]===
Team members: Li Jinhan; Liu Chenyang

We will use EA spectroscopy, which will include optical sensors, electrical sensors, and lock-in amplifiers, among other components as a highly sensitive, non-destructive optical sensing platform to measure the internal electric field modulation response of organic diodes under operating conditions, and to quantitatively extract carrier mobility based on this measurement. By systematically controlling the thin film preparation temperature and comparing the EA response characteristics of different samples, the project aims to reveal the influence of film preparation temperature on device mobility and its potential physical origins.

===[[Optical Sensor of Magnetic Dynamics: A Balanced-Detection MOKE Magnetometer]]===
Team members: LI Junxiang; Patricia Breanne Tan Sy

We will use a laser-based magneto-optical Kerr effect setup featuring a high-sensitivity differential photodiode array to measure the Kerr rotation angle induced by surface magnetism. This system serves as a versatile optical platform to investigate how external perturbations such as magnetic fields or radiation source alter the magnetic ordering of materials, allowing for the quantitative extraction of the magneto-optical coupling coefficients of various thin films.

==Resources==
===Books and links===
* A good textbook on the Physics of Sensors is Jacob Fraden: Handbook of Mondern Sensors, Springer, ISBN 978-3-319-19302-1 or [https://link.springer.com/book/10.1007/978-3-319-19303-8 doi:10.1007/978-3-319-19303-8]. There shoud be an e-book available through the NUS library at https://linc.nus.edu.sg/record=b3554643
* Another good textbook: John B.Bentley: Principles of Measurement Systems, 4th Edition, Pearson, ISBN: 0-13-043028-5 or https://linc.nus.edu.sg/record=b2458243 in our library.

===Software===
* Various Python extensions. [https://www.python.org Python] is a very powerful free programming language that runs on just about any computer platform. It is open source and completely free.
* [https://www.gnuplot.info Gnuplot]: A free and very mature data display tool that works on just about any platform used that produces excellent publication-grade eps and pdf figures. Can be also used in scripts. Open source and completely free.
* Matlab: Very common, good toolset also for formal mathematics, good graphics. Expensive. We may have a site license, but I am not sure how painful it is for us to get a license for this course. Ask if interested.
* Mathematica: More common among theroetical physicists, very good in formal maths, now with better numerics. Graphs are ok but can be a pain to make looking good. As with Matlab, we do have a campus license. Ask if interested.

===Apps===
Common mobile phones these days are equipped with an amazing toolchest of sensors. There are a few apps that allow you to access them directly, and turn your phone into a powerful sensor. Here some suggestions:

* Physics Toolbox sensor suite on [https://play.google.com/store/apps/details?id=com.chrystianvieyra.physicstoolboxsuite&hl=en_SG Google play store] or [https://apps.apple.com/us/app/physics-toolbox-sensor-suite/id1128914250 Apple App store].

===Data sheets===
A number of components might be useful for several groups. Some common data sheets are here:
* Photodiodes:
** Generic Silicon pin Photodiode type [[Media:Bpw34.pdf|BPW34]]
** Fast photodiodes (Silicon PIN, small area): [[Media:S5971_etc_kpin1025e.pdf|S5971/S5972/S5973]]
* PT 100 Temperature sensors based on platinum wire: [[Media:PT100_TABLA_R_T.pdf|Calibration table]]
* Thermistor type [[Media:Thermistor B57861S.pdf|B57861S]] (R0=10kΩ, B=3988Kelvin). Search for [https://en.wikipedia.org/wiki/Steinhart-Hart_equation Steinhart-Hart equation]. See [[Thermistor]] page here as well.
* Humidity sensor
** Sensirion device the reference unit: [[media:Sensirion SHT30-DIS.pdf|SHT30/31]]
* Thermopile detectors:
** [[Media:Thermopile_G-TPCO-035 TS418-1N426.pdf|G-TPCO-035 / TS418-1N426]]: Thermopile detector with a built-in optical bandpass filter for light around 4μm wavelength for CO2 absorption
* Resistor color codes are explained [https://en.wikipedia.org/wiki/Electronic_color_code here]

** Fluxgate magnetometer [[media:Data-sheet FLC-100.pdf|FCL100]]
* Lasers
** Red laser diode [[media:HL6501MG.pdf|HL6501MG]]
* Generic amplifiers
** Instrumentation amplifiers: [[media:Ad8221.pdf|AD8221]] or [[media:AD8226.pdf|AD8226]]
** Conventional operational amplifiers: Precision: [[media:OP27.pdf|OP27]], General purpose: [[media:OP07.pdf|OP07]]
** Transimpedance amplifiers for photodetectors: [[media:AD8015.pdf|AD8015]]

==Some wiki reference materials==
* [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:FAQ MediaWiki FAQ]
* [[Writing mathematical expressions]]
* [[Uploading images and files]]

==Old wikis==
You can find entries to the wiki from [https://pc5271.org/PC5271_AY2425S2 AY2024/25 Sem 2] and [https://pc5271.org/PC5271_AY2324S2 AY2023/24 Sem 2].

Inductor Sensors of Ultra-high Sensitivity Based on Nonlinear Exceptional Point

2026-01-27T05:49:47Z

Siyu: /* Team Members */

Inductor Sensors of Ultra-high Sensitivity Based on Nonlinear Exceptional Point

2026-01-27T05:48:09Z

Siyu: /* Setups */

==Introduction==
We are building two coupled oscillating circuits: one that naturally loses energy (lossy) and one that gains energy (active) using a specific amplifier that saturates at high amplitudes. When tuning these two circuits to a nonlinear Exceptional Point (NEP), the system becomes extremely sensitive to small perturbations in inductance, following a steep cubic-root response curve, while remaining resistant to noise.

==Team Members==
Wang Peikun E1538091@u.nus.edu

Zhu Ziyang E1583446@u.nus.edu

Yuan Si-Yu E1353381@u.nus.edu

Li Xunyu xunyu@u.nus.edu

==Setups==
[[File:Experimental_setup.png|500px|thumb|center|Experimental setups]]

In FIG. 1 adopted from Ref. 1, we present:

(a) Schematic of the circuit comprising inductors (<math>L</math>), capacitors (<math>C</math>), resistors (<math>R</math>), diodes (<math>D</math>), and an amplifier (<math>A</math>). The dashed rectangle indicates the negative resistance element, <math>-R_A(|V_A|)</math>.

(b) Photograph of the experimental setup. An oscilloscope records the waveforms of <math>V_A</math> and <math>V_B</math> to derive frequency, voltage, and relative phase data. A DC unit powers the amplifier, while an arbitrary waveform generator provides the external driving signal. A variable inductor allows for fine-tuning of <math>L_A</math> to control the resonance frequency of resonator <math>A</math>.

(c) Temporal dynamics of <math>V_A</math> (red) and <math>V_B</math> (blue) following the cessation of a 1 V, 70 kHz external driving signal. The system achieves a stable state at approximately 0.4 ms. Bold lines depict the signal envelopes, which remain constant after stabilization.

(d) Magnified view of the stable state dynamics beginning at 0.66 ms. Experimental parameters are: <math>L_A = 245.4~\mu\mathrm{H}</math>, <math>C_0 = 18.5~\mathrm{nF}</math>, <math>C_c = 3.9~\mathrm{nF}</math>, <math>R_c = 760~\Omega</math>, <math>R_B = 1314~\Omega</math>, and <math>L_B = 197.7~\mu\mathrm{H}</math>.

==Measurements==

...

==Results and Analysis==

...

==References==
[1] K. Bai, T.-R. Liu, L. Fang, J.-Z. Li, C. Lin, D. Wan, and M. Xiao, Observation of nonlinear exceptional points with a complete basis in dynamics, Phys. Rev. Lett. 132, 073802 (2024).

Inductor Sensors of Ultra-high Sensitivity Based on Nonlinear Exceptional Point

2026-01-27T05:46:45Z

Siyu: /* Setups */

==Introduction==
We are building two coupled oscillating circuits: one that naturally loses energy (lossy) and one that gains energy (active) using a specific amplifier that saturates at high amplitudes. When tuning these two circuits to a nonlinear Exceptional Point (NEP), the system becomes extremely sensitive to small perturbations in inductance, following a steep cubic-root response curve, while remaining resistant to noise.

==Team Members==
Wang Peikun E1538091@u.nus.edu

Zhu Ziyang E1583446@u.nus.edu

Yuan Si-Yu E1353381@u.nus.edu

Li Xunyu xunyu@u.nus.edu

==Setups==
[[File:Experimental_setup.png|500px|thumb|center|Experimental setups]]

In FIG. 1 adopted from Ref. 1, we present:

''(a)'' Schematic of the circuit comprising inductors (<math>L</math>), capacitors (<math>C</math>), resistors (<math>R</math>), diodes (<math>D</math>), and an amplifier (<math>A</math>). The dashed rectangle indicates the negative resistance element, <math>-R_A(|V_A|)</math>.

''(b)'' Photograph of the experimental setup. An oscilloscope records the waveforms of <math>V_A</math> and <math>V_B</math> to derive frequency, voltage, and relative phase data. A DC unit powers the amplifier, while an arbitrary waveform generator provides the external driving signal. A variable inductor allows for fine-tuning of <math>L_A</math> to control the resonance frequency of resonator <math>A</math>.

''(c)'' Temporal dynamics of <math>V_A</math> (red) and <math>V_B</math> (blue) following the cessation of a 1 V, 70 kHz external driving signal. The system achieves a stable state at approximately 0.4 ms. Bold lines depict the signal envelopes, which remain constant after stabilization.

''(d)'' Magnified view of the stable state dynamics beginning at 0.66 ms. Experimental parameters are: <math>L_A = 245.4~\mu\mathrm{H}</math>, <math>C_0 = 18.5~\mathrm{nF}</math>, <math>C_c = 3.9~\mathrm{nF}</math>, <math>R_c = 760~\Omega</math>, <math>R_B = 1314~\Omega</math>, and <math>L_B = 197.7~\mu\mathrm{H}</math>.

==Measurements==

...

==Results and Analysis==

...

==References==
[1] K. Bai, T.-R. Liu, L. Fang, J.-Z. Li, C. Lin, D. Wan, and M. Xiao, Observation of nonlinear exceptional points with a complete basis in dynamics, Phys. Rev. Lett. 132, 073802 (2024).

Inductor Sensors of Ultra-high Sensitivity Based on Nonlinear Exceptional Point

2026-01-27T05:43:04Z

2026-01-27T05:30:40Z

Siyu: /* Setups */

We are building two coupled oscillating circuits: one that naturally loses energy (lossy) and one that gains energy (active) using a specific amplifier that saturates at high amplitudes. When tuning these two circuits to a nonlinear Exceptional Point (NEP), the system becomes extremely sensitive to small perturbations in inductance, following a steep cubic-root response curve, while remaining resistant to noise.

==Team Members==
Wang Peikun E1538091@u.nus.edu

Zhu Ziyang E1583446@u.nus.edu

Yuan Si-Yu E1353381@u.nus.edu

Li Xunyu xunyu@u.nus.edu

==Introduction==
We are building two coupled oscillating circuits: one that naturally loses energy (lossy) and one that gains energy (active) using a specific amplifier that saturates at high amplitudes. When tuning these two circuits to a nonlinear Exceptional Point (NEP), the system becomes extremely sensitive to small perturbations in inductance, following a steep cubic-root response curve, while remaining resistant to noise.

==Setups==
[image:Experimental setup.png]

==Measurements==

==Results and Analysis==

==References==
[1] K. Bai, T.-R. Liu, L. Fang, J.-Z. Li, C. Lin, D. Wan, and M. Xiao, Observation of nonlinear exceptional points with a complete basis in dynamics, Phys. Rev. Lett. 132, 073802 (2024).

Inductor Sensors of Ultra-high Sensitivity Based on Nonlinear Exceptional Point

2026-01-27T05:26:19Z

Siyu:

We are building two coupled oscillating circuits: one that naturally loses energy (lossy) and one that gains energy (active) using a specific amplifier that saturates at high amplitudes. When tuning these two circuits to a nonlinear Exceptional Point (NEP), the system becomes extremely sensitive to small perturbations in inductance, following a steep cubic-root response curve, while remaining resistant to noise.

==Team Members==
Wang Peikun E1538091@u.nus.edu

Zhu Ziyang E1583446@u.nus.edu

Yuan Si-Yu E1353381@u.nus.edu

Li Xunyu xunyu@u.nus.edu

==Introduction==
We are building two coupled oscillating circuits: one that naturally loses energy (lossy) and one that gains energy (active) using a specific amplifier that saturates at high amplitudes. When tuning these two circuits to a nonlinear Exceptional Point (NEP), the system becomes extremely sensitive to small perturbations in inductance, following a steep cubic-root response curve, while remaining resistant to noise.

==Setups==
[image:Experimental_setup.png]

==Measurements==

==Results and Analysis==

==References==
[1] K. Bai, T.-R. Liu, L. Fang, J.-Z. Li, C. Lin, D. Wan, and M. Xiao, Observation of nonlinear exceptional points with a complete basis in dynamics, Phys. Rev. Lett. 132, 073802 (2024).

Inductor Sensors of Ultra-high Sensitivity Based on Nonlinear Exceptional Point

2026-01-27T05:24:56Z

Siyu:

==Team Members==
Wang Peikun E1538091@u.nus.edu

Zhu Ziyang E1583446@u.nus.edu

Yuan Si-Yu E1353381@u.nus.edu

Li Xunyu xunyu@u.nus.edu

==Introduction==
We are building two coupled oscillating circuits: one that naturally loses energy (lossy) and one that gains energy (active) using a specific amplifier that saturates at high amplitudes. When tuning these two circuits to a nonlinear Exceptional Point (NEP), the system becomes extremely sensitive to small perturbations in inductance, following a steep cubic-root response curve, while remaining resistant to noise.

==Setups==
[image:Experimental_setup.png]

==Measurements==

==Results and Analysis==

==References==
[1] K. Bai, T.-R. Liu, L. Fang, J.-Z. Li, C. Lin, D. Wan, and M. Xiao, Observation of nonlinear exceptional points with a complete basis in dynamics, Phys. Rev. Lett. 132, 073802 (2024).

Inductor Sensors of Ultra-high Sensitivity Based on Nonlinear Exceptional Point

2026-01-27T05:24:00Z

Siyu:

==Team Members==
Wang Peikun E1538091@u.nus.edu

Zhu Ziyang E1583446@u.nus.edu

Yuan Si-Yu E1353381@u.nus.edu

Li Xunyu xunyu@u.nus.edu

==Introduction==
We are building two coupled oscillating circuits: one that naturally loses energy (lossy) and one that gains energy (active) using a specific amplifier that saturates at high amplitudes. When tuning these two circuits to a nonlinear Exceptional Point (NEP), the system becomes extremely sensitive to small perturbations in inductance, following a steep cubic-root response curve, while remaining resistant to noise.

==Setups==

==Measurements==

==Results and Analysis==

==References==
[1] K. Bai, T.-R. Liu, L. Fang, J.-Z. Li, C. Lin, D. Wan, and M. Xiao, Observation of nonlinear exceptional points with a complete basis in dynamics, Phys. Rev. Lett. 132, 073802 (2024).

File:Experimental setup.png

2026-01-27T05:20:50Z

Siyu:

Inductor Sensors of Ultra-high Sensitivity Based on Nonlinear Exceptional Point

2026-01-27T05:17:59Z

Siyu:

==Team Members==
Wang Peikun E1538091@u.nus.edu

Zhu Ziyang E1583446@u.nus.edu

Yuan Si-Yu E1353381@u.nus.edu

==Introduction==
We are building two coupled oscillating circuits: one that naturally loses energy (lossy) and one that gains energy (active) using a specific amplifier that saturates at high amplitudes. When tuning these two circuits to a nonlinear Exceptional Point (NEP), the system becomes extremely sensitive to small perturbations in inductance, following a steep cubic-root response curve, while remaining resistant to noise.

==Setups==

==Measurements==

==References==
[1] K. Bai, T.-R. Liu, L. Fang, J.-Z. Li, C. Lin, D. Wan, and M. Xiao, Observation of nonlinear exceptional points with a complete basis in dynamics, Phys. Rev. Lett. 132, 073802 (2024).

==Results and Analysis==

Inductor Sensors of Ultra-high Sensitivity Based on Nonlinear Exceptional Point

2026-01-27T05:10:21Z

Siyu: /* Inductor Sensors of Ultra-high Sensitivity Based on Nonlinear Exceptional Point */

==[[Introduction]]==
We are building two coupled oscillating circuits: one that naturally loses energy (lossy) and one that gains energy (active) using a specific amplifier that saturates at high amplitudes. When tuning these two circuits to a nonlinear Exceptional Point (NEP), the system becomes extremely sensitive to small perturbations in inductance, following a steep cubic-root response curve, while remaining resistant to noise.

===[[Team Members]]===

Zhu Ziyang E1583446@u.nus.edu,
Yuan Si-Yu E1353381@u.nus.edu

===[[Setups]]===

===[[Measurements]]===

Inductor Sensors of Ultra-high Sensitivity Based on Nonlinear Exceptional Point

2026-01-27T05:09:33Z

Siyu: /* Inductor Sensors of Ultra-high Sensitivity Based on Nonlinear Exceptional Point */

==[[Inductor Sensors of Ultra-high Sensitivity Based on Nonlinear Exceptional Point]]==
We are building two coupled oscillating circuits: one that naturally loses energy (lossy) and one that gains energy (active) using a specific amplifier that saturates at high amplitudes. When tuning these two circuits to a nonlinear Exceptional Point (NEP), the system becomes extremely sensitive to small perturbations in inductance, following a steep cubic-root response curve, while remaining resistant to noise.

===[[Team Members]]===

Zhu Ziyang E1583446@u.nus.edu,
Yuan Si-Yu E1353381@u.nus.edu

===[[Setups]]===

===[[Measurements]]===