Introduction

Non-Hermitian degeneracy, known as exceptional points (EPs), has recently gained much attention because skewed vector space may increase symmetry by losing dimensionality1,2. Thus, breaking this symmetry may change the eigenfrequencies sharply, rendering a sensitive feature to external perturbations ε. The initial rationale for using EPs for sense is that the eigenfrequency splitting δω of coupled elements at Nth-order EPs scales as δω\(\propto\)ε1/N, in contrast to linear scaling δω\(\propto\)ε with diabolic points of Hermitian degeneracies3,4,5. However, such EP-enhanced sensitivity is available at a price of stringent requirements to fabrication errors6 or advanced fine-tune technology7,8. Motivated by engineering a robust EP, a new notion, exceptional surface, was proposed recently9. Nevertheless, the exceptional surface may provide less help in EPN with a higher sensitivity for N > 210,11, remaining an open question regarding the tradeoff between sensitivity and robustness.

In essence, addressing this tradeoff is challenging in the linear regime. The reason is attributed to the sensitivity of a linear system depending on its degeneracy. Specifically, a higher sensitivity must possess more degrees of freedom to maintain degeneracy, reducing a system’s robustness4,11,12; otherwise, a higher-order EP would split into several lower-order EPs, decreasing sensitivity13. Fortunately, the rapidly developed optical microcavity allows us to access nonlinear regimes by capturing light power within a tiny size14,15,16,17, and as a result, a pioneering theoretical report regarding optical parametric oscillators in resonators has generated four branches from EP2 degeneracy lifting, which offers an avenue to enhance sensitivity using fewer degrees of freedom18. However, such favorite nonlinearity-induced EP degeneracy lifting is difficult to observe in optics because of the sensitive phase matching condition and microcavity detune19,20. More recently, the trait of EP2 above the laser threshold predicted such multi-branches degeneracy lifting, but the experiments failed to observe21.

Due to the homeomorphism of the wave equation and Schrödinger equation, electrical microcavity has gradually developed as an essential platform for studying EP-based physics22. Compared to coupled optical counterparts, it possesses unique advantages like compact size and noise suppression. For example, a negative resistance circuit can serve as a gain directly without requiring an extra wavelength pump in optics, providing a compact size23,24,25. Besides, after a proper lowpass design, the electrical microcavity may reduce the thermal noise around a higher-order EP comparable to a diabolic point in the Hermitian situation26. More importantly, electrical microcavity can couple modes with a detuned frequency, offering an alternative protocol, anti-PT symmetry27, to harness EPs. In addition, the electrical system easily yields the nonlinear effect28. For example, the diode provides voltage-dependent resistor value, generating EP3 in two coupled electrical resonators29. However, most contributions have focused on the intensity-induced nonlinear effects akin to the Kerr effect in nonlinear optics28,29,30.

This study experimentally showcases a nonlinear phenomenon, multi-branch EP degeneracy lifting, using anti-PT electrical resonators, similar to the four-wave mixing in nonlinear optics. The study found that nonlinear interaction among eigenmodes under the anti-PT symmetric-breaking phase yields up to twelfth-order EP2 degeneracy lifting. In terms of identical limit of detection, a sixth-order nonlinear bifurcation amplifies the sensitivity by elevenfold compared to the conventional EP-based sensing operating in the linear regime. Unlike previous explorations of EPs in the linear regime, the nonlinearity-induced EP degeneration lifting produces a chaotic dynamic instead of random noise. This sensing protocol enhances sensitivity by over one order of magnitude and expands the scope of bifurcation into nonlinearity, providing a paradigm to exploit the benefits of EPs.

Results

Sensing scheme

To understand the role of the nonlinear effect in the EP degeneracy lifting, we compared three types of bifurcations on the EP2, EP3, and EP2 with nonlinearity involvement in Fig. 1. Figure 1a, b exhibit the sinusoidal waves dynamics of the most common situation of EP2 and EP3. From their spectra obtained by Fourier transformation, we find that after bifurcation their eigenfrequencies are composed of two or three branches in Fig. 1d, e, identical to their EPN’s degrees of freedom. In contrast, when the dynamics of the eigenmodes hold a non-sinusoidal fashion in Fig. 1c, its eigenfrequencies after bifurcation engender nonlinear components in Fig. 1f, producing more branches far beyond EPN’s degrees of freedom, and the extended gaps between each branch pair may merit higher sensitivity for sensing application.

Fig. 1: Schematic of eigenmode dynamics and corresponding eigenfrequencies real parts.
figure 1

Dynamics of a EP2, b EP3, and c EP2 with nonlinearity, where EP stands for exceptional point. M1, M2, and M3 are eigenmodes supported by a coupled system. Real parts of system eigenfrequencies \(\lambda\) of d EP2, e EP3, and f EP2 with nonlinearity, where \(\bar{\lambda }\) is the mean value of eigenvalues; \(\Delta\) and κ are the detuning and coupling strength. The color of solid curves represents the value of Re[\(\lambda\)-\(\bar{\lambda }\)]/κ, with red indicating a higher value and blue indicating a lower value.

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Circuits with nonlinearity-induced EP degeneracy lifting

To experimentally demonstrate this non-linearity in bifurcation, we have fabricated an anti-PT electric circuit with an engineered EP, as shown in Fig. 2a. The anti-PT electric circuit includes two LRC resonators, coupling by a resistor Rc. Each resonator is composed of a negative resistor unit \(-\widetilde{{R}_{j}}\), a capacitor \({C}_{j}\), and an inductor \({L}_{j}\), where j denotes resonator indexes. Applying Kirchhoff’s circuit laws, the circuit diagram can be simplified as

$${V}_{1}-{V}_{2}={R}_{C}{I}_{C}+{P}^{{NL}}$$
(1)
$${V}_{j}=-{L}_{j}\frac{d{I}_{{Lj}}}{{dt}}=-\frac{1}{{C}_{j}}{\int }_{\!\!\!0}^{t}{I}_{{Cj}}\left(\tau \right)d{\tau }={R}_{j}{I}_{{Rj}}$$
(2)
$${I}_{C}={I}_{L1}+{I}_{C1}+{I}_{R1}=-({I}_{L2}+{I}_{C2}+{I}_{R2})$$
(3)

where \({I}_{{Lj}}\), \({I}_{{Cj}}\), and \({I}_{{Rj}}\) are electrical currents at \({L}_{j}\), \({C}_{j}\), and \(-{R}_{j}\), respectively; \({P}^{{NL}}\) is the nonlinear component, contributed by \(-{R}_{j}\). With the ansatz of \({V}_{j}={v}_{j}(t){e}^{-i{\omega }_{0}t}\), we achieve the coupled differential equations

$$\frac{d}{dt}\left(\begin{array}{c}{v}_{1}\\ {v}_{2}\end{array}\right)=\underbrace{\frac{1}{\eta -2i}\left[\begin{array}{cc}-\varDelta & -i\alpha /2\\ -i\alpha /2 & \varDelta \end{array}\right]\left(\begin{array}{c}{v}_{1}\\ {v}_{2}\end{array}\right)}_{Linear}+\underbrace{\frac{i\alpha }{2{\omega }_{0}}{P}^{NL}{e}^{i{\omega }_{0}t}}_{Nonlinear}$$
(4)

where \(\eta =\beta /{\omega }_{0}\) and \(\Delta =({\omega }_{2}-{\omega }_{1})/2\), with the note of \({\omega }_{j}={({L}_{j}C)}^{-1/2}\), \({\omega }_{0}=({\omega }_{1}+{\omega }_{2})/2\), \(\alpha ={({R}_{c}C)}^{-1}\), and \(\beta ={C}^{-1}({R}_{C}^{-1}-{R}_{j}^{-1})\) in terms of \({C}_{j}=C\). Note that the linear component dominates if Rc = R, yielding η = 027; in contrast, we set Rc > R to stimulate the nonlinear components in the negative resistor (see methods). The solution achieved from the linear components should be \(v\left(t\right)={v}_{+}{e}^{-i{\omega }_{+}t}+{v}_{-}{e}^{-i{\omega }_{-}t}+c.c.\), where \({\omega }_{\pm }=\pm \sqrt{{\alpha }^{2}/4-{\Delta }^{2}}=\pm \delta\) are the eigenfrequencies. As a result, the nonlinear components can be expressed as

$${P}^{{NL}}(t)=\mathop{\sum}\limits_{k}\int {\chi }^{\left(k\right)}\left(t-\tau \right){v}^{k}\left(\tau \right)d{\tau }$$
(5)

where \({\chi }^{\left(k\right)}\) is the equivalent nonlinear susceptibility, and \(k\in Z\) is the nonlinear order. Thus, vsum(t) including linear and nonlinear components can be spanned by \({v}_{{sum}}\left(t\right)=\mathop{\sum}\limits_{k}{v}_{k}\cos k\delta t\), generating sharp changes in the time domain. Such a result is similar to the four-wave mixing in nonlinear optics, creating terms of \(\left(2{\omega }_{+}-{\omega }_{-}\right)+k\left({\omega }_{+}-{\omega }_{-}\right)\) with an extended gap between a branch pair from 2δ to 2(2k-1)δ. Note that linear components yield these nonlinear eigenfrequencies, thereby resulting in an inherent degeneracy. The degrees of freedom are identical to the EP2-based protocol, and thus, the increased sensitivity is no cost in robustness like high-order EP sensors.

Fig. 2: Anti-PT electric resonators and their response in the time domain.
figure 2

a The anti-PT circuit diagram, where R, C, L are the resistor, capacitor, and inductor; b Voltage dynamics of probes P1 and P2 at temperatures of 23.2, 23.1, 23.0, and 22.9 °C.

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To stimulate the nonlinear components, we selected the parameters of RC = 400 Ω, R1 = R2 = 100 Ω, C = 225 nF, L1 = 1.441 mH. Motivated by controlling Δ through \({\omega }_{2}\), a lab-made temperature controller with a resolution of 0.01 °C was deployed, adjusting the temperature of L2, and the measured value of L2 at ~1 mH is linearly dependent on temperature with a sensitivity of ~0.5 μH/°C. Later, we observe the circuit’s anti-PT phase transition by monitoring P1 and P2 in Fig. 2a. When the temperature of L2 was kept at 23.2 °C, the system entered an anti-PT symmetric phase. In contrast to the theoretically expected amplifying envelope, both probes remained flat due to the limited power supply. Later, several sharply changed peaks appeared in their envelopes if we cooled down the temperature to 23.1 °C, demonstrating the expected nonlinearity in Eq. (5). These peaks emerged in an aperiodic fashion when the system approached EP. After that, the system accessed the anti-PT symmetric-breaking phase when the temperature was reduced to 23.0 °C. The aperiodic peaks in the envelope nearly disappeared if we further decreased the temperature to 22.9 °C.

Motivated by observing the nonlinearity-induced EP degeneracy lifting, we have carried out a more accurate temperature control with a step of 0.05 °C and implemented Fourier transformation on P1 and P2, as shown in Fig. 3a, b. The anti-PT symmetric-breaking phase emerges multiple branches due to the contribution from nonlinearity when the temperature is below 23.1 °C, and the available nonlinear order is even up to twelve. Compared to Fig. 3a, b, the spectra vary with the temperature of L2, presenting a phase transition in the vicinity of 23.1 °C independent of orders, and the nearly identical evolution reveals the energy-difference conservation and synchronized power oscillation in the anti-PT symmetric phase31.

Fig. 3: Changes in the spectra with the temperature of the inductor L2.
figure 3

Spectra versus temperature at probes a P1 and b P2, where multiple branches emerge at the anti-PT symmetric-breaking phase. The numbers describe the order of nonlinearity degeneracy, and their counterpart at the blueshift are not labeled. c Allan deviation σ(τ) of P1 shown in blue curves versus the time of each neighboring peak τ at different temperatures, where dashed lines in red and orange are τ-1/2 and τ-1.

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We can note that the background spectra for the anti-PT symmetric-breaking phase increases, compared to the anti-PT symmetric phase. In general, the anti-PT symmetric-breaking phase generates the oscillated dynamics with increasing intensity, but the practical power cannot sustain the infinite value. As a result, the dynamics yield a sine oscillation, generating a single peak in the spectra with weak background spectra. In contrast, when the anti-PT symmetric phase dynamics evolve complicated due to the nonlinear contribution, the dynamics possess uneven peaks, causing a slight background spectra of <−25 dB.

To quantify the background components, we have analyzed the Allan deviation of P1 versus the time of neighboring peaks near the EP, as shown in Fig. 3c. At a temperature of 23.2 °C, σ(τ) is highly weak <~10−7, indicating the presence of a weak background. The decreased σ(τ) with a function between τ-1/2 and τ-1 demonstrates the coexistence of the quantization noise and white noise in the anti-PT symmetric-breaking phase. The white noise in P1 results from thermal noise from the circuit elements30, but the quantization noise comes from our treatment in math rather than sampling time because we employed the time gap between two neighboring peaks to consider nonlinear spectra components. As T decreased to 23.1 °C, the system approaches EP, where σ(τ) increased dramatically ~103, and σ(τ) approaches τ-1/2, indicating white noise, that is, thermal noise, becomes dominant. The white noise plays the leading role in the short-time readout when T further drops to 23.0 °C. However, its impact vanished for a longer readout, and bias insatiability appears at τ ~ 104, which stems from the random flickering of electronics or other system components. For T = 22.9 °C, the anti-PT symmetric phase disappears the bias insatiability, leading to a reduced σ(τ) and verifying the reduced signal noise ratio on EP.

Chaotic dynamics feature

Another interesting and significant characteristic of EP is the fluctuation near EP32,33. In general, EPs may cause random noise34 due to the dissipative essence under the linear theoretical frame according to the theory of quantum Fisher information35. In parallel to the progress of randomness around EP, chaotic dynamics are ubiquitously in various practical physical settings under nonlinear regimes36,37. Thus, two natural questions arose for the nonlinearity-induced degeneracy lifting: (I) Do the fluctuations occur around EP? (II) Is it chaotic or random?

For the first question, we have observed an aperiodic peak in the envelope near EP at ~23.1 °C in Fig. 2b. To quantify such fluctuations, we calculated the variance of spectra within 10 s in the step of 0.5 s, as shown in Fig. 4a, and we provided four typical spectral variances in Fig. 4b to aid visualization. The spectra variance in the anti-PT symmetric phase is nearly constant where the temperature of L2 is higher than 23.2 °C. By contrast, a sharp change appears when its temperature is slightly cooled to 23.15 °C, corresponding to an EP. A further decreased temperature generates a more extended spectra variance, and later, the spectra variance drops quickly to ~0.003.

Fig. 4: Chaotic characteristic analysis.
figure 4

a Variance of the spectra as a function of temperature; b Examples of time evolution of the spectra in a; c dependence of maximum Lyapunov exponent (MLE) on the temperature; d dependence of the correlation dimension (υ) on the temperature. The dimension in the calculation is taken from 2 to 30, marked by color from blue to red. The dots present a convergence behavior towards increasing dimensions at each temperature, confirming a nonrandom character.

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For the second question, we have calculated the most commonly used maximum Lyapunov index (MLE) in Fig. 4c. In general, a positive MLE indicates chaotic behavior, and experiments show that its MLE is above zero if the temperature closes EP38,39. Likewise, the correlation dimension (υ) is another landmark in estimating chaotic behavior, and a time series is chaotic if υ takes a non-integer40. Figure 4d displays that υ leaves integer one and takes non-integer when the temperature is slightly below 23.2 °C. A further reduction in temperature produces a transition, but υ maintains a non-integer if the temperature is above 22.55 °C, consistent with the chaotic estimation from MLE. Except for the region in the vicinity of EP, υ approaches integers one and three when the temperature is below 22.55 °C and above 23.2 °C. Generally, υ = 1 indicates sinusoidal dynamics, accounting for the oscillation in the anti-PT symmetric phase, and υ = 3 suggests more complex periodic dynamics, such as periodic oscillation with a rapidly changed envelope in the anti-PT symmetric-breaking phase.

Such chaotic behavior may bring a sensing protocol in utilizing EP for pursuing accuracy because chaos, in principle, is deterministic instead of random41,42. For example, reservoir computing has been used to forecast the time series with Lyapunov time ~10 43,44,45, and the continued rapid pace of improvement in computing performance may forecast a longer Lyapunov time. Thus, their combination would facilitate multidisciplinary cooperation among artificial intelligence, chaos, and sensing for the foreseeable future.

Results of enhanced sensitivity

To examine the nonlinear degeneracy lifting, we define the sensitivities of corresponding nonlinear order k by \(S\equiv \partial (2k\delta )/\partial {{{{{\rm{T}}}}}}\), where T is temperature, and plot the sensitivity depending on the temperature under different orders in Fig. 5a. The Puiseux series is a helpful tool for performing complex analysis around an EP, and several elegant reports have presented the bifurcation for an EP2 \(\propto {\varepsilon }^{1/2}\) 8,24. A log-log scale of the sensitivity to temperature in Fig. 5a may better visualize the theory, where the bottom curve describes the first-order degeneracy lifting of EP2, and the others depict that of higher-order. The responses are linearly dependent on the temperature perturbation, and the slopes of all curves in Fig. 5b approach 1/2 independently of the orders because higher-order nonlinearity is attributed to the first-order components, experimentally verifying the nonlinearity-induced degeneracy lifting.

Fig. 5: Results of enhanced sensitivity.
figure 5

a The log-log scale of sensitivity depends on the perturbation in temperature at probe P1. Squares and curves are the experiments and fitting results, where the nonlinear order increases from 1 to 12 from bottom to up. The beginning temperature is 22.95 °C, where spectra have a negligible impact from chaos. Here, we do not show the sensitivity of probe P2 because of the resembling result; b Slops, and c enhanced factor of various nonlinear (NL) orders.

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Such a degeneracy lifting mechanism merits a superior performance in sensing because of a more extended gap in spectra. To quantify the benefit of nonlinearity, we define an enhancement factor, that is, the ratio of the higher-order sensitivity and first-order sensitivity, and Fig. 5c shows that the enhancement factor is proportional to the order of nonlinearity, which is consistent with Eq. (5). In terms of the unchanged limit of detection, a sixth-order nonlinear bifurcation amplifies sensitivity by eleven folds than that of the conventional EP2 scenario in the linear regime. Except for the advantageous near EP, such a nonlinear signature compensates for the drawback of the EP-based method operating in the linear regime in a reduced sensing range. Specifically, the sensing strategy with the fashion of ε1/N must sacrifice the sensitivity for ε > 1 situation. However, for the nonlinearity-induced protocol, a greater enhancement is achieved for the higher order without losing the performance in larger perturbation, offering an extended sensing range.

Conclusions

In summary, this letter observed the nonlinear-induced EP degeneracy lifting at the electric resonators, where a bifurcation yields up to twenty-four branches, corresponding to twelfth-order nonlinearity. In terms of the unchanged limit of detection, a sixth-order nonlinear bifurcation amplifies the sensitivity eleven-fold compared to the conventional EP2-based sensing in the linear regime. Still, the degrees of freedom for the nonlinear bifurcation are identical to that of the EP2-based sensing protocol, thereby endowing a robust performance and breaking the constrict between sensitivity and robustness. Moreover, such nonlinearity-induced EP degeneracy lifting amplifies the sensitivity independent of the sensing range, compensating for the disadvantage of the EP-based method operating in the linear regime for ε > 1 situation. More interestingly, we discovered the determined chaotic dynamics near EPs due to the nonlinear contribution, which presents a distinct difference from the random noise character in the linear regime. This letter expands the scope of EP degeneracy lifting into nonlinearity and enhances the sensitivity beyond one order of magnitude. More importantly, our study bridges chaotic dynamics and bifurcation, providing a paradigm to exploit the benefit of EPs.

Methods

Experimental measurements

The anti-PT circuit diagram consists of two coupled LRC resonators, where a dual operational amplifier chip (Maxim Integrated, MAX44248ASA+) was employed to reduce experimental errors. We used a lab-made temperature controller with an accuracy of 0.01 °C to manipulate the temperature of L2. Experiments adopted the sampling rate of 220 dots/s and 10 s/measurement to distinguish these higher-order branches in spectra.

Nonlinearity stimulation

The four-wave mixing-like nonlinearity rises from the negative resistor, which can be stimulated when the Rc > R. Here, we simulated a series Rc from 100 Ω to 500 Ω using LTSpice to demonstrate such a phenomenon. Figure 6a shows the results, and Fig. 6b displays their spectra. When Rc = 100 Ω, the linear component dominates, which has also been verified by Ref. 27. In contrast, when Rc increases to 400 Ω, such four-wave mixing-like nonlinearity is generated. In principle, a larger Rc may generate a more significant nonlinearity, but it needs a matched inductor and capacitor to achieve an EP. Considering the achievable inductors and capacitors, we employed Rc = 400 Ω in experiments.

Fig. 6: Four-wave mixing-like nonlinearity stimulation.
figure 6

a Dynamics of probe P1 with various coupling resistor Rc from 100 Ω to 500 Ω. b corresponding spectra of Fig. 6a.

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