Introduction

The underlying mathematical structures of non-Hermitian wave systems [1][2][3] [4] have inspired the last few years new technologies [5][6][7] [8]. Many of these are reliant on the existence of exceptional point degeneracies (EPDs) [8].

These are non-Hermitian degeneracies where a set of 𝑁 eigenvalues and their corresponding eigenvectors coalesce [1] [2]. In the proximity of an 𝑁 -th order EPD (EPD-N), the eigenvalue detuning 𝛥𝑓 ≡ |𝑓 -𝑓 𝐸𝑃𝐷 |, due to a small external perturbation ε, follows a sublinear response (SLR) 𝛥𝑓 ∼ √𝜀 𝑁 ≫ 𝜀 that can be utilized for enhanced sensing [9][10] [11][12] [13][14] [15].

A principal requirement for efficient EPD sensing is the increase of the resolution limit via the narrowing of the resonance linewidth. This can be achieved by judicious design of cavity amplification mechanisms. The downside of this strategy is that it introduces additional noise that, in some EPD platforms, might offset the 𝜕ቀ 𝛥𝑓 + 𝑓 0 ቁ 𝜕ሺ𝛿𝑉ሻ is the sensitivity and 𝛼 𝑉𝑅𝑊 is the noise-equivalent voltage variations), diverges as 1 √𝛿𝑉 in the proximity of NLEPD.

enhanced signal sensitivity [13][16] [17][18] [19]. Furthermore, nonlinear effects might become important -requesting the development of theoretical tools that treat them on equal footing with the sensitivity enhancement near EPDs. However, most current studies rely on linear mathematical constructs, such as the Petermann factor [20][21] [22], which describes the linewidth enhancement near EPDs due to the biorthogonal nature of the eigenmodes of the underlying linear non-Hermitian systems [16]. Obviously, this approach is not suitable when the response of a system is influenced by nonlinearities. An example case is a laser at an EPD. Fortunately, an appropriate language exists from the area of dynamical systems and bifurcation theory [23][24] [25][26] [27], which can be adopted for the analysis of nonlinear EPDs (NLEPDs). Examples of systems that are amenable to such analysis are shown in Fig. 1a. In fact, some recent theoretical studies have utilized this approach to address issues like the formation of NLEPDs and the emulation of neuronal dynamic functions using parity-time ሺ𝒫𝒯ሻ symmetric systems that involve gain and loss nonlinear channels [26][27]. These neuromorphic functionalities, belong to the general category of oscillation quenching mechanisms, whose characteristics are determined by the underlying dynamical symmetries of the system [28][29] [30].

Oscillation quenching mechanisms occur in systems of coupled nonlinear oscillators, and can be classified in two different types [30]: oscillation death (OD), which is associated with the formation of an inhomogeneous steady-state, and amplitude death (AD), which results in a homogenous steady-state. These mechanisms have been observed in a variety of dynamical systems ranging from climate, lasers, electronic circuits, chemical systems, neurons, and more (for indicative references see review paper [30]). Recently, in the framework of non-Hermitian photonic systems with 𝒫𝒯-symmetry, oscillation quenching mechanisms have been proposed, theoretically, for topological protection, signal processing [26][27] and memory devices [31]. The interplay of 𝒫𝒯-symmetry and oscillation quenching were also theoretically discussed in the framework of electronic circuits. These theoretical studies highlighted a connection between a broken (exact) 𝒫𝒯-symmetric phase and an OD (AD) oscillation quenching mechanism, see Fig. 1b.I and Fig. 1b.II, respectively. The transition between these two regimes is characterized by the formation of a NLEPD (see Fig. 1b right and Fig. 1c). Can these NLEPDs be used for sensing and what is the signal-to-noise enhancement (SNE) factor in their proximity? A definite answer to this question requires not only a theoretical modeling [24][25] [32][33][34] [35][36] [24], but, most importantly, the establishment of controllable experimental platforms that will scrutinize the predictions of the theory, and guide the theoretical language as it is developing.

Here, we address the viability of NLEPD sensing protocols using two nonlinear RLC tanks whose capacitive coupling is used as a sensing platform for voltage variations. The RLC circuits have anharmonic parts consisting of a complementary amplifier (gain) and a dissipative conductor (loss), see Fig. 1a and Methods. The nonlinear supermodes (NS) are the fixed points of the dynamical equations, and their properties arise from the underlying dynamical symmetries and stability. We focus on stable NS that are experimentally accessible and these are identified from the (negative) real part of the eigenvalues {𝜆 𝑛 } of the Jacobian matrix -which describes the linearized dynamics around each of these fixed points [23].

Their properties lead to the partition of the parameter space into three distinct domains (see domains I, II, III in Fig. 1b

Results

Experimental Platform

The sensor consists of a pair of nonlinear RLC resonators [37][38][39] [40] (see Fig. 1a) with natural frequency 𝑓 0 =

1 2𝜋 1 √𝐿𝐶 ≈ 338 𝑘𝐻𝑧, and an impedance (at resonance) 𝑍 0 = √ 𝐿 𝐶 ≈ 424 𝛺. One of the resonators (gain-indicated with red in Fig. 1a) incorporates a nonlinear amplifier, -𝑅 1 ሺ𝑉 1 ሻ, which is characterized by an I-V curve of 𝐼 1 ሺ𝑉 1 ሻ = -𝑉 1 𝑅 1 ሺ0ሻ + 𝑏𝑉 1 3 while the other one (loss-indicated with green in Fig. 1a) incorporates a nonlinear loss, 𝑅 2 ሺ𝑉 2 ሻ, with an I-V curve of 𝐼 2 ሺ𝑉 2 ሻ = 𝑉 2 𝑅 2 ሺ0ሻ + 𝑏𝑉 2 3 (where 𝑏 ≈ 7 ⋅ 10 -4 𝐴𝑉 -3 and 𝑉 1ሺ2ሻ are the voltages at the nodes 1(2)). The two resonators are coupled together via a Capacitance Voltage Controlled (CVC) capacitor 𝐶 𝜅 ሺ𝑉ሻ = 𝜅 ⋅ 𝐶 where 𝜅 is a dimensionless parameter representing the strength of the coupling. The linear conductances, 1 𝑅 1 ሺ0ሻ > 1 𝑅 2 ሺ0ሻ , were tuned such that the system undergoes a transition from AD to OD as the voltage at the coupling capacitor varies (see Figs. 1b, 1c and Methods). A transmission line (TL) with impedance 𝑧 0 = 50 𝛺 is weakly coupled to each resonator via capacitors 𝐶 𝑒 ≪ 𝐶. The TLs were used to collect and direct the signal generated by the circuit to a VNA for further processing (see Methods). The NLEPD (occurring at 𝛿𝑉 = 0) can be experimentally identified as the point for which the voltage amplitudes 𝑉 1 𝑉 2 of each RLC resonator deviates from unity (AD domain) and begins to acquire larger values (OD domain) -see Fig. 2b. Theoretical Analysis of Nonlinear Supermodes The voltage dynamics 𝑉 𝑛 at each RLC tank (𝑛 = 1,2), is described via a temporal coupled mode theory (TCMT) (see Supplementary Notes 1 and 2) 𝑖 𝑑 𝑑𝜏 ቀ 𝑎 1 𝑎 2 ቁ = ( 𝜈 𝜅 + 𝑖𝛾 1 𝜅 2 𝜅 2 𝜈 𝜅 -𝑖𝛾 2 ) ቀ 𝑎 1 𝑎 2 ቁ; (1) where 𝑎 𝑛 ≡ √ 3 8 𝑏𝑍 0 ቀ𝑉 𝑛 + 𝑖 1 2𝜋 𝑉 ̇𝑛 𝑓 0 ቁ , 𝛾 𝑛 = 𝛾 𝑛 ሺ0ሻ + ሺ-1ሻ 𝑛 ሺ|𝑎 𝑛 | 2 + 𝜂ሻ, 𝜈 𝜅 = 1 -𝜅 2 -√ 𝜂 2 𝑍 0 𝑧 0 and 𝜏 ≡ 2𝜋𝑓 0 𝑡 is the rescaled time. The parameters 𝛾 𝑛 ሺ0ሻ are the linear gain (𝑛 = 1ሻ and loss (𝑛 = 2) coefficients associated with the gain and loss RLC resonator, respectively, while 𝜂 ≡ 1 2 𝑧 0 𝑍 0 ቀ 𝐶 𝑒 𝐶 ቁ 2 models the coupling of the circuit to the TLs. The global frequency shift 𝑓 𝜅 = 𝑓 0 ⋅ 𝜈 𝑘 is associated with the renormalization of the natural frequency 𝑓 0 of the RLC resonators due to the capacitive coupling between them (𝜅 -term) and the coupling with the TLs (𝜂 -term). The 𝜅-dependence could be, in principle, avoided if we choose another type of coupling (e.g., inductive coupling). Below, we analyze the steady-state properties of Eq. (1) in terms of the coupling parameter 𝜅 = 𝜅ሺ𝛿𝑉ሻ, which is used as a sensing platform of voltage variations 𝛿𝑉 (see Methods). The nonlinearities in our system have been chosen carefully to prevent the system from evolving towards undesirable unbounded states where 𝐴 1 and/or 𝐴 2 → ∞. This can be easily realized from Eq. (1) by recognizing that whenever the field intensity of the gain resonator exceeds a critical value |𝑎 1 | 2 > 𝛾 1 ሺ0ሻ -𝜂 the gain coefficient 𝛾 1 becomes negative, thus turning the gain RLC tank into a lossy one. The NS of Eq. (1) may be expressed in the polar representation 𝑎 𝑛 = 𝐴 𝑛 𝑒 𝑖𝜑 𝑛 𝑒 -𝑖𝑓𝜏 𝑓 0

and are identified as the fixed points of the dynamical system whose evolution is defined in a three-dimensional phase space (𝐴 1 , 𝐴 2 , 𝜑 ≡ 𝜑 2 -𝜑 1 ሻ. We classify these fixed points according to their underlying (dynamical) symmetry, and stability. The latter is determined by the eigenvalues {𝜆 1 , 𝜆 2 , 𝜆 3 } of the 3 × 3 Jacobian matrix, 𝐽, when evaluated at the fixed point (see Supplementary Note 5) [23]. When 𝑅𝑒ሺ𝜆 𝑛 ሻ ≠ 0 ሺ∀𝑛 = 1,2,3ሻ, the fixed point is a hyperbolic equilibrium and there is a homeomorphism that maps the phase portrait in its proximity onto solutions of its linearized system described by 𝐽 [41]. When all 𝑅𝑒ሺ𝜆 𝑛 ሻ < 0, the fixed point is stable, and it is unstable if at least one 𝑅𝑒ሺ𝜆 𝑛 ሻ > 0. Hyperbolic equilibria are robust to small variations which do not, qualitatively, change the phase portrait. The opposite scenario of non-hyperbolic equilibria is associated with cases where either one of the eigenvalues of the Jacobian matrix is zero or has a zero real part. These are structurally unstable cases, and one can numerically test the nature of the stability of these fixed points by direct dynamical simulations with Eq. (1). 2)) while the dashed lines indicate unstable solutions evaluated numerically using Eq. ( 1) together with the eigenvalue analysis of the Jacobian matrix in panel a (see also Supplementary Note 5). The fixed relative gain

46 is chosen in a way that the system undergoes a transition from oscillation death (OD) to amplitude death (AD) as the voltage variation 𝛿𝑉 increases. The nonlinear exceptional point degeneracy (NLEPD) occurs at 𝛿𝑉 = 0. (Inset) Measured average logarithmic voltage ratio (red symbols)

of the NS. Each point represents an average of five independent measurements and the error bars indicate ±1 standard deviation. The black line is the prediction of temporal coupled mode theory (TCMT). The blue line indicates the results from NGSPICE. In panels a and b, the gray shadow highlights the presence of nontrivial unstable NS. c Phase-space analysis and basins of attraction for the stable fixed points associated with the upper ሺ𝑓 + ሻ (blue highlighted domain) and lower ሺ𝑓 -ሻ (yellow highlighted domain) branches of the NS in the AD domain. The voltage variation is 𝛿𝑉 ≈ 2 𝑚𝑉 corresponding to a circuit configuration in the proximity of the NLEPD.

We have found that one fixed-point of Eq. ( 1) is a trivial state ሺ𝐴 1 , 𝐴 2 ሻ = ሺ0,0ሻ.

The dynamical simulations indicate that it is stable in the parametric domain III (see Fig. 1b) while it is unstable in the other two domains. Below, we analyze the stable non-trivial fixed points occurring in domains I and II of Fig. 1b. In these cases, the real-valued amplitudes 𝐴 𝑛 > 0 take the form (see Supplementary Note 4 and Supplementary Figure 2):

where the real-valued variable 𝜌 ≡

≥ 0) which determines the boundary between domains II and III (domains I and III), see Fig. 1b.

Finally, the relation 𝜅 𝑁𝐿𝐸𝑃𝐷 = 𝛾 1 ሺ0ሻ + 𝛾 2 ሺ0ሻ defines the transition between domains I and II -which is characterized by the formation of a NLEPD associated with the coalescence of two stable NS (see solid red line in Fig. 1b). This is further confirmed by evaluating the nonlinear eigenfrequencies 𝑓 ± ሺ𝜅ሻ associated with the solutions of Eq. ( 2). From the TCMT we have that (see Supplementary Notes 4a and 4b):

𝜅 𝑁𝐿𝐸𝑃𝐷 2 for 𝜅 ≤ 𝜅 𝑁𝐿𝐸𝑃𝐷 ሺdomain Iሻ; for 𝜅 ≥ 𝜅 𝑁𝐿𝐸𝑃𝐷 ሺdomain IIሻ; ሺ3ሻ where the square-root dependence of the eigenfrequencies from the coupling detuning 𝜅 reflects the presence of the NLEPD. In the range of voltage variations that have been used in our experiment (-1.1 𝑉 ≤ 𝛿𝑉 ≤ 2 𝑉 with resolution of 1 𝑚𝑉), the coupling is 𝜅ሺ𝛿𝑉ሻ ≈ 𝜅 𝑁𝐿𝐸𝑃𝐷 -ሺ0.0234 𝑉 -1 ሻ ⋅ 𝛿𝑉 with 𝜅 𝑁𝐿𝐸𝑃𝐷 ≡ 𝛾 1 ሺ0ሻ + 𝛾 2 ሺ0ሻ ≈ 0.30 (where 𝛾 1 ሺ0ሻ ≈ 0.18; 𝛾 2 ሺ0ሻ ≈ 0.12ሻ.

The two fixed points in domain II (𝜅 ≥ 𝜅 𝑁𝐿𝐸𝑃𝐷 ሻ, are associated with the AD phase where the field amplitudes at each resonator are the same and the two stable NS differ only by the relative phase 𝜑 ± (see Supplementary Note 4a). In this parameter range, the system respects an exact parity-time symmetry -where both the system and the corresponding NS are invariant under a joint parity (i.e., space inversion 1 ↔ 2ሻ and time-reversal (i.e., complex conjugation) symmetry. In domain I (𝜅 ≤ 𝜅 𝑁𝐿𝐸𝑃𝐷 ሻ, instead, there is only one non-trivial stable NS. This domain is associated with the OD phase where the field amplitudes at each RLC resonator differ from one another. A detailed fixed point numerical analysis using a MATLAB fsolve routine confirms the above theoretical predictions and provides more general information about other (unstable) fixed points, as well as the stability analysis of the nontrivial NS via the eigenvalues of the Jacobian (see Figs. 2a and 2b). In the inset of Fig. 2b, we also report some represented values of the measured voltage ratios

versus the voltage variation 𝛿𝑉. These results compare nicely with the TCMT predictions (black line) indicating that the NLEPD occurs at the transition between AD and OD phases. The slight deviations for large negative 𝛿𝑉 are attributed to the small detunings of various components in our circuit from its ideal (TCMT) parameters (see Supplementary Note 7 and Supplementary Figure 4) and to the limitations of the TCMT. The applicability of the latter is subject to a number of approximations: high-Q factors of each RLC resonator, weak coupling between the two RLC tanks, and the elimination of fast-oscillating terms (see Supplementary Note 1). Nevertheless, the overall agreement between the predictions of TCMT, the NGSPICE simulations (blue solid line) and the experimental results are satisfactory.

Additionally, the TCMT predicts that the voltage contrast follows a Puiseux expansion,

. This sublinear response of the voltage contrast suggests that it could also be used as another physical observable for sensing purposes. Further analysis on the SNE of such observable is necessary to establish it as sensing measurand and will be reported elsewhere.

The existence of a stable NS does not guarantee the evolution of the system to this specific state. Instead, the system may evolve either to a stable trivial state, or to another stable NS in case of bistabilities (AD domain). The former scenario is easily excluded by an appropriate choice of the relative gain parameter (see Fig. 1b).

The latter scenario can be controlled by realizing that the final state depends strongly on the initial conditions {𝐴 1 ሺ0ሻ, 𝐴 2 ሺ0ሻ, 𝜑ሺ0ሻ} (see Supplementary Note 6).

The phase-space volume that contains the initial conditions which converge to a specific stable fixed point constitute a basin of attraction, and its size provides a measure of how attractive this fixed point is. Detailed dynamical simulations using Eq. ( 1) -for various 𝜅 -values along with a fine mesh of initial conditions {𝐴 1 , 𝐴 2 , 𝜑} -allowed us to identify the basins of attraction of the two fixed points in the AD domain. We find that initial excitations with a relative phase |𝜑| > and Supplementary Figure 4). Away from the NLEPD the bistable nature at the AD domain persists. Either way, the square-root scaling of the NS frequency 𝑓 + from 𝑓 𝑁𝐿𝐸𝑃𝐷 (see Eq. ( 3)) is unaffected.

Sensing Protocol

In Fig. 3a, we report a density-plot of the voltage power spectrum |𝑉 1 ሺ𝜔ሻ| 2 for various voltage variations 𝛿𝑉 by performing a Fourier transform of the temporal field -which were evaluated via time-domain simulations of the TCMT Eq (1). To achieve the asymptotic states associated with the 𝑓 + ሺ𝑓 -ሻ supermodes in the AD domain, we have prepared the initial excitation at relative phase |𝜑| < 𝜋 2 (|𝜑| > 𝜋 2

) as discussed above. The numerical data agree nicely with the theoretical predictions of Eq. (3). In Fig. 3b we show a density plot of the measured power spectrum of the emitted signal together with the TCMT predictions of Eq. ( 3) for the 𝑓 + frequency (red dashed line).

The absence of 𝑓 -from the measured power spectrum, is associated with the fact that the experimental initial preparation favors a field excitation with a small relative phase |𝜑| < 𝜋 2 . In the same figure, we also report the 𝑓 + frequency (versus 𝛿𝑉) that has been extracted by a Fourier transform of the voltage 𝑉 1 ሺ𝑡ሻ using NGSPICE simulations (black dotted line).

The sublinear detuning is better appreciated by reporting 𝛥𝑓 + ≡ 𝑓 + -𝑓 𝑁𝐿𝐸𝑃𝐷 vs. 𝛿𝑉 in a double-logarithmic plot. The experimental data (cyan circles) nicely match the results from the NGSPICE (dashed black line) showing the predicted behavior 𝛥𝑓 + ∝ √𝛿𝑉 from TCMT, see Fig. 3c. This sublinear response offers an opportunity to develop an enhanced sensing protocol for detecting small voltage variations, 𝛿𝑉, using the coupling capacitor 𝐶 𝜅 ሺ𝛿𝑉ሻ as a sensing platform. At the same time, the square-root SLR extends the dynamical range (DR) of the sensing measurements up to relatively large values of 𝛿𝑉. The DR is the other important metric that characterizes the efficiency of a sensor, and it is defined as the ratio between the maximum and the minimum 𝛿𝑉 variation that the sensor can measure. Furthermore, the presence of gain elements guarantees the narrowing of the emission peaks and promotes an enhanced resolution. To further quantify the efficiency of our sensing protocol, we have introduced the sensitivity, 𝜒 ≡ 𝜕ቀ 𝛥𝑓 𝑓 0 ቁ 𝜕ሺ𝛿𝑉ሻ . In Fig. 3d we report the measured sensitivity (violet circles) together with the NGSPICE results (black dashed line). We find that, within the experimental resolution, 𝜒 ∼ 1 √𝛿𝑉 in the proximity of the NLEPD. Obviously, the frequency smoothening of 𝛥𝑓 + around the NLEPD (see Fig. 3b), whose origin is traced back to small unavoidable resonant mismatch between the resonances of the two RLC tanks (see Supplementary Note 7 and Supplementary Figure 4), will eventually set a saturation value for the sensitivity.

Noise Analysis

The sublinear frequency detuning Eq. ( 3) guarantees an enhanced transduction function from the voltage variation to the sensitivity 𝜒. However, it does not, address another important characteristic of high-performance sensors that is related to the precision of the measurements. The latter is identified with the smallest measurable variation in the input signal that can be identified by the sensor due to noise at the output signal.

To better understand the effects of noise in the measurement process, we have analyzed the Allan deviation, 𝜎 𝛥𝑓 ̃+ ሺ𝜏ሻ, of the normalized frequency detunings,

, as a function of the sampling time 𝜏 [42] [43]. The measured Allan deviation is reported in Fig. 4a for representative 𝛿𝑉 values -both well within the NLEPD-enhanced sensitivity regime and away from it. We observe that as 𝛿𝑉 decreases, and the system approaches towards the NLEPD, the noise increases. To decreases as we are approaching the NLEPD Each of them is described by a separate noise coefficient 𝛼 𝑐𝑖𝑟 , 𝛼 𝑇𝐿 , 𝛼 𝑑𝑒𝑡 , 𝛼 𝑎𝑑𝑑 , respectively, and contributes to the noise equivalent voltage variation given by 𝛼 𝑉𝑅𝑊 = √𝛼 𝐽𝑁 2 + 𝛼 𝑑𝑒𝑡 2 + 𝛼 𝑎𝑑𝑑 2 , where 𝛼 𝐽𝑁 2 = 𝛼 𝑐𝑖𝑟 2 + 𝛼 𝑇𝐿 2 . Since 𝛼 𝑑𝑒𝑡 can be actively minimized, the thermal noise, 𝛼 𝐽𝑁 , together with 𝛼 𝑎𝑑𝑑 , represent the best obtainable limit for 𝛼 𝑉𝑅𝑊 .

In Fig. 4c, we provide a panorama of 𝛼 𝑉𝑅𝑊 for all 𝛿𝑉-values that we have used in our measurements (blue circles). From the measurements, we conclude that

], indicating a robustness to the VRW noise, which is attributed to the stable hyperbolic nature of the 𝑓 + supermode. Specifically, the phase-space around these hyperbolic fixed points are structurally stable [41], and the associated basin of attraction is relatively broad -even for voltage variations close to the NLEPD (see Fig. 2c). Therefore, the VRW noise is not able to significantly push the phase-space trajectories out of the basin of attraction. In the same figure, we also show the noise coefficient 𝛼 𝐽𝑁 from NGSPICE (red diamonds),

where we have incorporated thermal noise at the resistors, the amplifier, and the TLs described by an ambient temperature of 𝑇 = 300 𝐾. The simulated value of 𝛼 𝐽𝑁 is over an order of magnitude smaller than our experimental measurements for 𝛼 𝑉𝑅𝑊 .

We conclude that the detection noise 𝛼 𝑑𝑒𝑡 combined with 𝛼 𝑎𝑑𝑑 overwhelms the thermal noise 𝛼 𝐽𝑁 . Eventually, the absolute bound of 𝛼 𝑉𝑅𝑊 will be determined by the noise due to coupling fluctuations that originate from applied voltage uncertainties and temperature-dependent capacitive effects.

Discussion

We have demonstrated a SNE sensing of an NLEPD-based voltmeter. The proposed sensing protocol is based on a square-root frequency detuning from the NLEPD induced by a small voltage variation that modifies the coupling between two nonlinear RLC tanks. The NLEPD occurs at the transition between two types of oscillation quenching regimes, i.e., OD and AD domains, in the parameter space, which are consequences of the nonlinear gain/loss channels assigned to each RLC tank. The underlying phase space is structurally stable in the proximity of the (stable) fixed points associated with the degenerate NS, while the corresponding basins of attraction are relatively broad, even for voltage variations that are close to the NLEPD. These characteristics, shield the sensing signal from noise resulting in a two orders of magnitude enhancement of signal-to-noise ratio in the proximity of the NLEPD. Our results establish EPD-sensing from self-oscillating systems as an efficient platform with a dramatically improved SNE factor in the proximity of the NLEPD. Our scheme can guide the design of oscillation quenching-based hypersensitive sensors with enhanced dynamical range that can be utilized in electroencephalography, electrocardiography and neuroprosthetics. Other applications that can utilize the current design include sensitive manometers, flow sensors, accelerometers, inclinometers for telemetry [44] and implantable microsensors [45]. Many of these applications could also be realized in other frameworks, such as photonics, where circuits with neuromorphic functionalities have already been reported [46].

Let us finally point out that while the specific form of the nonlinearity used in our design leads to an enhanced signal-to-noise ratio in the proximity of the NLEPD, this is not necessarily the case for just any self-oscillating system. For example, in

Refs. [13][16], the EPD-based Brillouin ring-laser gyroscope was found to enhance the noise in the proximity of the EPD, offsetting the enhancement in sensitivity. In this respect, it will be interesting to extend the current investigation of SNE sensing schemes to other self-oscillating systems whose nonlinear nature allows for the presence of limit cycles and Hopf bifurcations [47]. Sublinear sensing protocols based on nonlinear mechanisms are a promising direction for building hypersensitive SNE sensors. For example, a recent theoretical proposal that advocates for SNE [48] utilizes sublinear intensity variations due to a dynamic hysteresis occurring in nonlinear resonators. Its experimental implementation, however, could be problematic for fast sensing purposes, since it requires up-down sweeping of a control parameter which can be time-consuming. Further exciting future research includes the identification of other physical observables, such as scattering crosssection anomalies (e.g. Wigner cusps) [49][50] and transmission peak degeneracies [15], whose sublinear response for sensing purposes has thus far been utilized in linear settings.

Methods

Circuit Design and Fabrication

The circuit schematic used for this experiment can be seen in Supplementary

Figure

1. It features two RLC resonators which are coupled to each other via voltage-controlled capacitors. The main elements that make up each RLC resonator are a resistor, 𝑅 𝑖 , an inductor, 𝐿 𝑖 , and a pair of in-parallel grounded capacitors, 𝐶 𝑣 and 𝐶 𝑖 , where 𝑖 = 1,2, denotes the gain and loss RLC units, respectively. The inductors, 𝐿 𝑖 = 200 𝜇𝐻, used in both the gain and loss resonators are API Delevan 807-1537-90HTR. The total capacitance in each resonator is made up by a combination of a tunable capacitor, 𝐶 𝑣 , connected in parallel with a fixed capacitor, 𝐶 𝑖 . The tunable capacitor, 𝐶 𝑣 , is a Murata 81-LXRW19V201-058 with a capacitance range of 100 -200 𝑝𝐹. A voltage of 0.5 𝑉 was applied to the tunable capacitor in each RLC resonator, and it was controlled by a EG&G Instruments 7265 DSP lock-in amplifier, that was connected via a Bayonet Neill-Concelman (BNC) port to a resistor, 𝑅 𝑣 , model Yageo 603-RC0402FR-074K99L, with a resistance of 𝑅 𝑣 = 4.99 𝑘𝛺. This resistor was connected to a grounded fixed capacitor, 𝐶 𝑣1 , a Murata 81-GCM32EL8EH106KA7L, with a capacitance of 𝐶 𝑣1 = 10 𝜇𝐹. The fixed capacitors in each resonator unit, 𝐶 𝑖 , is a Kemet 80-C0603C911F5G with a capacitance of 910 𝑝𝐹. This gives a total capacitance in each RLC resonator of 200 𝑝𝐹 + 910 𝑝𝐹 = 1,110 𝑝𝐹. Each RLC resonator has resistive elements that collectively provide gain and loss. The former is provided by an operational amplifier (op-amp) model Analog Devices 584-ADA4862-3YRZ-R7. The power supply for the op-amp was connected by a standard 3 pin connector, with one going to ground, and the other two being connected to 𝑉 + = 6 𝑉 and 𝑉 -= -6 𝑉, respectively. To produce gain, the op-amp has a pair of internal resistances, 𝑅 𝐺1 and 𝑅 𝐺2 , of 550 Ω each. 𝑅 𝐺1 is connected in between the output of the op-amp and the inverting input of the op-amp. 𝑅 𝐺2 is connected on one end to the inverting input of the op-amp and, grounded on the other end. The mechanically tunable variable resistor in the gain RLC tank, 𝑅 1 , is a Vishay 71-PHPA1206E2001BST1 component which has a resistance of 𝑅 1 = 2000 𝛺. 𝑅 1 is connected on one end to the operational amplifier's non-inverting input. The other end of 𝑅 1 is connected to a capacitor, 𝐶 𝑒1 , model Kemet 80-C0805C100FDTACTU where 𝐶 𝑒1 = 10 𝑝𝐹, capacitively couples each RLC resonator to transmission lines. A mechanically tunable variable resistor, 𝑅 𝑇1 , is connected in parallel to 𝑅 1 , and the op-amp, is a Bourns 652-3269W-1-102GLF with 𝑅 𝑇𝑖 = 720 ± 200 𝛺. The set value of 𝑅 𝑇1 = 720 𝛺, and it is connected to a pair of grounded back-to-back of diodes, 𝐷 𝑖 , Onsemi 512-1N914BWS -which represent the nonlinear elements of this RLC dimer. The fixed resistor in the loss resonator, 𝑅 2 , is a Vishay 71-PCNM2512E2501BST5 component with a resistance of 𝑅 2 = 2500 𝛺. 𝑅 2 is connected in parallel to 𝑅 𝑇2 = 750 𝛺. On the other end, 𝑅 𝑇2 is connected to a pair of back-to-back grounded diodes -the same model of diodes as in the gain resonator was used. The coupling between the two resonators was achieved using two parallel variable capacitors, 𝐶 𝑣𝑐 . The component used in 𝐶 𝑣 , a Murata 81-LXRW19V201-058 is the same model as the variable capacitors in the resonator units, 𝐶 𝑣𝑐 . As in the RLC tanks, 𝐶 𝑣𝑐 is connected to a grounded 𝐶 𝑣1 which in turn is connected to 𝑅 𝑣 . The same lock-in amplifier is used to control the tuning voltage via a BNC port. One of the two capacitors is held fixed at 200 𝑝𝐹 with a constant applied voltage of 0.5𝑉, whereas the other is tuned in voltage range between 0.4 -3.5𝑉.

Voltage and Frequency Detuning Measurements

The emitted signal from the electronic circuit was collected for different applied

Allan Deviation Measurement

The Allan deviation 𝜎 Δ𝑓 ̃+ሺ𝜏ሻ of the frequency associated with the voltage power spectrum peak of the emitted signal is defined as

NGSPICE Simulations

We use NGSPICE, an open-source software for electronic circuits, to simulate the dynamical behavior of our experimental platform. We consider two 𝑅𝐿𝐶 tanks coupled by a capacitor using the same characteristics as the experimental platform, where, unless specified otherwise, uses the same parameters of the electronic components described in 'Circuit Design and Fabrication' sub-section of 'Methods'.

The op-amp in the gain resonator is represented by a high impedance Norton amplifier by designating its constituent components -a transconductance that quantifies gain, a capacitor and diode clippers. Nonlinearity in the gain and loss resonators is modeled with back-to-back diodes via 1N914 diodes -the same type as used in the experimental platform-using the appropriate parameters that describe its behavior. To compensate for the detuning due to the capacitor in the op-amp, the capacitor in the gain resonator, 𝐶 𝑣 + 𝐶 1 , was detuned slightly to 0.9955ሺ𝐶 𝑣 + 𝐶 1 ሻ, to fit the experimental data obtained. In conjunction with that 𝐿 𝑖 ሺ𝑖 = 1,2ሻ was set at 0.965𝐿 𝑖 , in the simulations. Using this setup, we evaluate the signal generated by the circuit for a total time of 𝑡 = 6000 𝑓 0 ≈ 0.017 𝑠 with the time steps of 𝑑𝑡 = 1 ሺ80𝑓 0 ሻ ≈ 37 𝑛𝑠.

In our analysis of the power spectrum, we dropped the first 0.0017 𝑠, which correspond to the short time transient, to consider only the steady-state behavior.

To add noise, we modeled Johnson-Nyquist noise -electronic noise due to thermal fluctuations. This was added to our simulations by adding random voltage sources at every resistor in our circuit (including the TLs). The amount of noise added at each resistor using the TRNOISE function of NGSPICE at these resistors are dictated by the root mean square of the voltage due to Johnson-Nyquist noise. This is given by 𝑣 𝑟𝑚𝑠 = √4𝑘 𝐵 𝑇𝑅𝐵 where 𝑘 𝐵 is Boltzmann's constant; 𝑅 is the value of the respective resistor at which the random voltage source was added; 𝐵 is the bandwidth of the noise for which the natural frequency of the resonator ≈ 338 𝑘𝐻𝑧 was assumed; and 𝑇 is the temperature for which the ambient value of 300 𝐾 was chosen.

is the noise-equivalent voltage variations), diverges as

1 √𝛿𝑉 in the proximity of NLEPD. from oscillation death (OD) to amplitude death (AD) as the voltage variation 𝛿𝑉 increases. The nonlinear exceptional point degeneracy (NLEPD) occurs at 𝛿𝑉 = 0. (Inset) Measured average logarithmic voltage ratio (red symbols) 𝑉 1 𝑉 2 of the NS. Each point represents an average of five independent measurements and the error bars indicate ±1 standard deviation. The black line is the prediction of temporal coupled mode theory (TCMT). The blue line indicates the results from NGSPICE. In panels a and b, the gray shadow highlights the presence of nontrivial unstable NS. c Phasespace analysis and basins of attraction for the stable fixed points associated with the upper ሺ𝑓 + ሻ (blue highlighted domain) and lower ሺ𝑓 -ሻ (yellow highlighted domain) branches of the NS in the AD domain. The voltage variation is 𝛿𝑉 ≈ 2 𝑚𝑉 corresponding to a circuit configuration in the proximity of the NLEPD. frequency domain for which each initial condition has been used. b Experimentally measured emitted spectrum as a function of voltage variations. The red dashed line in both panels a and b is the TCMT prediction Eq. (3) of the nonlinear frequencies. The magenta dotted line indicates the position 𝛿𝑉 = 0, where the nonlinear exceptional point degeneracy (NLEPD) is located. c The measured relative frequency detunings (circles) for the fixed point associated with the upper branch versus the applied voltage variations. The black dashed line is drawn to guide the eye and has a slope of 1 2 that is characteristic of a NLEPD of order N = 2. d The sensitivity of the active nonlinear circuit demonstrating two orders enhancement in the proximity of the NLEPD as opposed to a system configuration away from the NLEPD. The black dashed line in panels b, c and d indicates the numerical results using NGSPICE. Error bars in panels c and d indicate ±1 standard deviation obtained from ten independent measurements. decreases as we are approaching the NLEPD indicating that the sensitivity enhancement offsets the noise enhancement. c The measured (blue circles) voltage random walk coefficient 𝛼 𝑉𝑅𝑊 versus the voltage variations for all 𝛿𝑉 voltage variations that we have used. The red diamonds are the results of NGSPICE simulations where we have considered thermal (Johnson-Nyquist) noise at the resistors, amplifiers and at the transmission lines (TLs) described by an ambient temperature 𝑇 = 300 𝐾. In all cases the error bars are indicated by the shadow area and correspond to ±1 standard deviation evaluated over four different measurements.