Noise resilient exceptional-point voltmeters enabled by oscillation quenching phenomena

Non-Hermitian wave systems can host exceptional point degeneracies (EPDs), where multiple eigenvalues and eigenvectors coalesce. Near an EPD of order N, a small perturbation ε produces a sublinear eigenvalue detuning Δf ∝ ε^N, suggesting enhanced sensitivity. In many EPD sensor designs, improving resolution by amplification narrows linewidths but also introduces excess noise that can compromise the sensitivity advantage. Moreover, many platforms operate with significant nonlinearities (e.g., self-oscillating lasers), rendering linear tools like the Petermann factor inadequate for noise analysis. A nonlinear dynamical-systems framework based on bifurcation theory provides the appropriate language to analyze nonlinear EPDs (NLEPDs). Coupled nonlinear oscillators exhibit oscillation quenching via amplitude death (AD: homogeneous steady state) and oscillation death (OD: inhomogeneous steady state). The transition between AD and OD can host NLEPDs. The central question addressed here is whether NLEPDs in an active, self-oscillating system can be harnessed for sensing with a net signal-to-noise enhancement (SNE), and how noise behaves in the proximity of the NLEPD.

Prior works established EPD-enabled hypersensitive sensing in linear and photonic systems, including higher-order EPDs and microcavities, and discussed fundamental limits such as Petermann-factor-induced linewidth enhancement near EPDs. In some active systems (e.g., Brillouin ring-laser gyroscopes), increased noise near EPDs negated sensitivity gains. Nonlinear analyses using parity-time (PT) symmetric systems and bifurcation theory have explored NLEPD formation, neuromorphic functionalities, and connections between PT-symmetry phases and oscillation quenching (AD vs OD). Reviews of oscillation quenching in diverse systems (climate, lasers, electronics, chemistry, neuroscience) delineate AD and OD regimes. Recent proposals suggest leveraging sublinear responses from nonlinear mechanisms (e.g., dynamic hysteresis, Wigner cusps) for sensing, but experimental demonstrations with clear SNE near NLEPDs in self-oscillating platforms remained lacking.

Experimental platform: Two coupled nonlinear RLC resonators (natural frequency f0 ≈ 338 kHz; impedance Z0 ≈ 424 Ω) form an electronic dimer. The gain resonator includes a nonlinear amplifier with I–V characteristic I1(V1) = −γ1 + b V1^2; the loss resonator has I2(V2) = γ2 + b V2^2 with b = 7×10^−7 A V^−3. The resonators are coupled by a voltage-controlled capacitor C(V) = κ C; voltage variation δV tunes the coupling κ. Transmission lines (z0 = 50 Ω) couple weakly to each resonator via Ce ≪ C to read out the emission to a VNA. Linear conductances are tuned so the system transitions between AD and OD as δV varies. The NLEPD occurs at δV = 0 and is identified experimentally via transition of steady-state amplitudes from equal (AD) to unequal (OD). Theoretical model: A temporal coupled-mode theory (TCMT) describes the complex field amplitudes an = An e^{iΦn} at each tank with rescaled time τ = 2π f0 t, linear gain/loss parameters γn, coupling x = x(δV), and coupling to TLs η. The model includes nonlinear saturation to prevent unbounded growth; when a1^2 exceeds a threshold, the effective gain becomes negative. Nonlinear supermodes (NS) are steady-state fixed points of the 3D phase space (A1, A2, φ). Stability is determined by the Jacobian eigenvalues λ1,2,3; Re(λn) < 0 indicates stable hyperbolic equilibria. Analytical expressions for An depend on a real variable p ≥ 0 solving a quartic equation; boundaries between domains (AD, OD, trivial) follow from physical constraints, and the AD–OD transition satisfies κ_NLEPD = γ + η. Nonlinear eigenfrequencies f(κ) from TCMT show a square-root detuning from the NLEPD with κ-deviation δκ, implying Δf ∝ √δκ ∝ √δV. Numerical analysis: Fixed points and stability (including unstable NS) are computed via MATLAB fsolve and Jacobian analysis. Basins of attraction in the AD domain are mapped by sweeping initial conditions (A1, A2, φ); initial relative phase determines convergence to the f+ or f− NS. Small intrinsic detuning between the two tanks smooths the frequency splitting near the NLEPD and stabilizes the upper branch f+ while rendering f− unstable close to the NLEPD; away from the NLEPD, bistability persists. Measurements: Emitted spectra are acquired with a Keysight E5080A network analyzer. δV is controlled by a lock-in amplifier over −1.1 V ≤ δV ≤ 2 V with 1 mV resolution. Frequency sweeps (295–320 kHz, 4001 points) with IFBW 100 Hz yield sampling time 40.01 s for spectral readout and peak identification. For Allan deviation, spectra are collected at IFBW 10 kHz, 101 points over a 7 kHz window around Δfi, with 20,000 consecutive measurements over ~2700 s (sampling time 0.1337 s). Simulations: NGSPICE models the full circuit, including op-amp (Norton equivalent), back-to-back diodes (1N914), transmission lines, and minor parameter detunings to match experiment (e.g., Cg+C1 scaled by 0.9955; Li by 0.965). Time-domain signals are simulated with dt = 37 ns; transients are discarded before spectral analysis. Thermal Johnson–Nyquist noise sources are added at all resistors (including TLs) at T = 300 K with Urms = sqrt(4 kB T R B) using the resonator bandwidth (f0 = 338 kHz). Sensing protocol: The sensing measurand is the coupling-induced frequency detuning Δf = f+ − fNLEPD which follows Δf ∝ √δV near the NLEPD. Initial conditions are prepared to select the desired stable branch (typically f+). Sensitivity χ and dynamic range (DR) are quantified from measured Δf(δV). Noise analysis: Precision is assessed via Allan deviation στ(τ) of normalized frequency detunings vs sampling time τ. Noise types are identified by scaling regimes: drift rate ramp (σDRR = aDRR τ), bias instability (σBI = aBI), and voltage random walk (VRW) noise (σVRW = aVRW τ^−1/2). The VRW coefficient aVRW aggregates contributions from circuit thermal noise, TLs, detector/readout, and additional coupling/capacitance fluctuations.

• The electronic dimer exhibits an NLEPD at the transition between AD (homogeneous steady state) and OD (inhomogeneous steady state), controlled by the coupling tuned via δV.
• Nonlinear eigenfrequencies display a square-root detuning from the NLEPD: Δf = f+ − fNLEPD ∝ √δκ ∝ √δV, experimentally confirmed by double-log plots with slope 1/2 and by agreement with TCMT and NGSPICE.
• Two-orders-of-magnitude enhancement of signal-to-noise ratio for voltage variation measurements is demonstrated near the NLEPD relative to operation away from it.
• Sensitivity χ to δV is strongly enhanced near the NLEPD and approaches χ ≈ 1 (within experimental resolution) in the immediate vicinity before saturating due to slight resonant mismatch that smooths the frequency splitting.
• The stable NS near the NLEPD are hyperbolic fixed points with broad basins of attraction, making the dynamics structurally stable and robust to noise; initial-phase control selects the f+ branch in the AD domain, while small tank detuning stabilizes f+ and destabilizes f− near the NLEPD.
• Allan deviation increases as δV approaches the NLEPD (noise rises), but the normalized Allan deviation σ_α(τ) = σ_τ(τ)/χ decreases, showing that sensitivity enhancement outpaces noise increase, yielding net SNE near the NLEPD.
• Measured VRW noise coefficient across δV yields aVRW ≈ 0.0028 V^0.5 [V√s], indicating robustness of the readout to VRW noise; NGSPICE thermal noise estimates are over an order of magnitude lower, implying dominant contributions from detector/readout and coupling fluctuations rather than Johnson–Nyquist noise alone.
• Parameter examples: f0 ≈ 338 kHz; κNLEPD ≈ 0.30; δV range −1.1 to 2 V with 1 mV resolution; coupling calibration near the operating range consistent with κ(V) = κNLEPD(1 − 0.0234 V^−1) δV.

The study addresses whether NLEPDs in self-oscillating nonlinear systems can provide practical sensing advantages in the presence of noise. By operating at the AD–OD transition, the platform leverages a square-root sublinear response of the emission frequency to small voltage-induced coupling changes, achieving high transduction gain and extended dynamic range. Crucially, while noise generally increases near degeneracies, the structural stability of the hyperbolic fixed points and broad basins of attraction render the dynamics resilient, so the noise growth is slower than the signal enhancement, leading to a net signal-to-noise enhancement. This demonstrates that, contrary to some linear analyses and specific active sensors where EPDs amplify noise, properly engineered nonlinear self-oscillating systems can exploit NLEPDs for robust, high-SNR sensing. The findings validate a nonlinear dynamical-systems approach for analyzing sensitivity and noise near degeneracies, beyond linear constructs like the Petermann factor, and are relevant for designing EPD-based devices with improved precision and dynamic range.

This work demonstrates a noise-resilient NLEPD-based voltmeter using a nonlinear electronic dimer that undergoes an AD–OD transition. Near the NLEPD, the sensor exhibits Δf ∝ √δV scaling and achieves a two-orders-of-magnitude SNR enhancement, with experimental observations supported by TCMT theory and NGSPICE simulations. The hyperbolic, structurally stable fixed points and broad basins of attraction underpin the robustness to noise and contribute to improved precision. The results establish self-oscillating NLEPD platforms as viable for hypersensitive sensing with enhanced dynamical range and SNE. Future research directions include: exploring other self-oscillating nonlinear systems (with limit cycles/Hopf bifurcations) to generalize SNE behavior; identifying alternative observables with sublinear responses (e.g., scattering anomalies like Wigner cusps, transmission peak degeneracies); translating the approach to photonic implementations; and optimizing circuits to minimize detection and coupling fluctuation noise to approach thermal limits.

• The TCMT relies on approximations (high-Q resonators, weak inter-resonator coupling, rotating-wave-like elimination of fast oscillations), which introduce discrepancies at large negative δV and very close to the NLEPD.
• Small unintentional detuning between the two RLC tanks smooths the frequency splitting near the NLEPD, stabilizes only the upper branch f+, and sets a saturation limit to sensitivity.
• Noise analysis indicates that detection/readout noise and coupling fluctuations dominate over fundamental Johnson–Nyquist noise; the absolute precision is ultimately limited by applied-voltage uncertainties and temperature-dependent capacitance fluctuations.
• Bistability in the AD domain requires control of initial conditions (relative phase) to select the desired branch; although basins are broad, improper preparation may lead to alternate outcomes.
• The demonstrated SNE relies on the specific nonlinearities and operating regime; not all self-oscillating EPD systems yield SNE (e.g., Brillouin ring-laser gyroscopes show noise amplification near EPDs).