To $\mathcal{PT}$ or not to $\mathcal{PT}$: Noise-induced escape and nonlinear-damping stabilization in a parity-time dimer

Abstract: Parity-time ($\mathcal{PT}$) symmetric systems exhibit long-lived excitations by balancing gain and loss in coupled resonators, driving extensive theoretical interest and diverse experimental realizations. Realistic physical implementations, however, inevitably introduce nonlinearities and noise. This mandates a rigorous reevaluation of their global long-time dynamics. In this work, we show that Hamiltonian Duffing nonlinearity restricts the $\mathcal{PT}$-unbroken phase to a finite, nonattracting region of phase space. Consequently, unavoidable fluctuations drive first-passage escape into runaway trajectories. This renders the linearly $\mathcal{PT}$-unbroken phase a purely transient phenomenon. We then recover global stochastic stability by introducing two-photon loss on the gain oscillator. This nonlinear damping explicitly breaks exact $\mathcal{PT}$ symmetry while supplying genuine phase-space attraction, generating a bistable regime where a low-amplitude orbit mimicking the original linear state coexists with a high-amplitude limit cycle. Thus, we establish a revised origin for stability in non-Hermitian experiments: the observed long-time stochastic stability is governed by inherent restoring dissipation rather than the spectral $\mathcal{PT}$ symmetry itself.

• The paper demonstrates that nonlinear gain–loss dimers lack asymptotic attraction even below the linear exceptional point, so finite noise eventually drives trajectories out of their bounded region.
• TWA simulations of 10,000 trajectories find a mean escape time of 2.6 × 10⁴ at occupation scale 𝒩 = 1000, with lifetime scaling algebraically as 𝒩⁰·⁵⁹ rather than through exponentially suppressed Kramers activation.
• The paper shows that two-photon loss creates phase-space attraction and stabilizes low- and high-amplitude limit cycles, but this restoring mechanism explicitly breaks exact PT symmetry and can produce bistable or abruptly reorganized dynamics.

The paper by Tuquero, Seibold, and Zilberberg (2608.14468) addresses a foundational question in non-Hermitian physics: whether the linearly $\mathcal{PT}$-unbroken phase of a gain–loss dimer constitutes a genuinely stable long-time state once nonlinearities and reservoir fluctuations are included. The authors answer negatively for the idealized model and demonstrate that observed stability in nonlinear $\mathcal{PT}$ experiments must instead originate from restoring dissipation that itself breaks exact $\mathcal{PT}$ symmetry.

Model and framework

The system is a dimer of Duffing oscillators coupled linearly at rate $J$, with single-photon loss on oscillator 1 (rate $\gamma_1$), single-photon gain on oscillator 2 (rate $\gamma_2$), and an optional two-photon loss channel on oscillator 2 (rate $\beta$), all described by a Lindblad master equation. The open-system $\mathcal{PT}$ transformation maps each jump operator to its parity-conjugated adjoint, so the loss channel maps onto gain precisely when $\gamma_1=\gamma_2\equiv\gamma$. The two-photon-loss channel, however, maps onto an absent two-photon gain channel; hence any $\beta\neq0$ explicitly breaks the symmetry. In the linear limit $\mathcal{PT}$0, the squared eigenfrequencies are real below the exceptional point $\mathcal{PT}$1, with both branches positive in the weak-coupling regime $\mathcal{PT}$2.

Dynamics beyond mean field are treated via the truncated Wigner approximation (TWA), which maps the master equation onto stochastic equations for complex amplitudes $\mathcal{PT}$3. The scaling parameter $\mathcal{PT}$4 sets the characteristic occupation: noise amplitude scales as $\mathcal{PT}$5 while deterministic dynamics are unchanged, providing a controlled semiclassical limit.

Deterministic confinement without attraction

Finite-amplitude initial conditions split into two fates: bounded oscillations near the origin, and runaway growth of the gain oscillator for sufficiently large amplitudes. The mechanism is a differential Duffing frequency renormalization induced by population imbalance, which dynamically decouples the resonators. Crucially, at exact gain–loss balance the deterministic flow preserves phase-space volume — the Hamiltonian contribution is divergence-free and $\mathcal{PT}$6 — so no asymptotically attracting set can exist within the bounded sector. The bounded region therefore provides no restoring drift against fluctuations, and its boundary is a global property of the nonlinear flow, not derivable from local linearization about the origin. A supplementary similarity transformation establishes exactly that every boundary scale varies as $\mathcal{PT}$7, so the bounded region expands without limit as $\mathcal{PT}$8, recovering global boundedness only in the strictly linear limit.

Noise-induced first-passage escape

Because the bounded sector is nonattracting, escape under gain–loss noise is a first-passage process rather than Kramers-type activation out of an attracting basin. Simulating $\mathcal{PT}$9 TWA trajectories with escape defined as either oscillator exceeding $\mathcal{PT}$0, the authors find:

Quantity	Value
Mean escape time at $\mathcal{PT}$1	$\mathcal{PT}$2
Relative width	$\mathcal{PT}$3
Gamma shape parameter	$\mathcal{PT}$4
Scaling of mean lifetime	$\mathcal{PT}$5

The distribution is strongly nonexponential ($\mathcal{PT}$6), consistent with stochastic wandering through a nonattracting region rather than waiting-time statistics from a metastable attractor. Most significantly, the mean lifetime scales algebraically rather than exponentially in noise variance: reducing the noise variance by a factor of 20 increases survival time by only a factor of roughly 6. This fragility implies that the linearly $\mathcal{PT}$7-unbroken phase is a transient phenomenon for any finite fluctuation strength — apparently stable dynamics persist only over observation windows shorter than the characteristic escape time.

Stabilization via symmetry-breaking nonlinear damping

Introducing two-photon loss on the gain oscillator supplies amplitude-dependent phase-space contraction (divergence $\mathcal{PT}$8 per mode), generating genuine attraction. Trajectories then converge onto one of two coexisting limit cycles: a low-amplitude $\mathcal{PT}$9-like orbit or a high-amplitude, population-imbalanced cycle sustained by van der Pol-like self-saturation against runaway. With restoring attraction present, gain–loss fluctuations no longer drive irreversible escape over simulated timescales; the long-time phase-space distribution becomes bimodal around the two attractors.

Varying the damping strength $J$0 reveals a sharp reorganization: total intensity and population imbalance change discontinuously when the high-amplitude cycle disappears, leaving only the low-amplitude attractor. Prior to this transition, long-time observables depend only weakly on noise strength, confirming the robustness conferred by nonlinear damping. The essential tension is explicit: the mechanism that stabilizes the system necessarily breaks the very $J$1 symmetry whose unbroken phase it stabilizes.

Limitations and open questions

Several caveats bear directly on these results. First, the simulations deliberately omit the multiplicative noise generated by the two-photon-loss channel, retaining only the additive gain–loss fluctuations; the authors state plainly that they do not assume the omitted contribution is parametrically negligible on the high-amplitude branch, so quantitative features of the bistable regime may shift when it is included. Second, right-censored trajectories (at most 0.14% of realizations) are neglected in the escape-time fits. Third, the Gamma law is used as an empirical parametrization, not derived from microscopic stochastic dynamics, and the effective exponent $J$2 is established over a finite range of $J$3 without a theoretical derivation. Fourth, the two-photon Lindblad channel cannot be mapped exactly onto a phenomenological mechanical force $J$4 by Hermitian counterterms alone, which limits direct translation of the stabilization mechanism into conventional mechanical $J$5 models; the paper provides a squeezing-like Hamiltonian counterterm that recovers the standard mechanical spectrum but notes the spectral results refer to the original ladder-operator formulation. Open questions include the analytic origin of the algebraic escape scaling and how the bistability boundary depends on the multiplicative noise channel.

Conclusion

This work disentangles three distinct notions of stability — spectral $J$6 stability, deterministic confinement, and stochastic stability — in a canonical nonlinear gain–loss dimer. Its central claim is strict: long-time stochastic stability in such systems is governed by restoring phase-space attraction, not by the local spectrum, and the linearly $J$7-unbroken phase has finite lifetime under any finite noise. Observed stable $J$8-like dynamics in experiments must therefore reflect either finite-time survival within a nonattracting sector or additional restoring mechanisms — nonlinear damping, gain saturation, or engineered feedback — that independently break exact $J$9 symmetry. This reframes the interpretation of existing non-Hermitian experimental platforms and identifies global phase-space structure, rather than exceptional-point physics, as the operative criterion for assessing long-time stability.