Analytical phase boundary of a quantum driven-dissipative Kerr oscillator from classical stochastic instantons

TL;DR

The paper addresses the bistability and phase boundary of a two-photon driven Kerr oscillator in the thermodynamic limit, where a simple thermodynamic potential is hard to define. It maps the Keldysh path integral to a classical MSRJD description, treating the photon interaction $U$ as an effective temperature and using real-time instantons to estimate bistable tunneling rates and the phase boundary. An Ornstein-Uhlenbeck–type pseudo-potential and controlled approximations yield an implicit analytic expression for the first-order transition line as a function of $(\delta/\gamma, \epsilon/\gamma)$, with numerical validation showing errors below about 5%. The approach unifies semi-analytical methods such as Truncated Wigner and Fokker-Planck analyses and provides a scalable framework potentially applicable to driven-dissipative many-body optical systems like driven Bose-Hubbard arrays or Tavis-Cummings models.

Abstract

The framework of Keldysh path integral concisely describes quantum systems driven away from thermal equilibrium, such as the two-photon driven Kerr oscillator. Within the thermodynamic limit of diverging photon occupation, we map it to a Martin-Siggia-Rose-Janssen-de Dominicis path integral, and obtain a purely classical, stochastic equivalent where photon self-interaction plays the role of temperature. This perspective sheds light on the difficulties encountered in the search for an effective thermodynamic potential to describe the bistability of the model. It allows us to estimate the bistable tunneling rates using a real-time instanton technique leading to an analytical expression of the phase boundary, the first to our knowledge. It opens the way to powerful semi-analytical techniques to be applied to various quantum optics models displaying bistability.

Figures (3)

• Figure 1: Minimal action paths of the Ornstein-Uhlenbeck model.(a) Escape trajectory obtained by numerical integration (pink), non-conservative force field (blue) and equipotentials of $P$ (black). The steady-state is obtained by solving and averaging the Langevin dynamics (colormap). (b) Action along the instanton, using analytical expression based on Eq. \ref{['eq:OUpseudopotential']} (pink) and numerical integration (blue). (c) Given $\vec{f}$, $\vec{p}$ and $\dot{\vec{q}}$ are constrained to the grey circles. Grey arrows indicate the dynamical flow of $\theta(t)$, see Eq. \ref{['eq:theta']}. (d) In a pure potential force ($\delta=0$), $\theta_{\rm s}=\pi$, and the escape trajectory opposes $\vec{f}$.
• Figure 2: Minimal action paths of the 2-photon driven Kerr oscillator model at $\delta / \gamma = -10$ and $\epsilon / \gamma = 3.2$. (a) Instanton escape trajectories from the vacuum state (pink) and the bright state (blue). The streamlines represent the non-conservative force field $\vec{f}$. (b) Action along the bright state escape path, by numerical integration (blue) and analytical ansatz (orange). (c) Action along the vacuum state escape path, by numerical integration (pink) and analytical ansatz (green). (d)$\theta(t)$ along the bright state escape, by numerical integration (blue) and value of the instantaneous stationary angle $\theta_{\rm s}(\vec{q})$ (orange). (e)$\theta(t)$ along the vacuum state escape, by numerical integration (pink) and value of the instantaneous stationary angle $\theta_{\rm s}(\vec{q})$ (green).
• Figure 3: 2-photon driven Kerr oscillator phase diagram.(a) The mean-field phase diagram in detuning $\delta$ and drive amplitude $\epsilon$ shows a vacuum regime (blue), a cat regime (purple) and a bistable regime (colormap). Insets show stable (white) and unstable (black) states, and $\vec{f}$ aspect in each region. The colormap shows the photon number in the bistable regime at $U = 0.1 \gamma$ using exact expressions. A phase boundary appears, with an abrupt change of the photon number, separating a vacuum and a bright phase. The analytical phase boundary obtained at Eq. \ref{['eq:transitionLine']} (pink) matches. (b) The relative difference between transition boundary predicted by instanton method and numerically located on exact expression for photon number.