Keldysh approach for non-equilibrium phase transitions in quantum optics: beyond the Dicke model in optical cavities

TL;DR

This paper develops a Keldysh path integral approach to study non-equilibrium phase transitions in driven atom-cavity systems, extending beyond the Dicke model to include finite-size effects and nonlinearities, revealing thermal behavior and critical dynamics.

Contribution

It introduces a general Keldysh framework for analyzing nonlinearities and finite-size effects in non-equilibrium quantum optical phase transitions beyond the Dicke model.

Findings

Photonic mode exhibits thermal distribution with an effective temperature.

Finite N causes the photon mode to behave as a damped classical nonlinear oscillator.

A Dicke action for atoms captures depolarization due to dissipative dephasing.

Abstract

We investigate non-equilibrium phase transitions for driven atomic ensembles, interacting with a cavity mode, coupled to a Markovian dissipative bath. In the thermodynamic limit and at low-frequencies, we show that the distribution function of the photonic mode is thermal, with an effective temperature set by the atom-photon interaction strength. This behavior characterizes the static and dynamic critical exponents of the associated superradiance transition. Motivated by these considerations, we develop a general Keldysh path integral approach, that allows us to study physically relevant nonlinearities beyond the idealized Dicke model. Using standard diagrammatic techniques, we take into account the leading-order corrections due to the finite number of atoms N. For finite N, the photon mode behaves as a damped, classical non-linear oscillator at finite temperature. For the atoms, we propose a Dicke action that can be solved for any N and correctly captures the atoms' depolarization due to dissipative dephasing.