Keldysh approach for nonequilibrium phase transitions in quantum optics: Beyond the Dicke model in optical cavities

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.

Self-ordering dynamics of ultracold atoms in multicolored cavity fields

“…Even more complex results could be expected from a generalization to fermionic particles [5] or particles with several internal levels [30]. Of course, one of the most interesting aspects and motivations of this approach is the fact that only minor additions to current experimental setups are required and the theoretical model for the current generations of experiments have proven to be surprisingly accurate and reliable [6,8,17,31,32].…”

Discrete Time-Crystalline Order in Cavity and Circuit QED Systems

“…Our model can readily be generalized by taking into account the atomic motional degrees of freedom [83], interactions between atoms [84], local decoherence, and spontaneous emission [85][86][87]. In particular, our study raises an intriguing question of whether an intrinsically nonunitary DTC can possess absolute stability [82] against arbitrary nonunitary perturbation.…”

Collective phases of strongly interacting cavity photons

“…For example, thermal equilibrium is essentially never a reasonable assumption in photonic systems, where dissipation must be countered by active pumping [10]. Indeed, the inadequacy of equilibrium descriptions for photonic systems has long been recognized [11], even though close analogies to thermal systems sometimes exist [12][13][14][15][16].Until recently, photonic systems have been restricted to a weakly interacting regime. With notable progress towards generating strong optical nonlinearities at the few-photon level, for example with atoms coupled to small-mode-volume optical devices [17][18][19][20][21][22], Rydberg polaritons [23,24], and circuit-QED devices [25][26][27][28], this situation is rapidly changing.…”

“…For example, thermal equilibrium is essentially never a reasonable assumption in photonic systems, where dissipation must be countered by active pumping [10]. Indeed, the inadequacy of equilibrium descriptions for photonic systems has long been recognized [11], even though close analogies to thermal systems sometimes exist [12][13][14][15][16].…”