Noise-Resilient Phase Transitions and Limit-Cycles in Coupled Kerr Oscillators

Driven-dissipative quantum many-body systems have been the subject of many studies in recent years. They possess unique, novel classes of dissipation-stabilized quantum many-body phases including the limit cycle. For a long time it has been speculated if such a behavior, a recurring phenomenon in non-linear classical and quantum many-body systems, can be classified as a time crystal. However, the robustness of these periodic dynamics, against quantum fluctuations is an open question. In this work we seek the answer to this question in a canonical yet important system, i.e., a multi-mode cavity with self and cross-Kerr non-linearity, including the fluctuation effects via higher order correlations. Employing the Keldysh path integral, we investigate the Green's function and correlation of the cavity modes in different regions. Furthermore, we extend our analysis beyond the mean-field by explicitly including the effect of two-body correlations via the 2nd-cumulant expansion. Our results shed light on the emergence of dissipative phase transitions in open quantum systems and clearly indicate the robustness of limit-cycle oscillations in the presence of the quantum fluctuations.