Phase transition in open quantum systems is an important topic of current research. Due to more system parameters compared to the closed quantum systems with conserved dynamics, open quantum systems can have novel phases including multi-stability and limit-cycle.  

In this article, we theoretically investigate the first- and second-order quantum dissipative phase transitions of a three-mode cavity with a Hubbard interaction. In both types, there is a mean-field (MF) limit cycle phase where both the local U(1) symmetry and the time-translational symmetry of the Liouvillian superoperator are spontaneously broken. In MF, this spontaneous symmetry breaking manifests itself through the appearance of an unconditionally and fully squeezed state at the cavity output, connected to the well-known Goldstone mode. Our theoretical predictions suggest that interacting multimode photonic systems are rich, versatile test beds for investigating the crossovers between the mean-field picture and quantum phase transitions, a problem that can be investigated in various platforms including superconducting circuits, semiconductor microcavities, atomic Rydberg polaritons, and cuprite excitons.