  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 limit cycle phase where the
local U(1)symmetry and the time-translational symmetry (TTS) of the Liouvillian
super-operator are spontaneously broken (SSB). This SSB manifests itself
through the appearance of an unconditionally and fully squeezed state at the
cavity output, connected to the well-known Goldstone mode. By employing the
Wigner function formalism hence, properly including the quantum noise, we show
that away from the thermodynamic limit and within the quantum regime,
fluctuations notably limit the coherence time of the Goldstone mode due to the
phase diffusion. Our theoretical predictions suggest that interacting multimode
photonic systems are rich, versatile testbeds 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.
