Driven-Dissipative Time Crystalline Phases in a Two-Mode Bosonic System with Kerr Nonlinearity

For the driven-dissipative system of two coupled bosonic modes in a nonlinear cavity resonator, we demonstrate a sequence of phase transitions from a trivial steady state to two distinct dissipative time crystalline phases. These effects are already anticipated at the level of the semiclassical analysis of the Lindblad equation using the theory of bifurcations and are further supported by the full quantum treatment. The system is predicted to exhibit different dynamical phases characterized by an oscillating non -equilibrium steady state with nontrivial periodicity, which is a hallmark of time crystals. We expect that these phases can be directly probed in various cavity QED experiments.