Fluctuations and symmetry effects in many body self-organization in a dissipative cavity

We investigate the full quantum evolution of ultracold interacting bosonic atoms on a chain and coupled to an optical cavity. Extending the time-dependent matrix product state techniques and the many-body adiabatic elimination techniques to capture the global coupling to the cavity mode and the open nature of the cavity, we examine the long time behavior of the system beyond the mean-field elimination of the cavity field. We show that the fluctuations beyond the mean-field state give a mixed state character to the dissipative phase transition and self-organized steady states. In the case of ideal bosons coupled to the cavity, the open system exhibits a strong symmetry which leads to the existence of conservation laws and multiple steady states. We find that the introduction of a weak breaking of the strong symmetry by a small interaction term leads to a direct transition from multiple steady states to a unique steady state.