Fluctuations effects in many body self-organization in a 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 technique 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. Thus, beside the mean-field predicted density wave also excited states play an important role at long time.