Dissipative generation of highly entangled states of light and matter

We investigate the full quantum evolution of ultracold interacting bosonic atoms confined to a chain geometry and coupled to the field of an optical cavity. Extending the time-dependent matrix product state techniques to capture the global coupling to the cavity mode and the open nature of the cavity, we fully include the light-atom entanglement. We examine the long time behavior of the system beyond the mean-field elimination of the cavity field. We show that in the self-organized phase the steady state consists in a mixture of the mean-field predicted density wave states and coherent states with lower photon number, with a large entanglement between the atomic and photonic degrees of freedom. In the regime of large dissipation strengths we develop a variant of the many-body adiabatic elimination technique and obtain a highly entangled steady state with a fully mixed atomic sector. We observe numerically the crossover from the density wave state towards the fully mixed state.