Non-equilibrium dynamics of non-linear Jaynes-Cummings model in cavity arrays

We analyze in detail an open cavity array using mean-field description, where each cavity field is coupled to a number of three-level atoms. Such system is highly tunable and can be described by a Jaynes-Cummings like Hamiltonian with additional non-linear terms. In the single cavity case we provide simple analytic solutions and show, that the system features a bistable region. The extra non-linear term gives rise to a rich dynamical behaviour including occurrence of limit cycles through Hopf bifurcations. In the limit of large non-linearity, the system exhibits an Ising like phase transition as the coupling between light and matter is varied. We then discuss how these results extend to the two-dimensional case.