Photon solid phases in driven arrays of non-linearly coupled cavities

We introduce and study the properties of an array of QED cavities coupled by non-linear elements, in the presence of photon leakage and driven by a coherent source. The non-linear couplings lead to photon hopping and to nearest-neighbor Kerr terms. By tuning the system parameters, the steady state of the array can exhibit a photon crystal associated to a periodic modulation of the photon blockade. In some cases the crystalline ordering may coexist with phase synchronisation. The class of cavity arrays we consider can be built with superconducting circuits of existing technology.