Quantum phase transitions in coupled arrays of cavities

Recent experimental progress in cavity quantum-electrodynamics has lead to the fabrication of arrays of coupled cavities, which could be employed to create strongly correlated many-body systems, when operated in the strong coupling regime. In this work, we show that two models of interest to both, condensed matter physics and quantum information science, namely the Bose-Hubbard model and the anisotropic Heisenberg model (XYZ-model), could be realized in such systems with present or near future experimental capabilities. We also discuss the feasibility of realizing a Mott-insulator of photons in such systems. Crucial for this proposal is the possibility of creating very strong photonic Kerr nonlinearities, several oders of magnitude larger than previously considered feasible. We show how this could be achieved with the setting based on electromagnetically induced transparency and with a new scheme, which is experimentally less demanding than the former.