Effective Bose-Hubbard interaction with enhanced nonlinearity in an array of coupled cavities

An array of coupled cavities, each of which contains an N four-level atom, is investigated. When cavity fields dispersively interact with the atoms, an effective Bose-Hubbard model can be achieved. By numerically comparing the full Hamiltonian with the effective one, we find that within the parameters region, the effective Hamiltonian can completely account for the Mort-insulator as well as the phase transition from the similar Mort-insulator to superfluid. Through jointly adjusting the classical Rabi frequency and the detuning, the nonlinearity can be improved.