Quantum Phase Transitions Of Light In A Dissipative Dicke-Bose-Hubbard Model

The impacts that the environment has on the quantum phase transition of light in the Dicke-Bose-Hubbard model have been studied in our paper.Firstly,based on the quasibosonic approach,the interaction of cavity field and atoms with the environment can be regarded as a whole system and the infinite degree of freedom of the environment can be eliminated.Therefore,the equivalent Hamiltonian of the open system is obtained using the mean-field theory.Then,the eigenvalues and the eigenstates are derived for two two-levels atoms in each cavity as a example based on the perturbation theory.And the analytical solutions of the superfluid order parameter for the dressed states are given.Compared with the ideal case,the order parameter of the system evolves with time that because the cavities and atoms naturally decay in their environment.When the system starts with the superfluid states,the interaction with the environment makes the photons more likely to be localized and a quantum phase transition that from superfluid to Mott-insulation occurs.However,the system of ideal case remains in the superfluid sates.In contrast,if the system starts with the Mott-insulation states,in both cases the system will restore the long-range phase coherence as the hopping energy of photons increases.And a greater hopping energy of photons is required for the dissipative system.The critical value of the hopping energy of photons to restore the long-range phase coherence also depends on the resonant frequency,the number of photons in per cavity and the time that system interacts with its environment.In what follows,we extend the model to an arbitrary number of two-level atoms in each cavity and diagonalize the effective Hamiltonian by numerical computation.Then,the phase diagram is alsogiven accordingly.The dissipation destroys the coherence of the system,the structure of the Mott lobes changes and the area of the coherent phase decreases.Furthermore,the Mott lobes depend crucially on the numbers of atoms and photons.When the number of photons is constant,with an increase in the number of atoms,the effective repulsive potential gets bigger,the Mott lobes become smaller and may disappear and the system tends to be classical.The effect of the dissipation will become more obvious than the one for the ideal cases.With the increase of the photon number and keeping the number of atoms constant in each cavity,more and more Mott lobes emerge in the phase diagrams.