Entangled States and Super-radiant Phase Transition

The Dicke spin-boson model is composed by a single bosonic mode and an ensemble of $N$ identical two-level atoms. Assuming thermal equilibrium with a reservoir at temperature $\beta^{-1}$, we consider the situation where the coupling between the bosonic mode and the atoms generates resonant and non-resonant processes. The thermodynamic of the model is investigated. Next we introduce dipole-dipole interaction between the atoms. We investigate the transition from fluorescent to super-radiant phase and the quantum phase transition in a situation where the dipole-dipole interaction between the atoms generates entangled states in the atomic system. We proved that, the critical behavior is not modified by the introduction of the dipole-dipole interaction.