Finite-temperature Dicke phase transition and the collapse of superradiant phase in an optomechanical-atomic cavity

We mainly investigate Dicke phase transition at finite temperature in an optomechanical-atomic cavity by means of the functional path-integral approach. By the thermodynamic equilibrium equation, the mean photon number, the scaled atomic population, the mean phonon number, the average energy and the rich phase diagram are depicted in the existence of the nonlinear atom–photon interaction, the nonlinear photon–photon coupling strength, the pump–cavity detuning and the finite temperature. Beside Dicke phase transition from the normal phase (NP) to the superraidant phase (SP) at finite temperature, the collapse of the SP and the dynamically unstable state (DUS) are revealed showing the effects of the photon–phonon coupling strength ξ and nonlinear atom–photon interaction U. Meanwhile, a new phase transition occurs from the SP to the dynamically unstable state (DUS) at the turning point gtwith the existence of ξ or U. The region of the SP can be completely eliminated by the mechanically and thermally stimulated photon-emission. The entropy and corresponding specific heats are presented to demonstrate the corresponding critical behavior at the phase transition point.