In this paper, we theoretically investigate Dicke phase transition at finite temperature in a hybrid optical-mechanical system. The thermodynamic equilibrium states are presented by means of the functional path-integral approach. We plot the mean photon number of the two cavities and the corresponding phase diagram showing the effects of the coupling strength between the two cavities, the nonlinear photon-phonon interaction strength, the finite temperature, the detuning of the two cavities. The coupling strength between the two cavities can make the phase boundary of the normal phase (NP) to the superradiant phase (SP) shift left with the increase of the atom-field coupling strength. The region of the SP is suppressed completely and the SP disappears completely with the increase of the coupling strength between the two cavities or the nonlinear photon-phonon interaction strength, finally just the NP and the dynamically unstable state (DUS) exist. The nonlinear photon-phonon coupling strength just affects the turning curve, but not the phase boundary. The unstable non-zero photon number state and the SP are coexistent, similar to the optical bistability.

在本文中，我们从理论上研究了混合光机械系统中有限温度下的 Dicke 相变。热力学平衡状态通过泛函路径积分方法呈现。我们绘制了两个腔的平均光子数和相应的相图，显示了两个腔之间的耦合强度、非线性光子-声子相互作用强度、有限温度、两个腔的失谐。两个腔之间的耦合强度可以使正相（NP）到超辐射相（SP）的相边界随着原子场耦合强度的增加而左移。随着两腔间耦合强度或非线性光子-声子相互作用强度的增加，SP区域被完全抑制，SP完全消失，最终只存在NP和动态不稳定态（DUS）。非线性光子-声子耦合强度只影响转向曲线，而不影响相界。不稳定的非零光子数态和 SP 共存，类似于光学双稳态。