Dissipative Phase Transition in the two-photon Dicke Model*

The Dicke model has long been a focal point of research due to its intriguing collective phenomena, particularly superradiance. A notable extension of this model is the two-photon Dicke model, where atomic transitions are mediated by two photons instead of one, as in the conventional Dicke model. In this study, we demonstrate a method to stabilize the previously unstable two-photon Dicke model. Our findings reveal that the superradiant phase can be recovered by incorporating two-photon losses into the system. Utilizing second-order mean-field calculations, we derive an analytical expression, which is then compared with numerical simulations employing Exact Diagonalization and the Quantum Trajectory Method for systems with a small number of emitters. The results from both approaches exhibit strong agreement, thereby reinforcing the validity of our findings. Additionally, we analyze the Wigner distribution to confirm the Z4 symmetry of the model and provide insights into the nature of the phase transition occurring in this system.