First- and second-order phase transitions in the Dicke model: Relation to optical bistability

The authors consider a system comprised of a collection of identical two-level atoms interacting with the electromagnetic field in the dipole approximation and with an externally applied steady-state coherent driving field. The atoms are considered to interact with each other only via the electromagnetic field and are assumed to be contained within a volume much smaller than a resonance wavelength. The system is treated as a quantum-statistical ensemble in the rotating frame, rotating at the carrier frequency of the externally applied coherent field. The Hamiltonian in this frame is explicitly time independent and the exact analog of a spin temperature is defined. The thermodynamic-Green's-function method is used to determine the thermodynamic equilibrium properties of the system in the rotating frame. This results in a nonlinear relation between the applied field and the field in the volume containing the atoms. The expression is a nonlinear function of the effective or "spin" temperature as well, and has the form of a first-order phase transition with the macroscopic atomic polarization as order parameter. In the low-"spin"-temperature limit, and for perfect tuning, the expression reduces to the exact form of the nonlinear relation for optical bistability derived by Bonifacio and Lugiato and relating the incident and transmitted fields for atoms in a ring cavity in the mean-field approximation. The results contain absorptive as well as dispersive contributions. These results predict essentially different behavior as a function of off tuning compared with behavior predicted by others using statistical steady-state models far from thermodynamic equilibrium. In the limit of zero value for the applied field amplitude, the results reduce to the conditions for the well-known second-order superradiant" phase transition. The condition for the existence of the second-order phase transition is shown to depend on the cavity tuning and the photon escape rate. The authors also show why, in light of the experiments in optical bistability, this second-order phase is difficult to observe.