Bose condensation of cavity polaritons beyond the linear regime: The thermal equilibrium of a model microcavity

We consider a generalization of the Dicke model. This model describes localized, physically separated, saturable excitations, such as excitons bound on impurities, coupled to a single long-lived mode of an optical cavity. We consider the thermal equilibrium of this model at a fixed total number of excitons and photons. We find a phase in which both the cavity field and the excitonic polarization are coherent. This phase corresponds to a Bose condensate of cavity polaritons, generalized to allow for the fermionic internal structure of the excitons. It is separated from the normal state by an unusual reentrant phase boundary. We calculate the excitation energies of the model, and hence the optical absorption spectra of the cavity. In the condensed phase the absorption spectrum is gapped. The presence of this gap distinguishes the polariton condensate from the normal state and from a conventional laser, even when the inhomogeneous linewidth of the excitons is so large that there is no observable polariton splitting in the normal state.