  We consider a generalization of the Dicke model. It 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 the 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 a gap distinguishes
the polariton condensate from the normal state and from a conventional laser,
even when the inhomogeneous linewidth is so large that there is no observable
polariton splitting in the normal state.
