  We study a model of a photon mode dipole-coupled to a medium of two-level
oscillators in a microcavity in the presence of dephasing processes introduced
by coupling to external baths. Decoherence processes can be classified as
pair-breaking or non-pair-breaking in analogy with magnetic or non-magnetic
impurities in superconductors. In the absence of dephasing, the ground state of
the model is a polariton condensate with a gap in the excitation spectrum.
Increase of the pair-breaking parameter $\gamma$ reduces the gap, which becomes
zero at a critical value $\gamma_{C1}$; for large $\gamma$, the conventional
laser regime is obtained in a way that demonstrates its close analogy to a
gapless superconductor. In contrast, weak non-pair-breaking processes have no
qualitative effect on the condensate or the existence of a gap, although they
lead to inhomogeneous broadening of the excitations.
