Bose-Einstein Condensation of Photons and Photon Pairs

We investigate the Bose-Einstein condensation of photons and photon pairs in a two-dimension optical microcavity. We find that in the paraxial approximation, the mixed gas of photons and photon pairs is formally equivalent to a two dimension system of massive bosons with non-vanishing chemical potential, which implies the existence of two possible condensate phase. We also discuss the quantum phase transition of the system and obtain the critical point analytically. Moreover, we find that the quantum phase transition of the system can be interpreted as second harmonic generation.