  We propose a two-dimensional model consisting of photons and photon pairs. In
the model, the mixed gas of photons and photon pairs is formally equivalent to
a two-dimensional system of massive bosons with non-vanishing chemical
potential, which implies the existence of two possible condensate phases. Using
the variational method, we discuss the quantum phase transition of the mixed
gas and obtain the critical coupling line analytically. Moreover, we also find
that the phase transition of the photon gas can be interpreted as second
harmonic generation. We then discuss the entanglement between photons and
photon pairs. Additionally, we also illustrate how the entanglement between
photons and photon pairs can be associated with the phase transition of the
system.
