Keldysh field theory for nonequilibrium condensation in a parametrically pumped polariton system

We develop a quantum field theory for parametrically pumped polaritons using Keldysh Green's function techniques with which the occupations of the excitation spectra can be calculated. By considering the mean field and Gaussian fluctuations, we find that the highly nonequilibrium phase transition to the optical parametric oscillator regime is in some ways similar to equilibrium condensation. In particular, we show that this phase transition can be associated with an effective chemical potential, at which the system's bosonic distribution function diverges, and an effective temperature for low energy modes. As in equilibrium systems, the transition is achieved by tuning this effective chemical potential to the energy of the lowest normal mode. Since the nonequilibrium occupations of the modes are available, we determine experimentally observable properties such as the luminescence and absorption spectra