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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. By considering the mean-field and Gaussian fluctuations, we find that the low energy physics of the highly non-equilibrium phase transition to the optical parametric oscillator regime is in many 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. As in equilibrium systems, the transition is achieved by tuning this effective chemical potential to the energy of the lowest normal mode. Since the occupations of the modes are available, we determine experimentally observable properties, such as the luminescence and absorption spectra.