Non-equilibrium phase transition to the polariton OPO regime

Exciton-polaritons are the quasi-particles that form when cavity photons couple strongly to quantum well excitons in semiconductor microcavities. When a pump laser is applied near the point of inflection of the lower polariton dispersion, a phase transition to the polariton optical parametric oscillator regime where two additional, `signal' and `idler', modes with macroscopic occupation appear can occur. The steady state of the non-equilibrium polariton system is maintained by continuous pumping and the Keldysh functional integral approach is used to study the phase transition. Despite its highly non-equilibrium nature, an effective chemical potential is identified and the phase transition occurs when the effective chemical potential crosses the normal modes. The Keldysh formalism also gives access to the occupations of the modes and experimentally observable properties such as the incoherent luminescence and absorption spectra are calculated. One of the key properties of the signal mode is that it occurs near the minimum of the lower polariton dispersion with zero momentum. To calculate the mean field occupation of the three mode optical parametric oscillator regime analytically, the signal momentum has to be chosen explicitly. A simple method to determine the signal momentum by using linear response analysis for any system parameters is proposed and the predictions compared with numerical integration of the complex Gross-Pitaevskii equations describing the system. At weak pump strengths, the signal momentum is found best by the linear response analysis of the three mode description, while at higher pumping, a linear response analysis of the single pump mode gives best agreement with the numerical simulations