Polariton parametric photoluminescence in spatially inhomogeneous systems

A general theory of polariton parametric photoluminescence in spatially inhomogeneous systems is developed. The quantum Langevin equations are solved in a generalized Bogoliubov de Gennes approximation. We apply the formalism to the specific case of a disordered microcavity. In this case, we numerically solve the equations for the coherent emission and the photoluminescence. We describe the effect of the exciton and photon disorder on the photoluminescence pattern exhibited in momentum space, finding a good agreement with the experimental observations.