Quantum kinetic derivation of the nonequilibrium Gross-Pitaevskii equation for nonresonant excitation of microcavity polaritons

The space and time dependent nonequilibrium Keldysh-Green functions are employed to derive the scattering rates between the condensed microcavity polaritons described by a Gross-Pitaevskii equation and an uncondensed higher lying exciton reservoir. Slowly varying center coordinates and rapidly varying relative coordinates are assumed. For particle-particle and particle-phonon interactions the scattering rates which provide gain to the condensate are calculated explicitly. These processes result in scattering rates which are quadratic and linear in the density of reservoir excitons, respectively. The resulting quantum Boltzmann equation for the reservoir is simplified by assuming local thermal equilibrium to rate equations for the exciton density and their temperature. Using the microscopically calculated (not phenomenologically chosen) transition amplitudes for CdTe microcavity polaritons we demonstrate that our model is able to describe the spontaneous pattern formation for a ring-shaped nonresonant excitation as seen in recent experiments