Kinetics and luminescence of the excitations of a nonequilibrium polariton condensate

We extend the description of the nonequilibrium Gross-Pitaevskii equation for microcavity polaritons by including not only the kinetics of the nonresonantly pumped exciton reservoir but also that of the collective excitations---called bogolons---above the condensate. Quasiequilibrium Hartree-Fock-Bogoliubov-Popov self-energies are used to describe the condensate and the bogolons. With nonequilibrium Green functions, the quantum Boltzmann equation of the bogolons in a frame moving with the condensate, together with the coupled Gross-Pitaevskii equation can be derived. The resulting scattering rates contain vertex corrections in form of symmetrized products of the $u$ and $v$ pairing functions of all involved scattering states. Corresponding results have been derived for ultracold atoms by A. Griffin, T. Nikuni, and E. Zaremba [Bose-Condensed Gases at Finite Temperatures, 1st ed. (Cambridge University Press, Cambridge, New York, 2009)] Furthermore, we show that the results of the quantum kinetics also determine the luminescence spectrum including the ghost branch. The kinetics is solved numerically, the resulting condensate, the bogolon distribution, and the luminescence are calculated temporally and spatially resolved after the switch-on of a $\mathit{cw}$ nonresonant pump beam with a Gaussian profile. It is studied in detail to which extent the bogolons approach a local thermal equilibrium.