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We present an extended version of the projector-based renormalization methodthat can be used to address not only equilibrium but also non-equilibriumsituations in coupled fermion-boson systems. The theory is applied tointeracting electrons, holes and photons in a semiconductor microcavity, wherethe loss of cavity photons into vacuum is of particular importance. The methodincorporates correlation and fluctuation processes beyond mean-field theory ina wide parameter range of detuning, Coulomb interaction, light-matter couplingand damping, even in the case when the number of quasiparticle excitations islarge. This enables the description of exciton and polariton formation, andtheir possible condensation through spontaneous phase symmetry breaking byanalyzing the ground-state, steady-state and spectral properties of a rathergeneric electron-hole-photon Hamiltonian, which also includes the coupling totwo fermionic baths and a free-space photon reservoir. Thereby, thesteady-state behavior of the system is obtained by evaluating expectationvalues in the long-time limit by means of the Mori-Zwanzig projectiontechnique. Tracking and tracing different order parameters, the fullyrenormalized single-particle spectra and the steady-state luminescence, wedemonstrate the Bose-Einstein condensation of excitons and polaritons and itssmooth transition when the excitation density is increased.

Projector-based renormalization approach to electron-hole-photon systems in nonequlibrium steady-state