Generating Functional Approach for Spontaneous Coherence in Semiconductor Electron-Hole-Photon Systems

Electrons, holes, and photons in semiconductors are interacting fermions and bosons. In this system, a variety of ordered coherent phases can be formed through the spontaneous phase symmetry breaking because of their interactions. The Bose-Einstein condensation (BEC) of excitons and polaritons is one of such coherent phases, which can potentially crossover into the Bardeen-Cooper-Schrieffer (BCS) type ordered phase at high densities under quasi-equilibrium conditions, known as the BCS-BEC crossover. In contrast, one can find the semiconductor laser, superfluorescence (SF), and superradiance as relevant phenomena under nonequilibrium conditions. In this paper, we present a comprehensive generating functional theory that yields nonequilibrium Green's functions in a rigorous way. The theory gives us a starting point to discuss these phases in a unified view with a diagrammatic technique. Comprehensible time-dependent equations are derived within the Hartree-Fock approximation, which generalize the Maxwell-Semiconductor-Bloch equations under the relaxation time approximation. With the help of this formalism, we clarify the relationship among these cooperative phenomena and we show theoretically that the Fermi-edge SF is directly connected to the e-h BCS phase. We also discuss the emission spectra as well as the gain-absorption spectra.