Mean-field theory and fluctuation spectrum of a pumped decaying Bose-Fermi system across the quantum condensation transition

We study the mean-field theory, and the properties of fluctuations, in an out of equilibrium Bose-Fermi system, across the transition to a quantum condensed phase. The system is driven out of equilibrium by coupling to multiple baths, which are not in equilibrium with each other, and thus drive a flux of particles through the system. We derive the self-consistency condition for a uniform condensed steady state. This condition can be compared both to the laser rate equation and to the Gross-Pitaevskii equation of an equilibrium condensate. We study fluctuations about the steady state and discuss how the multiple baths interact to set the system's distribution function. In the condensed system, there is a soft phase (Bogoliubov, Goldstone) mode, diffusive at small momenta due to the presence of pump and decay, and we discuss how one may determine the field-field correlation functions properly including such soft phase modes. In the infinite system, the correlation functions differ both from the laser and from an equilibrium condensate; we discuss how in a finite system, the laser limit may be recovered.