Keldysh Greens function approach to coherence in a non-equilibrium steady state: connecting Bose-Einstein condensation and lasing

Solid state quantum condensates often differ from previous examples of condensates (such as Helium, ultra-cold atomic gases, and superconductors) in that the quasiparticles condensing have relatively short lifetimes, and so as for lasers, external pumping is required to maintain a steady state. On the other hand, compared to lasers, the quasiparticles are generally more strongly interacting, and therefore better able to thermalise. This leads to questions of how to describe such non-equilibrium condensates, and their relation to equilibrium condensates and lasers. This chapter discusses in detail how the non-equilibrium Greens function approach can be applied to the description of such a non-equilibrium condensate, in particular, a system of microcavity polaritons, driven out of equilibrium by coupling to multiple baths. By considering the steady states, and fluctuations about them, it is possible to provide a description that relates both to equilibrium condensation and to lasing, while at the same time, making clear the differences from simple lasers.