Quantum hydrodynamics of a polariton fluid: Pure energy relaxation terms

Cavity polaritons, hybrid half-light half-matter excitations in quantum microcavities in the strong-coupling regime, demonstrate clear signatures of quantum collective behavior, such as analogs of Bose-Einstein condensates and superfluidity at remarkably high temperatures. The analysis of the formation of these states demands an account of the relaxation processes in the system. Although there are well-established approaches for the description of some of them, such as finite lifetime polaritons, an external optical pump, and coupling with an incoherent excitonic reservoir, the treatment of pure energy relaxation in a polariton fluid still remains a puzzle. Here, based on the quantum hydrodynamics approach, we derive the corresponding equations where the energy relaxation term appears naturally. We analyze in detail how it affects the dynamics of polariton droplets and the dispersion of elementary excitations of a uniform polariton condensate. Although we focus on the case of cavity polaritons, our approach can be applied to other cases of bosonic condensates, where the processes of energy relaxation play an important role.