Pseudo-thermalization in driven-dissipative non-Markovian open quantum systems

We investigate a `pseudo thermalization' effect, where an open quantum system coupled to several non-Markovian reservoirs presents an emergent thermal behaviour in spite of its coupling to a non-equilibrated environment. The thermal behaviour is visible at both static and dynamical levels and the system satisfies the fluctuation-dissipation theorem. Our analysis is focused on the exactly solvable model of a weakly interacting driven-dissipative Bose gas in presence of frequency-dependent particle pumping and losses, and is based on a quantum Langevin theory, which we derive starting from a microscopical quantum optics model. For generic non-Markovian reservoirs, we demonstrate that the emergence of thermal properties occurs in the range of frequencies corresponding to low-energy excitations. For the specific case of non-Markovian baths verifying the Kennard-Stepanov relation, we show that pseudo-thermalization can instead occur at all energy scales. The possible implications regarding the interpretation of thermal laws in exciton-polariton low temperature experiments are discussed. We finally show that the presence of either a saturable pumping or a dispersive environment leads to a breakdown of the pseudo-thermalization effect.