A mobile impurity immersed into a degenerate Fermi system is a paradigmatic problem in many-body physics. The dressing of the impurity by particle-hole excitations of the environment leads to formation of quasi-particles called polarons. Recent observations of these quasi-particles in experiments with ultracold atomic gases or with two-dimensional monolayer semiconductors renewed interest in the field of quantum impurities. The description of polaron physics in these systems was based on equilibrium quantum field theory or wave function techniques. However, a non-equilibrium treatment is required that includes not only coherent processes of quasi-particle formation but also gain and loss in the theoretical description, since, for example, in semiconductors, due to finite lifetime of excitons, the system is far from equilibrium.
In this talk I will describe our approach to non-equilibrium polaron physics. Based on non-equilibrium quantum field theory, we derived a kinetic equation that consistently includes dissipation and drive, and it allows for studying relaxation of polarons and their non-equilibrium distributions. In particular, in the context of two-dimensional materials we found that due to the bosonic nature of polarons, around the point where effective drive overcomes the radiative and non-radiative loss, stimulated processes lead to a transition towards a lasing regime, where many-body polaronic effects reduce the emission linewidth below the bare linewidth.