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We study properties of a periodically driven system coupled to a thermalbath. As a nontrivial example, we consider periodically driven metallic systemcoupled to a superconducting bath. The effect of the superconductor on thedriven system is two-fold: it (a) modifies density of states in the metal viathe proximity effect and (b) acts as a thermal bath for light-excitedquasi-particles. Using Keldysh formalism, we calculate, nonpertubatively in thesystem-bath coupling, the steady-state properties of the system and obtainnon-equilibrium distribution function. The latter allows one to calculateobservable quantities which can be spectroscopically measured in tunnelingexperiments.

Keldysh approach to periodically driven systems with a fermionic bath: non-equilibrium steady state, proximity effect and dissipation