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

We study properties of a periodically driven system coupled to a thermal bath. As a nontrivial example, we consider a periodically driven metallic system coupled to a superconducting bath. The effect of the superconductor on the driven system is twofold: it (a) modifies density of states in the metal via the proximity effect and (b) acts as a thermal bath for light-excited quasiparticles. Using Keldysh formalism, we calculate, nonperturbatively in the system-bath coupling, the steady-state properties of the system and obtain nonequilibrium distribution function. The latter allows one to calculate observable quantities which can be spectroscopically measured in tunneling experiments.