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Microwave response of a superconductor beyond the Eliashberg theory

We review recent progress in the theory of electromagnetic response of dirty superconductors subject to microwave radiation. The theory originally developed by Eliashberg in 1970 and soon after that elaborated in a number of publications addressed the effect of superconductivity enhancement in the vicinity of the transition temperature. This effect originates from nonequilibrium redistribution of quasiparticles and requires a minimal microwave frequency depending on the inelastic relaxation rate and temperature. In a recent series of papers we generalized the Eliashberg theory to arbitrary temperatures $T$, microwave frequencies $ω$, dc supercurrent, and inelastic relaxation rates, assuming that the microwave power is weak enough and can be treated perturbatively. In the phase diagram ($ω,T$) the region of superconductivity enhancement occupies a finite area located near $T_c$. At sufficiently high frequencies and low temperatures, the effect of direct depairing prevails over quasiparticle redistribution, always leading to superconductivity suppression.