Superconductivity in the presence of microwaves: Full phase diagram

We address the problem of non-equilibrium superconductivity in the presence of microwave irradiation. We refine the old Eliashberg theory and generalize it to arbitrary temperatures $T$ and frequencies $\omega$. Microwave radiation is shown to stimulate superconductivity in a bounded region in the $(\omega,T)$ plane. In particular, for $T<0.47\, T_c$ and for $\hbar\omega>3.3\, k_BT_c$ superconductivity is always suppressed by a weak \emph{ac} driving. We also study the supercurrent in the presence of microwave irradiation and establish the criterion for the critical current enhancement. Our results can be qualitatively interpreted in terms of the interplay between the kinetic ("stimulation" vs. "heating") and spectral ("depairing") effects of the microwaves.