Intrinsic dissipative Floquet superconductors beyond mean-field theory

We study the intrinsic superconductivity in a dissipative Floquet electronic system in the presence of attractive interactions. Based on the functional Keldysh theory beyond the mean-field treatment, we find that the system shows a time-periodic bosonic condensation and reaches an intrinsic dissipative Floquet superconducting (SC) phase. Due to the interplay between dissipations and periodic modulations, the Floquet SC gap becomes ``soft'' and contains the diffusive fermionic modes with finite lifetimes. However, bosonic modes of the bosonic condensation are still propagating even in the presence of dissipations.