Floquet topological phases coupled to environments

We consider the fate of a helical edge state of a spin Hall insulator and its topological transition in presence of a circularly polarized light when coupled to various forms of environments. A Lindblad type equation is developped to determine the fermion occupation of the Floquet bands. We find that non-secular terms, corresponding to 2-photon transitions lead to a mixing of the band occupations, hence the light induced photocurrent is in general not perfectly quantized, although deviations are small in the adiabatic limit. Sharp crossovers are identified at frequencies $\Omega$ and 1/2 $\Omega$ ($\Omega$ is the strength of light-matter coupling) with the former corresponding to a phase transition on the level of the 2nd order theory.