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We consider the fate of a helical edge state of a spin Hall insulator and itstopological transition in presence of a circularly polarized light when coupledto various forms of environments. A Lindblad type equation is developed todetermine the fermion occupation of the Floquet bands. We find by usinganalytical and numerical methods that non-secular terms, corresponding to2-photon transitions, lead to a mixing of the band occupations, hence the lightinduced photocurrent is in general not perfectly quantized in the presence offinite coupling to the environment, although deviations are small in theadiabatic limit. Sharp crossovers are identified at frequencies $\Omega$ and$\frac{1}{2}\Omega$ ($\Omega$ is the strength of light-matter coupling) withthe former resembling to a phase transition.

Floquet topological phases coupled to environments and the induced photocurrent