  We study long-time asymptotic states of periodically driven quantum systems
coupled to a thermal bath. In order to describe a class of such a system, we
introduce the Floquet-Gibbs state, i.e. the state whose density matrix is
diagonalized in the basis of the Floquet state of the system Hamiltonian, and
its diagonal element obeys the Boltzmann distribution over its Floquet
quasienergy. We obtain sufficient conditions for the realization of the
Floquet-Gibbs state in a system with infinitesimal system-bath coupling [T.
Shirai, et al., Phys. Rev. E 91, 030101 (2015)]. These conditions severely
restrict a class of suitable physical models attaining the Floquet-Gibbs state.
We also show that some of the conditions can be lifted by imposing conditions
on timescales of the thermal bath with the aid of the truncated Floquet
Hamiltonian in the Floquet-Magnus expansion [T. Shirai, et al., New Journal of
Physics 18, 053008 (2016)]. In this paper we give a overview of this theory and
reconsider it by looking at the dynamics from a rotating frame.
