  A periodically driven quantum system, when coupled to a heat bath, relaxes to
a non-equilibrium asymptotic state. In the general situation, the retrieval of
this asymptotic state presents a rather non-trivial task. It was recently shown
that in the limit of an infinitesimal coupling, using so-called rotating wave
approximation (RWA), and under strict conditions imposed on the time-dependent
system Hamiltonian, the asymptotic state can attain the Gibbs form. A
Floquet-Gibbs state is characterized by a density matrix which is diagonal in
the Floquet basis of the system Hamiltonian with the diagonal elements obeying
a Gibbs distribution, being parametrized by the corresponding Floquet
quasi-energies. Addressing the non-adiabatic driving regime, upon using the
Magnus expansion, we employ the concept of a corresponding effective Floquet
Hamiltonian. In doing so we go beyond the conventionally used RWA and
demonstrate that the idea of Floquet-Gibbs states can be extended to the
realistic case of a weak, although finite system-bath coupling, herein termed
effective Floquet-Gibbs states.
