  We theoretically investigate basic properties of nonequilibrium steady states
of periodically-driven open quantum systems based on the full solution of the
Maxwell-Bloch equation. In a resonantly driving condition, we find that the
transverse relaxation, also known as decoherence, significantly destructs the
formation of Floquet states while the longitudinal relaxation does not directly
affect it. Furthermore, by evaluating the quasienergy spectrum of the
nonequilibrium steady states, we demonstrate that the Rabi splitting can be
observed as long as the decoherence time is as short as one third of the
Rabi-cycle. Moreover, we find that Floquet states can be formed even under
significant dissipation when the decoherence time is substantially shorter than
the cycle of driving, once the driving field strength becomes strong enough. In
an off-resonant condition, we demonstrate that the Floquet states can be
realized even in weak field regimes because the system is not excited and the
decoherence mechanism is not activated. Once the field strength becomes strong
enough, the system can be excited by nonlinear processes and the decoherence
process becomes active. As a result, the Floquet states are significantly
disturbed by the environment even in the off-resonant condition. Thus, we show
here that the suppression of heating is a key condition for the realization of
Floquet states in both on and off-resonant conditions not only because it
prevents material damage but also because it contributes to preserving
coherence.
