  Floquet engineering of isolated systems is often based on the concept of the
effective time-independent Floquet Hamiltonian, which describes the
stroboscopic evolution of a periodically driven quantum system in steps of the
driving period and which is routinely obtained analytically using
high-frequency expansions. The generalization of these concepts to open quantum
systems described by a Markovian master equation of Lindblad type turns out to
be non-trivial: On the one hand, already for a two-level system two different
phases can be distinguished, where the effective time-independent Floquet
generator (describing the stroboscopic evolution) is either again Markovian and
of Lindblad type or not. On the other hand, even though in the high-frequency
regime a Lindbladian Floquet generator (Floquet Linbladian) is numerically
found to exist, this behaviour is, curiously, not correctly reproduced within
analytical high-frequency expansions. Here, we demonstrate that a proper
Floquet Lindbladian can still be obtained from a high-frequency expansion, when
treating the problem in a suitably chosen rotating frame. Within this approach,
we can then also describe the transition to a phase at lower driving
frequencies, where no Floquet Lindbladian exists, and show that the emerging
non-Markovianity of the Floquet generator can entirely be attributed to the
micromotion of the open driven system.
