  We present the stochastic thermodynamics analysis of an open quantum system
weakly coupled to multiple reservoirs and driven by a rapidly oscillating
external field. The analysis is built on a modified stochastic master equation
in the Floquet basis. Transition rates are shown to satisfy the local detailed
balance involving the entropy flowing out of the reservoirs. The first and
second law of thermodynamics are also identified at the trajectory level.
Mechanical work is identified by means of initial and final projections on
energy eigenstates of the system. We explicitly show that this two step
measurement becomes unnecessary in the long time limit. A steady-state
fluctuation theorem for the currents and rate of mechanical work is also
established. This relation does not require the introduction of a time reversed
external driving which is usually needed when considering systems subjected to
time asymmetric external fields. This is understood as a consequence of the
secular approximation applied in consistency with the large time scale
separation between the fast driving oscillations and the slower relaxation
dynamics induced by the environment. Our results are finally illustrated on a
model describing a thermodynamic engine.
