  We investigate the long-time behavior of quantum N-level systems that are
coupled to a Markovian environment and subject to periodic driving. As our main
result, we obtain a general algebraic condition ensuring that all solutions of
a periodic quantum master equation with Lindblad form approach a unique limit
cycle. Quite intuitively, this criterion requires that the dissipative terms of
the master equation connect all subspaces of the system Hilbert space during an
arbitrarily small fraction of the cycle time. Our results provide a natural
extension of Spohn's algebraic condition for the approach to equilibrium to
systems with external driving.
