  We study a dynamical phase transition in optical bistable systems subject to
a time-periodic driving field. The phase transition occurs in the structure of
limit cycle as a function of the frequency of the driving field. In the
thermodynamic limit, a single limit cycle is divided into two separated limit
cycles at the transition point. In finite-size systems, however, there is
always a single limit cycle due to the quantum tunneling effect. We use a
Floquet dissipative map, which is a time-evolution operator over one period in
a dynamics given by a quantum master equation, and discuss the decay rate of
relaxation dynamics into the limit cycle based on the dominant eigenvalue of
the map. We found that the decay rate exhibits qualitatively different
system-size dependence before and after the phase transition, and it shows a
finite-size scaling of spinodal phenomena around the transition point. The
present work provides a systematic way of studying dynamical phase transition
observed in time-periodically driven open systems in terms of the Floquet
dissipative map.
