Generalized Floquet theory for open quantum systems

For a periodically driven open quantum system, the Floquet theorem states that the time evolution operator Λ(t,0) of the system can be factorized as Λ(t,0)=mathcal{D}(t)e^{mathcal{L}_efft} with micro-motion operator mathcal{D}(t) possessing the same period as the external driving, and time-independent operator mathcal{L}_eff. In this work, we extend this theorem to open systems that follow a modulated periodic evolution, in which the fast part is periodic while the slow part breaks the periodicity. We derive a factorization for the time evolution operator that separates the long time dynamics and the micro-motion for the open quantum system. High-frequency expansions for the effective evolution operator control the long time dynamics, and the micro-motion operator is also given and discussed. It may find applications in quantum engineering with open systems following modulated periodic evolution.