  For a periodically driven open quantum system, the Floquet theorem states
that the time evolution operator $\Lambda(t,0)$ of the system can be factorized
as $\Lambda(t,0)=\mathcal{D}(t)e^{\mathcal{L}_{eff}t}$ 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.
