  The Born-Oppenheimer (BO) approximation has shaped our understanding on
molecular dynamics microscopically in many physical and chemical systems.
However, there are many cases that we must go beyond the BO approximation,
particularly when strong light-matter interactions are considered. Floquet
theory offers a powerful tool to treat time-periodic quantum systems. In this
overview, we briefly review recent developments on Floquet nonadiabatic
dynamics, with a special focus on open quantum systems. We first present the
general Floquet Liouville von-Neumann (LvN) equation. We then show how to
connect Floquet operators to real time observables. We proceed to outline the
derivation of the Floquet quantum master equation in treating the dynamics
under periodic driving in open quantum systems. We further present the Floquet
mixed quantum classical Liouville equation (QCLE) to deal with coupled
electron-nuclear dynamics. Finally, we embed FQCLE into a classical master
equation (CME) to deal with Floquet nonadiabatic dynamics in open quantum
systems. The formulations are general platforms for developing trajectory based
dynamical approaches. As an example, we show how Floquet QCLE and Floquet CME
can be implemented into a Langevin dynamics with Lorentz force and surface
hopping algorithms.
