Periodically driven quantum dynamics has recently attracted much attention in experimental as well as theoretical studies as it offers a promising way for exploring novel quantum phenomena which would be difficult or impossible to observe otherwise. However, analyzing periodically driven systems is in general a hard problem. With a few exceptions, recent studies mostly focus on driving simple (often non-interacting particles) Hamiltonians, whereas such a condition can be satisfied only approximately in realistic experimental conditions. From recent studies, even small integrability-breaking terms are known to be relevant to the long-time dynamics and eventually lead to a state of infinite temperature (a random state) as a final steady state. In the experimental context, this final state is no longer intriguing since one cannot get any information reflected from the system.
The experiments on many-body quantum systems, however, do not focus on the long-time limit; rather, they are interested in the transient dynamics within experimental measurement time scale. So far, most studies on transient ynamical properties are based on numerical calculations with phenomenological arguments. In the present work, we establish a new paradigm that describes transient dynamics for generic quantum many-body systems. We show that as long as we consider a finite-time scale, the time-evolution of the system is approximately governed by a well-defined effective Hamiltonian based on the Floquet-Magnus (FM) theory. We give a prescription to obtain the effective Hamiltonian and reveal the time scale for which the time-evolution of the system is governed by this Hamiltonian.