Floquet theory and applications in open quantum and classical systems

This article reviews theoretical methods for analyzing Floquet engineering (FE) phenomena in open (dissipative) quantum or classical systems, with an emphasis on our recent results. In many theoretical studies for FE in quantum systems, researchers have used the Floquet theory for closed (isolated) quantum systems, that is based on the Schr\"odinger equation. However, if we consider the FE in materials driven by an oscillating field like a laser, a weak but finite interaction between a target system and an environment (bath) is inevitable. In this article, we describe these periodically driven dissipative systems by means of the quantum master (GKSL) equation. In particular, we show that a nonequilibrium steady state appears after a long driving due to the balance between the energy injection by the driving field and the release to the bath. In addition to quantum systems, if we try to simply apply Floquet theory to periodically driven classical systems, it failed because the equation of motion (EOM) is generally nonlinear, and the Floquet theorem can be applied only to linear differential equations. Instead, by considering the distribution function of the classical variables (i.e., Fokker-Planck equation), one can arrive at the effective EOM for the driven systems. We illustrate the essence of the Floquet theory for classical systems. On top of fundamentals of the Floquet theory, we review representative examples of FEs (Floquet topological insulators, inverse Faraday effects in metals and magnets, Kapitza pendulum, etc.) and dissipation-assisted FEs.