Floquet theory and applications in open quantum and classical systems

TL;DR

This paper surveys Floquet engineering for driven open quantum and classical systems, foregrounding GKSL (Lindblad) dynamics and Fokker-Planck formalisms to describe nonequilibrium steady states. It develops and applies high-frequency expansions to derive effective Floquet Hamiltonians and Floquet-Lindbladians, clarifying when Floquet-Gibbs or canonical (CFSS) states emerge and how dissipation shapes FE. By presenting microscopic derivations (Floquet-GKSL), phenomenological approaches, and classical FP-based FE, the work offers a practical toolkit for predicting and controlling FE in realistic dissipative settings, including NV centers, Rashba metals, Kapitza pendulums, and LLG spin systems. The findings have implications for designing laser-driven materials and devices, enabling robust FE effects under finite-temperature and environment-coupled conditions. Overall, the framework bridges idealized closed-system FE with experimentally relevant open-system dynamics, facilitating targeted FE applications with dissipative support.

Abstract

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 Schrödinger 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.

Figures (16)

• Figure 1: Schematic images of isolated (closed) and dissipative (open) systems driven by laser. In the closed system, a long enough application of laser makes the system burned, while a nonequilibrium steady state (NESS) is expected to appear in the open (dissipative) system.
• Figure 2: Image of a time-periodic system (e.g., a laser-driven system) and the effective eigenvalue problem of Eq. \ref{['eq:GeneralEigen']} mapped from the original Schrödinger equation of Eq. \ref{['eq:Sch']}.
• Figure 3: Typical time evolution of a Floquet engineered quantity in closed quantum systems. We start the application of periodic external field at $t=0$. The physical quantity first grows up and approaches a constant value (Floquet pre-thermalization), whereas the engineered quantity finally disappears because the system approaches a featureless state after a long driving. FE is at least partially succeeded during the blue regime including the pre-thermalization (green) range as the engineered quantity is finite there.
• Figure 4: (a) Honeycomb lattice structure of graphene with a staggered potential. Green diamond including A and B sublattice sites represents the unit cell. Vectors ${\bm d}_{1,2,3}$ are the unit vectors connecting two nearest-neighboring sites, while ${\bm a}_{1,2,3}$ are the vectors connecting two second-nearest-neighboring sites. (b) Energy dispersion of the model \ref{['eq:Graphene']} with $\mu_{\rm s}=0$ around the half-filled level. The gray honeycomb regime corresponds to the Brillouin zone.
• Figure 5: Image of inverse Faraday effect. The photo-induced effective magnetic field $\bm B_{\rm IFE}$ changes its sign by changing the helicity of light from (a) left circularly polarization (CPL+) to (b) right circularly polarization (CPL-). If the system initially possesses a magnetization along the $x$ direction, a precession motion occurs due to $\bm B_{\rm IFE}$. Reproduced from Ref. Tanaka2024 with permission.