Universal High-Frequency Behavior of Periodically Driven Systems: from Dynamical Stabilization to Floquet Engineering

TL;DR

This paper reviews the high-frequency behavior of periodically driven quantum systems, classifying different driving protocols, analyzing Floquet Hamiltonians, and discussing theoretical frameworks including gauge structures and finite-frequency corrections.

Contribution

It provides a comprehensive classification of high-frequency regimes and extends Floquet theory with a focus on gauge structures and finite-frequency effects in driven systems.

Findings

Identification of three classes of driving protocols with non-trivial Floquet Hamiltonians

Analysis of finite-frequency corrections to the infinite-frequency limit

Comparison of Floquet theory with Schrieffer-Wolff transformation

Abstract

We give a general overview of the high-frequency regime in periodically driven systems and identify three distinct classes of driving protocols in which the infinite-frequency Floquet Hamiltonian is not equal to the time-averaged Hamiltonian. These classes cover systems, such as the Kapitza pendulum, the Harper-Hofstadter model of neutral atoms in a magnetic field, the Haldane Floquet Chern insulator and others. In all setups considered, we discuss both the infinite-frequency limit and the leading finite-frequency corrections to the Floquet Hamiltonian. We provide a short overview of Floquet theory focusing on the gauge structure associated with the choice of stroboscopic frame and the differences between stroboscopic and non-stroboscopic dynamics. In the latter case one has to work with dressed operators representing observables and a dressed density matrix. We also comment on the application of Floquet Theory to systems described by static Hamiltonians with well-separated energy scales and, in particular, discuss parallels between the inverse-frequency expansion and the Schrieffer-Wolff transformation extending the latter to driven systems.