Universal high-frequency behavior of periodically driven systems: from dynamical stabilization to Floquet engineering

TLDR

The paper reviews Floquet theory, highlighting gauge choices, stroboscopic versus non‑stroboscopic dynamics, and applications to systems such as the Kapitza pendulum, Harper–Hofstadter model, and Haldane Floquet Chern insulator, and draws parallels to the Schrieffer–Wolff transformation. The authors aim to classify high‑frequency driving protocols where the infinite‑frequency Floquet Hamiltonian differs from the time‑averaged Hamiltonian. They analyze the infinite‑frequency limit and leading finite‑frequency corrections, discuss dressed operators and density matrices, and relate the inverse‑frequency expansion to the Schrieffer–Wolff transformation in driven systems.

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.