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.

中文翻译：

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周期性驱动系统的通用高频行为：从动态稳定到 Floquet 工程

我们概述了周期性驱动系统中的高频状态，并确定了三类不同的驱动协议，其中无限频率 Floquet 哈密顿量不等于时间平均哈密顿量。这些类别涵盖系统，例如 Kapitza 摆、磁场中中性原子的 Harper-Hofstadter 模型、Haldane Floquet Chern 绝缘体等。在考虑的所有设置中，我们讨论了 Floquet 哈密顿量的无限频率限制和领先的有限频率校正。我们简要概述了 Floquet 理论，重点关注与频闪框架选择相关的规范结构以及频闪和非频闪动力学之间的差异。在后一种情况下，必须使用表示可观察量的修饰运算符和修饰的密度矩阵。我们还评论了 Floquet 理论在由具有良好分离的能量尺度的静态哈密顿量描述的系统中的应用，特别是讨论了逆频扩展和 Schrieffer-Wolff 变换之间的相似之处，将后者扩展到驱动系统。