Floquet theory is an elegant mathematical formalism originally developed to solve time-dependent differential equations. Besides other fields, it has found applications in optical spectroscopy and nuclear magnetic resonance (NMR). This review attempts to give a perspective of the Floquet formalism as applied in NMR and shows how it allows one to solve various problems with a focus on solid-state NMR. We include both matrix- and operator-based approaches. We discuss different problems where the Hamiltonian changes with time in a periodic way. Such situations occur, for example, in solid-state NMR experiments where the time dependence of the Hamiltonian originates either from magic-angle spinning or from the application of amplitude- or phase-modulated radiofrequency fields, or from both. Specific cases include multiple-quantum and multiple-frequency excitation schemes. In all these cases, Floquet analysis allows one to define an effective Hamiltonian and, moreover, to treat cases that cannot be described by the more popularly used and simpler-looking average Hamiltonian theory based on the Magnus expansion. An important example is given by spin dynamics originating from multiple-quantum phenomena (level crossings). We show that the Floquet formalism is a very general approach for solving diverse problems in spectroscopy.

Floquet 理论是一种优雅的数学形式，最初是为求解时间相关的微分方程而开发的。除其他领域外，它还在光谱学和核磁共振（NMR）中得到应用。这篇评论试图给出一个在 NMR 中应用的 Floquet 形式主义的观点，并展示它如何允许人们解决各种问题，重点是固态 NMR。我们包括基于矩阵和基于运算符的方法。我们讨论了哈密顿量随时间周期性变化的不同问题。例如，在固态 NMR 实验中会出现这种情况，其中哈密顿量的时间依赖性源自魔角旋转或振幅或相位调制射频场的应用，或两者兼而有之。具体情况包括多量子和多频率激励方案。在所有这些情况下，Floquet 分析允许人们定义一个有效的哈密顿量，此外，还可以处理基于马格努斯展开的更普遍使用和看起来更简单的平均哈密顿量理论无法描述的情况。源自多量子现象（水平交叉）的自旋动力学给出了一个重要的例子。我们表明，Floquet 形式主义是解决光谱学中各种问题的一种非常通用的方法。源自多量子现象（水平交叉）的自旋动力学给出了一个重要的例子。我们表明，Floquet 形式主义是解决光谱学中各种问题的一种非常通用的方法。源自多量子现象（水平交叉）的自旋动力学给出了一个重要的例子。我们表明，Floquet 形式主义是解决光谱学中各种问题的一种非常通用的方法。