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In quantum mechanics, Floquet theory provides a framework to treat the time-dependent Schrödinger equation for $$T$$-periodic Hamiltonians $$\hat{H}(t) = \hat{H}(t + T)$$. Mathematically, it is founded on Floquet's theorem that applies to linear differential equations with periodic functions.

Introduction
Floquet theory applies to all time-periodic problems in quantum mechanics, such as the time-evolution of a spin-1/2 in an oscillating magnetic field (rotating at angular frequency $$\omega$$), the paradigmatic problem in nuclear magnetic resonance (NMR). It separates the time-evolution into a slow "effective" motion (the Rabi flopping in the NMR example) and a fast "micromotion" on the timescale $$2\pi/\omega$$. This separation of timescales allows a convenient description of the slow motion as if using a time-independent Hamiltonian, thus greatly simplifying the ensuing dynamics.

The slow time evolution governed by the effectively static Hamiltonian, often called "Floquet Hamiltonian" $$\hat{H}_{\text{F}}$$, is the basis for the field of Floquet engineering. In Floquet engineering, a fast periodic drive allows to realise novel effects that would be impossible to achieve in static systems, such as Synthetic gauge fields.

Floquet theory provides a formal ground for the rotating wave approximation.

Floquet's theorem
Floquet's theorem makes a statement on the unitary time-evolution operator
 * $$\hat{U}(t, t_0) = \mathcal{T} \exp\left[-\frac{i}{\hbar}\int_{t_0}^{t} \mathcal{H}(t') \text{d} t' \right]$$,

where $$\mathcal{T}$$ denotes time ordering.

Given a time-periodic Hamiltonian $$\hat{H}(t) = \hat{H}(t + T)$$, Floquet's theorem can be stated as


 * $$\hat{U}(t,t_0) = e^{-i \hat{K}_{\text{F}}[t_0](\tau)} e^{-i\hat{H}_{\text{F}}[t_0] \times (t - t_0)/\hbar}$$

with
 * $$\hat{K}_{\text{F}}[t_0](t + T) = \hat{K}_{\text{F}}[t_0](t)$$.

This implies that the time-evolution $$\hat{U}(t,t_0)$$ from a starting time $$t_0$$ to some final time $$t$$ separates into two parts: a slow evolution with the time-independent Floquet Hamiltonian $$\hat{H}_{\text{F}}[t_0]$$ (which depends on the starting time $$t_0$$), and a fast micromotion governed by the "stroboscopic kick operator" $$\hat{K}_{\text{F}}[t_0](t)$$, which is periodic in time.