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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.