The Floquet-Boltzmann equation

TL;DR

This paper introduces the Floquet-Boltzmann equation, a formalism combining Floquet theory and Boltzmann dynamics to describe quasiparticle behavior and heating in periodically driven quantum systems, with applications to cold-atom experiments.

Contribution

It develops a novel Floquet-Boltzmann formalism to analyze long-time dynamics and heating in driven quantum systems, bridging Floquet theory with semiclassical Boltzmann equations.

Findings

Derived the Floquet-Boltzmann equation for quasiparticle dynamics.

Applied the formalism to calculate heating rates in cold-atom systems.

Demonstrated the approach with the Haldane model under periodic shaking.

Abstract

Periodically driven quantum systems can be used to realize quantum pumps, ratchets, artificial gauge fields and novel topological states of matter. Starting from the Keldysh approach, we develop a formalism, the Floquet-Boltzmann equation, to describe the dynamics and the scattering of quasiparticles in such systems. The theory builds on a separation of time-scales. Rapid, periodic oscillations occurring on a time scale $T_0=2 \pi/\Omega$, are treated using the Floquet formalism and quasiparticles are defined as eigenstates of a non-interacting Floquet Hamiltonian. The dynamics on much longer time scales, however, is modelled by a Boltzmann equation which describes the semiclassical dynamics of the Floquet-quasiparticles and their scattering processes. As the energy is conserved only modulo $\hbar \Omega$, the interacting system heats up in the long-time limit. As a first application of this approach, we compute the heating rate for a cold-atom system, where a periodical shaking of the lattice was used to realize the Haldane model.