Scattering Theory for Floquet-Bloch States

TL;DR

This paper develops a formalism for understanding scattering processes of particles in time-periodic Hamiltonians using Floquet theory, with implications for cold-atom experiments involving artificial gauge fields.

Contribution

It introduces a general scattering theory for Floquet-Bloch states, highlighting the role of quasi-energy conservation and analyzing elastic and inelastic scattering in driven systems.

Findings

Scattering cannot generally be described by an effective time-independent Hamiltonian.

Inelastic scattering can cause heating but can be suppressed by additional confinement.

The formalism applies to single-particle and two-particle scattering in Floquet systems.

Abstract

Motivated by recent experimental implementations of artificial gauge fields for gases of cold atoms, we study the scattering properties of particles that are subjected to time-periodic Hamiltonians. Making use of Floquet theory, we focus on translationally invariant situations in which the single-particle dynamics can be described in terms of spatially extended Floquet-Bloch waves. We develop a general formalism for the scattering of these Floquet-Bloch waves. An important role is played by the conservation of Floquet quasi-energy, which is defined only up to the addition of integer multiples of $\hbar\omega$ for a Hamiltonian with period $T=2\pi/\omega$. We discuss the consequences of this for the interpretation of "elastic" and "inelastic" scattering in cases of physical interest. We illustrate our general results with applications to: the scattering of a single particle in a Floquet-Bloch state from a static potential; and, the scattering of two particles in Floquet-Bloch states through their interparticle interaction. We analyse examples of these scattering processes that are closely related to the schemes used to general artifical gauge fields in cold-atom experiments, through optical dressing of internal states, or through time-periodic modulations of tight-binding lattices. We show that the effects of scattering cannot, in general, be understood by an effective time-independent Hamiltonian, even in the limit $\omega \to \infty$ of rapid modulation. We discuss the relative sizes of the elastic scattering (required to stablize many-body phases) and of the inelastic scattering (leading to deleterious heating effects). In particular, we describe how inelastic processes that can cause significant heating in current experimental set-up can be switched off by additional confinement of transverse motion.