Universal scattering with general dispersion relations

TL;DR

This paper demonstrates that in waveguide QED with general dispersion relations, the scattering matrix approaches a universal limit at zero energy, depending only on the dispersion power, and extends Levinson's theorem to these settings.

Contribution

It establishes universal scattering behavior at zero energy for systems with arbitrary dispersion relations and generalizes Levinson's theorem to higher dimensions and non-quadratic dispersions.

Findings

Scattering matrix converges to a universal limit at zero energy depending only on the dispersion power m.

Generalization of Levinson's theorem to waveguide QED with arbitrary dispersion relations.

Extension of results to higher dimensions and separable potential scattering.

Abstract

Many synthetic quantum systems allow particles to have dispersion relations that are neither linear nor quadratic functions. Here, we explore single-particle scattering in general spatial dimension $D\geq 1$ when the density of states diverges at a specific energy. To illustrate the underlying principles in an experimentally relevant setting, we focus on waveguide quantum electrodynamics (QED) problems (i.e. $D=1$) with dispersion relation $\epsilon(k)=\pm |d|k^m$, where $m\geq 2$ is an integer. For a large class of these problems for any positive integer $m$, we rigorously prove that when there are no bright zero-energy eigenstates, the $S$-matrix evaluated at an energy $E\to 0$ converges to a universal limit that is only dependent on $m$. We also give a generalization of a key index theorem in quantum scattering theory known as Levinson's theorem -- which relates the scattering phases to the number of bound states -- to waveguide QED scattering for these more general dispersion relations. We then extend these results to general integer dimensions $D \geq 1$, dispersion relations $\epsilon(\boldsymbol{k}) = |\boldsymbol{k}|^a$ for a $D$-dimensional momentum vector $\boldsymbol{k}$ with any real positive $a$, and separable potential scattering.