Universal scattering with general dispersion relations

• URL: http://arxiv.org/abs/2103.09830v2
• Date: Tue, 19 Oct 2021 01:29:46 GMT
• Title: Universal scattering with general dispersion relations
• Authors: Yidan Wang, Michael J. Gullans, Xuesen Na, Seth Whitsitt and Alexey V. Gorshkov
• Abstract summary: We show that when there are no bright zero-energy eigenstates, the $S$-matrix evaluated at an energy $Eto 0$ converges to a universal limit. We extend these results to general integer dimensions $D geq 1$, dispersion relations $epsilon(boldsymbolk) = |boldsymbolk|a$ for a $D$-dimensional momentum vector.
• Score: 0.15749416770494704
• License: http://creativecommons.org/publicdomain/zero/1.0/
• 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.

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