Title | Universal scattering with general dispersion relations |

Publication Type | Journal Article |

Year of Publication | 2022 |

Authors | Wang, Y, Gullans, M, Na, X, Whitsitt, S, Gorshkov, AV |

Journal | Phys. Rev. Research |

Volume | 4 |

Start Page | 023014 |

Date Published | 4/6/2022 |

Keywords | FOS: Physical sciences, Mathematical Physics (math-ph), Quantum Physics (quant-ph) |

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≥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 ϵ(k)=±|d|km, where m≥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→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≥1, dispersion relations ϵ(k)=|k|a for a D-dimensional momentum vector k with any real positive a, and separable potential scattering. |

URL | https://arxiv.org/abs/2103.09830 |

DOI | 10.1103/PhysRevResearch.4.023014 |