2021/03/17 by Yidan Wang, Wang, Yidan, Michael J. Gullans +7 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Quantum Information and Cryptography #Quantum and electron transport phenomena #Quantum optics and atomic interactions #math-ph #math.MP #quant-ph
paper · pdf · doi:10.48550/arxiv.2103.09830
arxiv created 2021/10/19 · arxiv updated 2021/10/20
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(\boldsymbolk) = |\boldsymbolk|a for a D-dimensional momentum vector \boldsymbolk with any real positive a, and separable potential scattering.