Quantum walks in external gauge fields

authored by: C. Cedzich, T. Geib, A. H. Werner, R. F. Werner
Abstract: Describing a particle in an external electromagnetic field is a basic task of quantum mechanics. The standard scheme for this is known as "minimal coupling" and consists of replacing the momentum operators in the Hamiltonian by the modified ones with an added vector potential. In lattice systems, it is not so clear how to do this because there is no continuous translation symmetry, and hence, there are no momenta. Moreover, when time is also discrete, as in quantum walk systems, there is no Hamiltonian, but only a unitary step operator. We present a unified framework of gauge theory for such discrete systems, keeping a close analogy to the continuum case. In particular, we show how to implement minimal coupling in a way that automatically guarantees unitary dynamics. The scheme works in any lattice dimension, for any number of internal degrees of freedom, for walks that allow jumps to a finite neighbourhood rather than to nearest neighbours, is naturally gauge invariant, and prepares possible extensions to non-abelian gauge groups.
Organisation(s): Faculty of Mathematics and Physics
Institute of Theoretical Physics
CRC 1227 Designed Quantum States of Matter (DQ-mat)
External Organisation(s): University of Cologne
University of Copenhagen
Type: Article
Journal: Journal of Mathematical Physics
Volume: 60
ISSN: 0022-2488
Publication date: 01.2019
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Statistical and Nonlinear Physics, Mathematical Physics
Electronic version(s): https://doi.org/10.48550/arXiv.1808.10850 (Access: Open)
https://doi.org/10.1063/1.5054894 (Access: Closed)

BibTeX

@article{4a4598a899d248609cbffa7f0e54e7a7,
title = "Quantum walks in external gauge fields",
abstract = "Describing a particle in an external electromagnetic field is a basic task of quantum mechanics. The standard scheme for this is known as {"}minimal coupling{"} and consists of replacing the momentum operators in the Hamiltonian by the modified ones with an added vector potential. In lattice systems, it is not so clear how to do this because there is no continuous translation symmetry, and hence, there are no momenta. Moreover, when time is also discrete, as in quantum walk systems, there is no Hamiltonian, but only a unitary step operator. We present a unified framework of gauge theory for such discrete systems, keeping a close analogy to the continuum case. In particular, we show how to implement minimal coupling in a way that automatically guarantees unitary dynamics. The scheme works in any lattice dimension, for any number of internal degrees of freedom, for walks that allow jumps to a finite neighbourhood rather than to nearest neighbours, is naturally gauge invariant, and prepares possible extensions to non-abelian gauge groups.",
author = "C. Cedzich and T. Geib and Werner, {A. H.} and Werner, {R. F.}",
note = "Funding information: A. H. Werner thanks the Humboldt Foundation for its support with a Feodor Lynen Fellowship and the VILLUM FONDEN via the QMATH Centre of Excellence (Grant No. 10059). C. Cedzich acknowledges support by the Excellence Initiative of the German Federal and State Governments (ZUK 81) and the DFG (Project No. B01 of CRC 183). T. Geib and R. F. Werner acknowledge support from the ERC grant DQSIM, the DFG SFB 1227 DQmat, and the European project SIQS.",
year = "2019",
month = jan,
doi = "10.48550/arXiv.1808.10850",
language = "English",
volume = "60",
journal = "Journal of Mathematical Physics",
issn = "0022-2488",
publisher = "American Institute of Physics",
number = "1",
}