Quantum walks in external gauge fields

TL;DR

This paper develops a unified gauge theory framework for quantum walks in discrete lattice systems, enabling the implementation of minimal coupling that preserves unitarity and gauge invariance across various dimensions and internal degrees of freedom.

Contribution

It introduces a novel, unified approach to gauge theory in quantum walks, extending minimal coupling to discrete, non-continuous systems with guarantees of unitarity and gauge invariance.

Findings

Framework works in any lattice dimension

Automatically guarantees unitary dynamics

Naturally gauge invariant and extendable to non-abelian groups

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 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, 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 degree of freedom, for walks that allow jumps to a finite neighborhood rather than to nearest neighbours, is naturally gauge invariant, and prepares possible extensions to non-abelian gauge groups.