Discrete-time quantum walks in synthetic dimensions

TL;DR

This paper introduces a framework for discrete-time quantum walks in Fock-state lattices within synthetic dimensions, using Lie algebraic methods to explore novel quantum dynamics and interference effects.

Contribution

It develops a general formalism based on Lie algebras for implementing quantum walks in synthetic Fock-state lattices, revealing new dynamical behaviors and interference phenomena.

Findings

Demonstrated ballistic spreading in Fock-state lattices

Observed coin-walker entanglement and interference patterns

Identified algebraic structures leading to super-ballistic spreading and localization

Abstract

In this work we introduce discrete-time quantum walks in state space, more precisely on Fock-state lattices. Fock-state lattices provide a natural and clean setting for implementing lattice models, particularly in quantum optical systems. Thus, contrary to the common setting where the walker resides in real space or phase space, here the walk takes place in a synthetic space. We present a general formalism based on Lie algebras and their properties. For each Lie algebra one can associate both a phase space and a Fock-state lattice, and by understanding how these spaces are related, together with the action of generalized displacement operators, we construct the discrete unitary operator that generates the walk. In this framework the displacement operators replace the usual nearest-neighbor shifts and lead to state-dependent tunneling on the lattice. By considering several examples we demonstrate ballistic spreading and other characteristic features of discrete-time quantum walks, such as coin-walker entanglement and symmetry-induced interference patterns. We also show that different algebraic structures can give rise to qualitatively different dynamics, including anomalous behavior such as super-ballistic spreading as well as localization effects.