Quantum Intermittency for Sparse CMV Matrices with an Application to Quantum Walks on the Half-Line

TL;DR

This paper analyzes quantum walks on the half-line with sparse barriers, demonstrating quantum intermittency and explicitly calculating transport exponents and spectral measure dimensions.

Contribution

It provides the first rigorous example of quantum intermittency in quantum walks using CMV matrices with sparse barriers and computes related spectral properties.

Findings

Quantum intermittency observed in a specific quantum walk model.

Explicit formulas for transport exponents in terms of model parameters.

Exact Hausdorff dimension of the spectral measure calculated.

Abstract

We study the dynamics given by the iteration of a (half-line) CMV matrix with sparse, high barriers. Using an approach of Tcheremchantsev, we are able to explicitly compute the transport exponents for this model in terms of the given parameters. In light of the connection between CMV matrices and quantum walks on the half-line due to Cantero-Gr\"unbaum-Moral-Vel\'azquez, our result also allows us to compute transport exponents corresponding to a quantum walk which is sparsely populated with strong reflectors. To the best of our knowledge, this provides the first rigorous example of a quantum walk which exhibits quantum intermittency, i.e., nonconstancy of the transport exponents. When combined with the CMV version of the Jitomirskaya-Last theory of subordinacy and the general discrete-time dynamical bounds from Damanik-Fillman-Vance, we are able to exactly compute the Hausdorff dimension of the associated spectral measure.