Dynamical Localization of Quantum Walks in Random Environments

TL;DR

This paper investigates how randomness in the coin operator of a one-dimensional quantum walk causes localization or diffusion, demonstrating Anderson localization in spatially random cases and calculating diffusion constants in temporally random cases.

Contribution

It proves dynamical Anderson localization for quantum walks with spatially random coin operators and derives diffusion constants for temporally random coin operators.

Findings

Anderson localization occurs with spatial randomness in the coin operator.

Diffusive behavior with computed diffusion constants occurs with temporal randomness.

The paper provides a rigorous mathematical framework for these phenomena.

Abstract

The dynamics of a one dimensional quantum walker on the lattice with two internal degrees of freedom, the coin states, is considered. The discrete time unitary dynamics is determined by the repeated action of a coin operator in U(2) on the internal degrees of freedom followed by a one step shift to the right or left, conditioned on the state of the coin. For a fixed coin operator, the dynamics is known to be ballistic. We prove that when the coin operator depends on the position of the walker and is given by a certain i.i.d. random process, the phenomenon of Anderson localization takes place in its dynamical form. When the coin operator depends on the time variable only and is determined by an i.i.d. random process, the averaged motion is known to be diffusive and we compute the diffusion constants for all moments of the position.