Quantum Walk with Jumps

TL;DR

This paper investigates how different types of jumps in 1-D quantum walks, caused by imperfect multi-ports, affect the walker's probability distribution, revealing distinct behaviors for static and dynamic disorder.

Contribution

It introduces a model of quantum walks with jumps due to multi-port errors, analyzing static and dynamic disorder effects on probability distributions and variance behavior.

Findings

Dynamic disorder results in Gaussian-like distributions.

Static disorder shows parity-dependent behaviors: three-peak or Laplace-like distributions.

Universal variance behavior emerges when scaled by jump size.

Abstract

We analyze a special class of 1-D quantum walks (QWs) realized using optical multi-ports. We assume non-perfect multi-ports showing errors in the connectivity, i.e. with a small probability the multi- ports can connect not to their nearest neighbor but to another multi-port at a fixed distance - we call this a jump. We study two cases of QW with jumps where multiple displacements can emerge at one timestep. The first case assumes time-correlated jumps (static disorder). In the second case, we choose the positions of jumps randomly in time (dynamic disorder). The probability distributions of position of the QW walker in both instances differ significantly: dynamic disorder leads to a Gaussian-like distribution, while for static disorder we find two distinct behaviors depending on the parity of jump size. In the case of even-sized jumps, the distribution exhibits a three-peak profile around the position of the initial excitation, whereas the probability distribution in the odd case follows a Laplace-like discrete distribution modulated by additional (exponential) peaks for long times. Finally, our numerical results indicate that by an appropriate mapping an universal functional behavior of the variance of the long-time probability distribution can be revealed with respect to the scaled average of jump size.