Electric quantum walks in two dimensions

TL;DR

This paper investigates electric quantum walks in two dimensions, analyzing how different coin operators and field directions influence trapping phenomena and dispersion relations.

Contribution

It provides a comparative analysis of Grover, Alternate, Hadamard, and DFT quantum walks under electric fields, revealing conditions for perfect 2D trapping and the role of conical intersections.

Findings

Grover walk exhibits directional trapping under electric fields along axes.

Removing conical intersections enables perfect 2D trapping.

Hadamard walk can also show perfect 2D trapping.

Abstract

We study electric quantum walks in two dimensions considering Grover, Alternate, Hadamard, and DFT quantum walks. In the Grover walk the behaviour under an electric field is easy to summarize: when the field direction coincides with the x or y axes, it produces a transient trapping of the probability distribution along the direction of the field, while when it is directed along the diagonals, a perfect 2D trapping is frustrated. The analysis of the alternate walk helps to understand the behaviour of the Grover walk as both walks are partially equivalent; in particular, it helps to understand the role played by the existence of conical intersections in the dispersion relations, as we show that when these are removed a perfect 2D trapping can occur for suitable directions of the field. We complete our study with the electric DFT and Hadamard walks in 2D, showing that the latter can exhibit perfect 2D trapping.