Fractional recurrence in discrete-time quantum walk

TL;DR

This paper investigates fractional recurrence phenomena in discrete-time quantum walks, revealing that interference effects prevent complete recurrence but allow fractional revival characterized by the quantum Pólya number.

Contribution

It introduces the concept of fractional recurrence in quantum walks and demonstrates its occurrence through the quantum Pólya number, expanding understanding of quantum recurrence phenomena.

Findings

Fractional recurrence occurs due to wave packet revival.

Complete quantum recurrence does not hold in quantum walks.

Fractional revival can be characterized using the quantum Pólya number.

Abstract

Quantum recurrence theorem holds for quantum systems with discrete energy eigenvalues and fails to hold in general for systems with continuous energy. We show that during quantum walk process dominated by interference of amplitude corresponding to different paths fail to satisfy the complete quantum recurrence theorem. Due to the revival of the fractional wave packet, a fractional recurrence characterized using quantum P\'olya number can be seen.