Universal dynamical scaling laws in three-state quantum walks

TL;DR

This paper investigates the scaling laws governing the transition from trapping to spreading in a three-state quantum walk, revealing universal behaviors and clarifying previous observations of sublinear growth.

Contribution

It introduces a finite-time scaling analysis of the detrapping transition in three-state quantum walks, identifying universal scaling laws and clarifying the growth behavior of the participation ratio.

Findings

Participation ratio grows linearly with time with a logarithmic correction.

Survival probability and participation ratio follow single parameter scaling near the detrapping point.

The study clarifies previous reports of sublinear growth in quantum walk spreading.

Abstract

We perform a finite-time scaling analysis over the detrapping point of a three-state quantum walk on the line. The coin operator is parameterized by $\rho$ that controls the wavepacket spreading velocity. The input state prepared at the origin is set as symmetric linear combination of two eigenstates of the coin operator with a characteristic mixing angle $\theta$, one of them being the component that results in full spreading occurring at $\theta_c(\rho)$ for which no fraction of the wavepacket remains trapped near the initial position. We show that relevant quantities such as the survival probability and the participation ratio assume single parameter scaling forms at the vicinity of the detrapping angle $\theta_c$. In particular, we show that the participation ratio grows linearly in time with a logarithmic correction, thus shedding light on previous reports of sublinear behavior.