Tailoring discrete quantum walk dynamics via extended initial conditions

Germán J De Valcárcel,Eugenio Roldán,Alejandro Romanelli

doi:10.1088/1367-2630/12/12/123022

Germán J De Valcárcel, Eugenio Roldán + Show 1 more

Open Access

https://doi.org/10.1088/1367-2630/12/12/123022

Copy DOI

Abstract

We study the evolution of initially extended distributions in the coined quantum walk (QW) on the line. By analysing the dispersion relation of the process, continuous wave equations are derived whose form depends on the initial distribution shape. In particular, for a class of initial conditions, the evolution is dictated by the Schrödinger equation of a free particle. As that equation also governs paraxial optical diffraction, all of the phenomenology of the latter can be implemented in the QW. This allows us, in particular, to devise an initially extended condition leading to a uniform probability distribution whose width increases linearly with time, with increasing homogeneity.

Highlights

Introduction.– The discrete, or coined, quantum walk (QW) [1] is a process originally introduced as the quantum counterpart of the classical random walk (RW)
In [6] extended distributions, with top-hat profile, were considered in the context of the superfluid-Mott insulator transition in optical lattices, but no general conclusions were drawn on the influence of these extended initial conditions on the long time state
The situation can be improved by multiplying the initial condition by a Gaussian of convenient width, as shown in Figure 2(c), which is another main result of this Letter: Almost uniform distributions can be obtained in the QW by making a judicious choice of the initial condition

Summary

Introduction

Introduction.– The discrete, or coined, quantum walk (QW) [1] is a process originally introduced as the quantum counterpart of the classical random walk (RW). QW group velocity has been used for determining hitting times [20], and it will allow us to make simple but relevant predictions about the QW dynamics when the initial state is a wavepacket close to some of the eigensolutions above, say |Ψx,0 = fx(s)eik0x Φ(ks0) with fx(s) a smooth envelope.

Results

Conclusion