Eur. Phys. J. D 64, 119-129 (2011)
https://doi.org/10.1140/epjd/e2011-20138-8

Regular Article

Quantum walk with jumps

¹ Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical Engineering, Department of Physics, Břehová 7, 115 19 Praha 1, Czech Republic
² Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical Engineering, Doppler Institute for Mathematical Physics and Applied Mathematics, Břehová 7, 115 19 Praha 1, Czech Republic
³ Research Institute for Solid State Physics and Optics, Hungarian Academy of Sciences, 1525 Budapest, P.O. Box 49, Hungary
⁴ Department of Physics, University of Augsburg, 86135 Augsburg, Germany

^a e-mail: hynek.lavicka@fjfi.cvut.cz

Received: 28 February 2011
Received in final form: 2 May 2011
Published online: 27 July 2011

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 a universal functional behavior of the variance of the long-time probability distribution can be revealed with respect to the scaled average of jump size.