Universal behavior of quantum walks with long-range steps
Oliver Mülken, Volker Pernice, A. Blumen · Physical Review E · 2008
Continuous-time quantum walks with long-range steps ${R}^{\ensuremath{-}\ensuremath{\gamma}}$ ($R$ being the distance between sites) on a discrete line behave in similar ways for all $\ensuremath{\gamma}\ensuremath{\ge}2$. This is in contrast to classical random walks, which for $\ensuremath{\gamma}>3$ belong to a different universality class than for $\ensuremath{\gamma}\ensuremath{\le}3$. We show that the average probabilities to be at the initial site after time $t$ as well as the mean square displacements are of the same functional form for quantum walks with $\ensuremath{\gamma}=2$, $4$, and with nearest neighbor steps. We interpolate this result to arbitrary $\ensuremath{\gamma}\ensuremath{\ge}2$.