| 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. |
