Multiple transitions between normal and hyperballistic diffusion in quantum walks with time-dependent jumps

Abstract

We extend to the gamut of functional forms of the probability distribution of the time-dependent step-length a previous model dubbed Elephant Quantum Walk, which considers a uniform distribution and yields hyperballistic dynamics where the variance grows cubicly with time, σ2 ∝ t3, and a Gaussian for the position of the walker. We investigate this proposal both locally and globally with the results showing that the time-dependent interplay between interference, memory and long-range hopping leads to multiple transitions between dynamical regimes, namely ballistic → diffusive → superdiffusive → ballistic → hyperballistic for non-hermitian coin whereas the first diffusive regime is quelled for implementations using the Hadamard coin. In addition, we observe a robust asymptotic approach to maximal coin-space entanglement.

Enhancing entanglement with the generalized elephant quantum walk from localized and delocalized states

“…This counterintuitive result was demonstrated mathematically in 2013 [26] and experimentally in 2018 [22]. Further works have provided an analysis of the entanglement considering several protocols of disorder embedded either in the coin operator [27][28][29][30][31][32][33][34][35][36][37][38][39][40][41] or in the step operator [42][43][44]. With some exceptions, the overwhelming majority of these works focused on QW dynamics from local initial states.…”

“…Even though QWs with temporal disorder in the coin operator have been studied for a much longer time [4][5][6][7][8][9], the QWs with temporal disorder in the displacement operator exhibits an interesting phenomenology [42][43][44][46][47][48][49][50][51][52][53][54][55][56]. In some circumstances, both types of models share some similarities, for instance, the achievement of maximal asymptotic entanglement from local states [25,26,42,43]. However, up to now, only QWs with dynamic disorder in the displacement operator are able to produce hyperballistic dynamics [51].…”

“…Such impasse can be avoided when the temporal disorder is present in the shift operator. Specifically, recently [42] it was presented the first protocol -the generalized elephant quantum walk (gEQW) -that is able to exhibit both amplification of entanglement and tunable spreading from slower-thanballistic to faster-than ballistic. However, the authors have only considered some specific and local initial states.…”

“…The quantum walk model we consider consists of a noisy unitary evolution of a DTQW on a lattice where the steps sizes are randomly chosen, a model devised in Ref. [42], called the generalized elephant quantum walk (gEQW). This model is an extension of the elephant quantum walk [51], a quantum walk based on the classical non-Markovian elephant walk where the walker remember its previous steps, giving a variety of diffusion characteristics [59].…”

“…In order to compare the degree of dispersion of a quantum walker it was analyzed the asymptotic limit of the variance of the position [42]. It is expected that its limiting behavior obeys…”