We analyze a set of discrete-time quantum walks for which the displacements on a chain follow binary aperiodic jumps according to three paradigmatic sequences: Fibonacci, Thue-Morse, and Rudin-Shapiro. We use a generalized Hadamard coin, ${\widehat{C}}_H$, as well as a generalized Fourier coin, ${\widehat{C}}_K$. We verify the QW experiences a slowdown of the wave packet spreading, ${\sigma }^2\left(t\right)\sim t^{\alpha }$, by the aperiodic jumps whose exponent, $\alpha$, depends on the type of aperiodicity. Additional aperiodicity-induced effects also emerge, namely, (1) while the superdiffusive regime ($1<\alpha <2$) is predominant, $\alpha$ displays an unusual sensibility with the type of coin operator where the more pronounced differences emerge for the Rudin-Shapiro and random protocols and (2) even though the angle $\theta$ of the coin operator is homogeneous in space and time, there is a nonmonotonic dependence of $\alpha$ with $\theta$. Fingerprints of the aperiodicity in the hoppings are also found when distributional measures such as the Shannon and von Neumann entropies, the Inverse Participation Ratio, the Jensen-Shannon dissimilarity, and the kurtosis are computed, which allow assessing informational and delocalization features arising from these protocols and understanding the impact of linear and nonlinear correlations of the jump sequence in a quantum walk as well. Finally, we argue the spin-lattice entanglement is enhanced by aperiodic jumps.

• Received 5 October 2019
• Revised 8 April 2020
• Accepted 3 June 2020

DOI:https://doi.org/10.1103/PhysRevE.102.012104