Quantum walks with sequential aperiodic jumps

Abstract

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 ${\hat{C}}_H$ as well as a generalized Fourier coin ${\hat{C}}_K$. We verify the QW experiences a slowdown of the wavepacket spreading --- ${\sigma }^2(t)\sim t^{\alpha }$ --- by the aperiodic jumps whose exponent, $\alpha$, depends on the type of aperiodicity. Additional aperiodicity-induced effects also emerge, namely: (i) 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 protocol; (ii) 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 additional distributional measures such as Shannon entropy, IPR, Jensen-Shannon dissimilarity, and kurtosis are computed. Finally, we argue the spin-lattice entanglement is enhanced by aperiodic jumps.

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