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 $\widehat C_{H}$ as well as a generalized Fourier coin $\widehat C_{K}$. We verify the QW experiences a slowdown of the wavepacket spreading --- $σ^2 (t) \sim t^α$ --- by the aperiodic jumps whose exponent, $α$, depends on the type of aperiodicity. Additional aperiodicity-induced effects also emerge, namely: (i) while the superdiffusive regime ($1<α<2$) is predominant, $α$ 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 $θ$ of the coin operator is homogeneous in space and time, there is a nonmonotonic dependence of $α$ with $θ$. 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.