One-dimensional quantum walks with a time and spin-dependent phase shift

Abstract

We investigate the role of a time and spin-dependent phase shift on the evolution of one-dimensional discrete-time quantum walks. By employing Floquet engineering a time and spin-dependent phase shift (ϕ) is stroboscopically imprinted onto the walker's wave function at the end of each step of the walk. For a quantum walk driven by the standard protocol and with rational values of the phase factor, i.e., ϕ/2π=p/q we show with our numerical simulations that revivals with equal periods occur in the probability distribution of the walk. In the split-step quantum walk we employ two different coin operators which are parametrized by a parameter δθ. By making use of δθ as a control parameter we demonstrate that for ϕ/2π=p/q the walk exhibits three major types of dynamics: ballistic spreading, revivals, and localization. For an irrational value of ϕ/2π our results show revivals with unpredictable periods, and the walker remains localized in a small region of the lattice. Furthermore, in view of an experimental realization we investigate the robustness of revivals against noise in the phase shift. Our results show that signatures of revivals persist for smaller values of the noise parameter while these vanish for larger values. Our work is important in the context of quantum computation and simulation with quantum walks where different types of the walk dynamics can be engineered by making use of the time and spin-dependent phases shift, and the coin parameter as control knobs.