Quantum walk with quadratic position-dependent phase defects

A comprehensive study of the property of one-dimensional quantum walks, via position distribution of the walker, is adopted herein by considering position-dependent quadratic phase defect. We have explored the origins of this property by introducing a designated position quadratic phase defect conditional shift operator for a discrete-time quantum walk. Numerical simulations conclude that the revival of the walker depends on the phase modulation parameter and the number of steps. In addition, we have found that the localization effect appears after the threshold point where the amplitude of the walker reaches its maximum value, at a constant interval corresponding to the revival period. Furthermore, numerical results show that for rational parabolic coefficients, $$2\pi \frac{q}{p}$$, of the phase defect profile, revivals occur with period 2p for odd p and period p for even p. Furthermore, the period of revival as a function of p is the same as in the case of a linear (instead of quadratic) phase gradient, which was previously investigated in many studies [for instance, see Wojcik et al. (Phys Rev Lett 93:180601, 2004) and Cedzich et al. (Phys Rev Lett 111:160601, 2013)]. The results of this approach therefore shed light on the analysis of discrete quantum processes and the potential relevant for physical implementations of quantum computing with various mesoscopic systems.