Discrete-time quantum walk with time-correlated noise

We investigate the dynamics of discrete-time quantum walk subject to time correlated noise. Noise is described as an unitary coin-type operator before each step, and attention is focused on the noise generated by a Gaussian Ornstein Uhlenbeck process, going beyond the usual telegraph noise, where the random variables are consist of only -1 and 1. Under the first-order approximation of BCH formula, the master equation of noisy discrete-time quantum walk is derived. The dynamics given by the master equation are in good agreement with those given by numerical simulations within a certain period of steps, which is controlled by noise parameters. Two remarker behaviors of long time noisy dynamics are observed in numerical simulations, corresponding to two opposite noise regimes: in slow noise regime, with the increase of the noise amplitude, the quantum coherence is suppressed, and the dynamics of noisy discrete-time quantum walk gradually transits to that of classical random walk. In fast noise regime, the walker is confined into few lattice sites, and the width of wave packet is much narrower compared with that in slow noise regime.