Mimicking spatial localization in dynamical random environment

We study the role played by noise on the QW introduced in [1], a 1D model that is inspired by a two particle interacting QW. The noise is introduced by a random change in the value of the phase during the evolution, from a constant probability distribution within a given interval. The consequences of introducing such kind of noise depend on both the center value and the width of that interval: a wider interval manifests as a higher level of noise. For some range of parameters, one obtains a quasi-localized state, with a diffusive speed that can be controlled by varying the parameters of the noise. The existence of this (approximately) localized state for such kind of time-dependent noise is, to the best of our knowledge, totally new, since localization (i.e., Anderson localization) is linked in the literature to a spatial random noise.