Anderson localization in a discrete time quantum walk: Interplay of dispersion and disorder

A theoretical study of the phenomenon of Anderson localization in a generalized discrete time quantum walk will be reported. The generalized discrete time quantum walk is controlled by a quantum coin unitary matrix on each step of particle motion. The quantum coin matrix depends on four angles that can be engineered with microwave pulses in qubit chains. In the absence of disorder in quantum coins parameters the quantum walk dynamics is determined by two-bands dispersion relationship which becomes completely flat in the limit of vanishing kinetic energy. Various types of disorder result in the localization of quantum particle. In particular, kinetic energy disorder leads to logarithmic divergence of the localization length at spectral symmetry points. By making use of the transfer matrix technique we were able to obtain analytical expressions for the frequency dependent localization length in two limits, i.e. weak and strong disorder, and we anticipate that it will be useful for various applications.