Dynamics of the extended Aubry-Andr\'e-Harper model with localization transition

We show the localization phase transition and its effect on three dynamical processes for an extended Aubry-Andr\'e-Harper model with incommensurate on-site and hopping potentials. With the extended Aubry-Andr\'e-Harper model, we illustrate the localization transition of all eigenstates and fractal characters of the eigenenergy band versus system parameter. To examine the effect of localization transition to dynamical process, an adiabatic pumping of the edge states are examined. In the dynamical process, the system acts as conductor for the excitation in the nonlocal phase and insulator in the localized phase. By Lyapunov control, we propose a protocol to prepare the edge localized state in the nonlocal phase, and find that the control effect is suppressed in the localized phase. By introducing Kerr-type nonlinearity, we examine the dynamical localization behavior in a statistical way. An experimental apparatus to observe the prediction is also suggested.