  We study the localization phenomena in a one-dimensional lattice system with
a uniformly moving disordered potential. At a low moving velocity, we find a
sliding localized phase in which the initially localized matter wave
adiabatically follows the moving potential without diffusion, thus resulting in
an initial state memory in the many-body dynamics. Such an intriguing localized
phase distinguishes itself from the standard Anderson localization in two
aspects: it is not robust against interaction, but persists in the presence of
slowly varying perturbations. Such a sliding localized phase can be understood
as a consequence of interference between the wavepacket paths under moving
quasi-periodic potentials with various periods that are incommensurate with the
lattice constant. The experimental realization and detection were also
discussed..
