Deriving currents on the one-dimensional lattice in the regime of dynamic localization

In this paper one deals with a transparent derivation of dynamic localization conditions for the electron on the 1D lattice proceeding irrespective of the concrete form of the periodic modulation of the time dependent electric field. This amounts to account for periodic zero minima generated in the time dependence of the mean square displacement (MSD). Related DL conditions have also been established by looking for the extremal points of the MSD. Interestingly enough, the conditions established in this manner are similar to the ones written down before for pure ac fields but exhibiting a rather unusual modulation. Related time dependent currents can be readily established, now by resorting to discrete derivatives. Our typical example is the dc-ac electric field. We found that in the regime of dynamic localization the MSD and the current get characterized by the same period, i.e by the period of the time dependent modulation. Moreover, the current crosses the time axis just in the maxima-locations of the MSD.