Time Discretization Approach to Dynamic Localization Conditions

Abstract

An alternative wavefunction to the description of the dynamic localization of a charged particle moving on a one-dimensional lattice under the influence of a periodic time dependent electric field is written down. For this purpose the method of characteristics such as applied by Dunlap and Kenkre [Phys. Rev. B 34, 3625 (1986)] has been modified by using a different integration variable. Handling this wavefunction one is faced with the selection of admissible time values. This results in a conditionally exactly solvable problem, now by accounting specifically for the implementation of a time discretization working in conjunction with a related dynamic localization condition. In addition, one resorts to the strong field limit, which amounts to replace, to leading order, the large order zeros of the Bessel function J 0 (z), used before in connection with the cosinusoidal modulation, by integral multiples of π. Here z stands for the ratio between the field amplitude and the frequency. The modulation function of the electric field vanishes on the nodal points of the time grid, which stands for an effective field-free behavior. This opens the way to propose quickly tractable dynamic localization conditions for arbitrary periodic modulations. We have also found that the present time discretization approach produces the minimization of the mean square displacement characterizing the usual exact wavefunction. Other realizations and comparisons have also been presented.

Interplays between Mesoscopic Josephson Junctions and the Harper Equation

“…This results, however, in tight binding descriptions going beyond the capabilities of the Harper equation. Nevertheless, we can look for further generalizations of the Harper equation which should be able to account for (5) in terms of suitable time discretization schemes [29]. The incorporation of the spin-orbit coupling [30], of coherence effects [12,31], as well as of strong couplings to quantized microwave fields [32], deserve further attention, too.…”

The influence of the dynamic localization on the time dependent oscillations of currents on the one dimensional lattice

“…Dealing with periodic modulations for which f(t+T)=f(t), indicates that the period of the above sequence of zero's is T, too. The minimization of the MSD has been discussed before by resorting to a related time lattice description [3]. This amounts to account systematically for periodic zero minima located at selected time points for which t=nT, which is reminiscent to a time lattice.…”

“…Accordingly, one begins with the derivation of DL conditions just referred to above for arbitrary time-periodic modulations, mixed dc-ac fields included [3]. Having established the DL-conditions opens the way to discuss current oscillations in the regime of DL one deals with, now by resorting to discrete derivatives [4].…”