Nonsmooth and level-resolved dynamics illustrated with a periodically driven tight-binding model

Jiang Min Zhang,Masudul Haque

doi:10.14293/s2199-1006.1.sor-phys.a2cem4.v2

Jiang Min Zhang, Masudul Haque

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https://doi.org/10.14293/s2199-1006.1.sor-phys.a2cem4.v2

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Abstract We point out that in the first-order time-dependent perturbation theory, the transition probability may behave nonsmoothly in time and have kinks periodically. Moreover, the detailed temporal evolution can be sensitive to the exact locations of the eigenvalues in the continuum spectrum, in contrast to coarse-graining ideas. Underlying this nonsmooth and level-resolved dynamics is a simple equality about the sinc function sinc x ≡ sin x/x. These physical effects appear in many systems with approximately equally spaced spectra, and are also robust for larger amplitude coupling beyond the domain of perturbation theory. We use a one-dimensional periodically driven tight-binding model to illustrate these effects, both within and outside the perturbative regime.

Highlights

Coarse graining is a common trick in physics
The predictions are based on the first-order time-dependent perturbation theory, the signatures persist even outside of the perturbative regime
The main difficulty might come from the relatively large number of sites required for a clean manifestation of the effects

Summary

INTRODUCTION

Coarse graining is a common trick in physics. In principle, it is invoked whenever one replaces a summation by an integral. In the General Formalism Section, we derive the effect from the first-order perturbation theory, assuming that the energy levels in the target region are spaced and the couplings to them are equal too. In the Driven Tightbinding Model Section, we demonstrate the effect by taking two examples of tight-binding lattices and driving a local parameter (potential of a single site) In these realistic models, the two assumptions are only approximately satisfied, but we still see sharp kinks (nonsmooth behavior) and sudden bifurcations (level resolution). The two assumptions are only approximately satisfied, but we still see sharp kinks (nonsmooth behavior) and sudden bifurcations (level resolution) All these effects persist even beyond the perturbative regime, i.e., for large amplitude driving. Defining d 1⁄4 EnÃþ1 À EnÃ and a 1⁄4 ðEf À EnÃ Þ=d, we can rewrite Equation (3) as: ð4Þ

À pðtÞ ’ 4dg22WaðTÞ

CONCLUSION AND DISCUSSION