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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.

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P. A. M. Dirac (1927)

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S. J. Longhi (2009)

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G.C. Stey, R.W. Gibberd (1972)

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References: 10,
Pages: 6

jiang min zhang wrote:

Thanks a lot for the comment! Some short reply: (1) Possibly you mean when the first order perturbation theory breaks down? By some simply scaling argument, we believe the second order term will overtake the first order term when the driving amplitude reaches some critical value, which is independent of the lattice size. But we do not have a quantitative estimate of this value. (2) Our emphasize is on the mathematics of Eq.(10). Then we believe that the effect predicted in the idealized case will persist in the non-ideal tight-binding model. We did not intend to study the quantitative effect of the non-idealness. Possibly the intrinsic error with the perturbation calculation is the overwhelming error? We are not sure. (3) We have no plan to generalize it to higher dimensions. We do not see the possibility, to be frank. (4) There are several different definitions of the sinc function. But they differ just by some linear scaling, as far as we know. Each of them will be equally convenient, we think.

2015-04-07 15:33 UTC

jiang min zhang wrote:

Thanks a lot for the comment! First, I am surprised that you found another reference with the kinks! It took us a while to find the paper by Giessen et al. [PRA 53, 2816(1996)] As for your questions, (1) Yes, equation 10 is the central result. I feel so stupid that we did not carry out the summation thoroughly! (2) I do not think the line with \alpha = 0.4 will coincide with the line with \alpha=0.2. It will coincide with the line with \alpha = 0.6. (3) The period of the kinks is simply linearly proportional to the DoS, as the period is inversely proportional to the level spacing. Yes, the tight binding model has the merit of providing varying level spacing. But it also allows \alpha to very almost randomly. We emphasized the latter but not the former. We will modify the manuscript accordingly. Thanks again!

2015-04-01 11:41 UTC

jiang min zhang wrote:

Sorry, now I think you mean the perturbative regime, which is the regime our result is relevant. Then you are right, I think.

2015-03-24 18:40 UTC

jiang min zhang wrote:

I doubt the validity of this statement: ======== By shining a well-defined frequency on a semiconductor, with $\hbar\omega$ less than the gap, one cannot lift an electron from the valence band to the conduction band because the electron would land in the forbidden band where there are no states available. ======== I guess this kind of argument is popular because of Einstein's explanation of the photoelectric effect. But Einstein's paper was in 1905, far ahead of the invention of the Schroedinger equation, which was in 1926. Once we have the Schroedinger equation, we should study the dynamics of a system by solving the Schroedinger equation, instead of using the simple, arithmetic argument of Einstein. I think it is totally possible to excite an electron from the conduction band to the valence band by using a laser with frequency below the critical frequency, as long as the strength of the laser is strong enough. The point is that, one should not only count the frequency, one should also take into account the amplitude of the driving. Just consider a hydrogen atom in a static electric field. Its frequency is zero, yet if the electric field is strong enough, the atom can get ionized. That is, you get a photoelectric electron with zero frequency driving. The statement is valid only when the laser is weak enough. This is also the regime of the Fermi golden rule. Fermi golden rule is indeed widely used, but I am not sure how good quantitatively it is. In our paper, we need to go to $t> t_c$ to have the level resolution effect. This is somehow consistent with the usual (again hand-waving) argument based on the energy-time uncertainty relationship.

2015-03-22 16:54 UTC