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

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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.

### Most cited references10

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### The Quantum Theory of the Emission and Absorption of Radiation

(1927)
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### Quantum-optical analogies using photonic structures

(2009)
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### Decay of quantum states in some exactly soluble models

(1972)
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### Author and article information

###### Affiliations
[1 ]Max Planck Institute for the Physics of Complex Systems, Dresden, Germany
###### Author notes
[* ]Corresponding author's e-mail address: wdlang06@ 123456gmail.com
###### Contributors
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###### Journal
SOR-PHYS
ScienceOpen Research
ScienceOpen
2199-1006
23 December 2014
03 September 2015
: 0 (ID: cdf576df-bf02-47a2-b17e-21283b806b11 )
: 0
: 1-6
3076:XE
10.14293/S2199-1006.1.SOR-PHYS.A2CEM4.v2
© 2014 Jiang Min Zhang and M. Haque.

This work has been published open access under Creative Commons Attribution License CC BY 4.0 , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Conditions, terms of use and publishing policy can be found at www.scienceopen.com .

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Figures: 4, Tables: 0, References: 10, Pages: 6
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Original article

Recently, we realized that the fact that sinc^2 has a finite support in the fourier space is a consequence of the Paley-Wiener theorem. The function is entire and is of exponential type 2. Therefore, by the Paley-Wiener theorem, its fourier transform is supported on [-2, +2].

2015-05-13 08:06 UTC
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I found this paper an interesting work on perturbation theory. I think, for a better readership, the paper could benefit from 1. a brief discussion on the regime of linearity of perturbations and when this linearity breaks down and 2. a quantitative clarification of such propositions as assumptions being "only approximately satisfied", the level spacing and coupling strength being "slowly varying", etc... As indicated by the authors' own admission, the results of the paper are relevant only to single particle models in one dimension. It might be worthwhile mentioning if there are any attempts/prospects to generalize this work. I also wonder if using the normalized sinc function, rather than the sinc function, would be any more convenient.
2015-04-03 00:05 UTC
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One person recommends this
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
Although the paper seem to be a bit confusing, as a condensed matter physicist, I do find the results interesting. I have noticed that the kinks seen in the paper is an old topics in the field of AMO which goes back to as early as 1986 (e.g. Phys. Rev. A. 35, 4226 (1987) by J. Parker. ). Nonetheless, I have following comments and questions, 1. The central result of the paper seems to be rewriting Eq. 5 in the form of Eq. 10. I suggest to emphasize more on the advantage of the new formula. Furthermore, I can see that Eq 10 can be summed over to obtain an analytic expression. 2. In Fig. 1c it would have been better to include \alpha=0.4 and, if possible, explain why there is an overlap with \alpha=0.2. 3. If I understand correctly, the motivation for considering tight binding Hamiltonian is to have varying energy level spacing. I would suggest the author clarify how the DoS can alter the periodicity of the kinks.
2015-03-31 23:38 UTC
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One person recommends this
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
This seems to me a somewhat provocative, hence interesting, paper. Now, I have the following question related to the results for $t>2\pi/\delta$ (LaTex for formulas and Greek letters): When $t$ is larger than the "Heisenberg time" which is directly related to the level spacing, it seems to me that one must ask the following question: Depending on the value of $\alpha$, in general there will be no state which can be reached by the sinusoidal perturbation, whose frequency $\omega$ becomes defined with an accuracy better than the level spacing. Expressed differently on an elementary example taken from semiconductor physics: 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. But if the perturbation occurs during a short enough time, indeed there will be a (perhaps small) Fourier component in it with a frequency high enough to trigger such a transition to the valence band. The case of equally spaced levels with well-defined energies which the authors consider seems to me analogous, just more subtle because they consider level spacings of the same order of magnitude as the inverse duration of the perturbation (in units where $\hbar=1$). Viewed this way, their results seem to me less surprising, without losing their interest, at least from a pedagogical point of view.
2015-03-19 10:14 UTC
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2 people recommend this
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
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
As one of the authors, I would like to stress that I appreciate any comment, positive or negative! So, please do not hesitate to tell me whatever you have in mind.
2015-01-24 17:30 UTC
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