Quantum Revivals in Periodically Driven Systems close to nonlinear resonance

Farhan Saif; Mauro Fortunato

REVIEW

Quantum Revivals in Periodically Driven Systems close to nonlinear resonance

Farhan Saif, Mauro Fortunato

Quantum Physics · Submitted 28 Feb 2006 · 30 references

Pith's verdict: A compact but thinly sourced formula for driven revival times, plausible within its stated approximations but not independently checkable as submitted.

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CitationFarhan Saif, Mauro Fortunato. "Quantum Revivals in Periodically Driven Systems close to nonlinear resonance." arXiv:quant-ph/0602236 (2006). https://pith.science/paper/KGNPYBO7

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Editor's note

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The paper gives a clean closed form [Eq. (22)] for how a periodic drive modifies quantum revival times near nonlinear resonance, and the Mathieu-equation route is standard. The λ=0 limit correctly reduces to the undriven quadratic expansion, which is a good check. The gravitational-cavity example is concrete and the formula's qualitative behavior matches what you would expect. That is the genuine content.

The soft spots are real, and they are in the load-bearing parts. First, the central expression Eq. (20) is quoted from the authors' own thesis [15] without derivation. The jump from the Mathieu expansion Eq. (19) to Eq. (20) is nontrivial; the paper skips the algebra and does not state the approximations beyond 'small λ'. Second, the secular approximation that drops all couplings except n=m±N is not quantitatively validated. For the modulation strengths in Fig. 1 (λ up to 0.25), the neglected terms have amplitudes comparable to the frequency denominators, so they may contribute at the same order as the retained correction. The paper gives no estimate of this error.

Third, the numerical comparison is not independently checkable. No values of N or α are given, no numerical method is described, and there is no error metric. The stress-test point holds up: from the reported 8% and 40% changes, Eq. (25) implies (1−α) ≈ 0.07 for Er=104.1 and ≈ 0.06 for Er=70.28, which do not correspond to a single resonance energy if α is defined with a fixed E_N. Either the resonance order changes in a way the paper does not state, or the formula is being applied in a regime where its approximations are not controlled. That inconsistency needs to be addressed before the claim of quantitative agreement can be taken seriously.

On the positive side, the formula is plausible and the paper is honest that the method is limited to weak modulations and weak-binding potentials. The self-citation to [15] is not itself a flaw, but the paper should delimit what is actually new relative to [15, 16, 25, 26] — right now it reads like a condensed restatement with a new parameterization. The right audience is researchers working on driven wave-packet dynamics, and the paper deserves a serious referee because the compact formula could be useful if the derivation is filled in and the numerics made reproducible. I would send it to peer review, but with a clear expectation of major revision.

Bring it to the reading group if you want to work through the Mathieu expansion together; otherwise one look at the figure is enough to see the gaps.

Referee report

Major objections (4)

[Eq. (19) to Eq. (22)] The derivation of the central formula is not self-contained. Eq. (20) is quoted from the authors' prior thesis [15], and the passage from Eq. (19) to Eq. (22) is not shown. In particular, the second derivative of E_n with respect to n, the use of nu_r = 2 mu / N, and the small-q expansion all need to be written out. Since Eq. (22) is the main result, this algebra must be reproducible from the text.
[After Eq. (3)] The secular approximation that reduces Eq. (3) to Eq. (4) is not quantitatively validated. The dropped terms with n = m +/- (N-1) oscillate with frequency of order 1/N and have amplitude lambda V / 2; the condition lambda V N / 2 << 1 should be stated and checked for the parameters used. With lambda up to 0.25, V = 1, and N not given, the reader cannot judge whether the retained terms dominate. Please provide N and an estimate of the rotating-wave approximation error, for example by comparing the exact Eq. (3) with the approximate Eq. (4) for the parameters of Fig. 1.
[Eq. (23)] The replacement of V_{m,m+/-N} by a single constant V is checked only by the statement "for large m". For the gravitational cavity the matrix element in Eq. (23) varies with m through E_m, which scales as m^{2/3} for the triangular-well limit; over a finite wave-packet width Delta m this variation may change the O(lambda^2) correction. The paper should estimate Delta m and show that V is constant over that interval, or average V over the packet.
[Fig. 1 and Eq. (25)] The numerical validation is not reproducible. The text gives E_r, V_0, kappa, and k_-, but not the resonance order N, the value of alpha used in Eq. (25), the numerical method, or an error metric. The reported 8% and 40% reductions together with Eq. (25) imply (1-alpha) approximately 0.072 and 0.059 for E_r = 104.1 and 70.28, respectively; the corresponding E_N values and the resulting consistency with the chosen drive frequency should be given explicitly, including whether N is the same for both initial energies.

Minor objections (4)

[Eq. (3)] The phase factors in Eq. (3) appear to have n - m in the exponent, whereas substituting the ansatz Eq. (2) gives m - n in the exponent; please check the algebra and verify that Eq. (4) follows with the stated sign convention.
[Eq. (12) and Eq. (14)] The Mathieu functions P_nu(k), the range of k, and the relation between the coefficients xi_n in Eq. (14) and the initial C_n(0) are not defined; the basis expansion should be made explicit.
[Abstract and Fig. 1] The abstract and text contain typos ("modul ating", "va lues", "hyperthesis"), and the caption of Fig. 1 is garbled; the vertical axis should be labeled T_lambda / T_0.
[Section "Application"] The phrase "complete qualitative and quantitative agreement" is stronger than what a comparison without error bars supports; please state the maximum relative deviation between Eq. (25) and the numerical curves.

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The paper

The paper claims

Periodic driving shortens a quantum wave packet's revival time by an amount that grows with the square of the modulation strength, and the paper's analytic formula matches numerical simulations for cesium atoms in a modulated…

The reading

Periodic driving changes the moment when a localized quantum wave packet reassembles, and this paper derives a closed formula for that change. Working near a nonlinear resonance of a one-dimensional system, it shows that the revival time is shortened by an amount proportional to the square of the modulation strength and inversely proportional to the square of the initial energy. The derivation maps the driven problem onto the Mathieu equation, extracts the quasi-energy spectrum, and reads the revival time off its second derivative. Applied to cesium atoms bouncing in a modulated gravitational cavity, the formula agrees with direct numerical simulations, with shifts of roughly 8% at high initial energy and 40% at lower initial energy over the same modulation range.

What carries the argument

The load-bearing object is the quasi-energy spectrum of the driven system near a nonlinear resonance, obtained by a secular approximation that keeps only the resonant couplings $n = m \pm N$, treats the coupling matrix element as a constant $V$, and expands the unperturbed energy to second order around the resonant level. Fourier-transforming the probability amplitudes into an angle variable turns the time-dependent Schrödinger equation into a time-independent Mathieu equation, the standard linear differential equation with a periodic coefficient whose solutions are periodic or quasi-periodic. The Mathieu characteristic parameter $a_\nu(q)$ supplies the quasi-energies $E_n$, and the second derivative of $E_n$ with respect to $n$ fixes the revival time; Eq. (22) assembles these pieces into a closed expression.

Extended reading notes

Core claim

At the paper's center is the claim that close to an $N$th-order nonlinear resonance, the revival time of a periodically driven one-dimensional system is $T_\lambda = T_0 [1 - \frac{1}{2}(\lambda V / E''_r)^2 \frac{3\mu^2 + N^2/4}{(\mu^2 - N^2/4)^3}]$, where $T_0$ is the unmodulated revival time, $\lambda$ is the modulation strength, $V$ is the resonant coupling matrix element, and $\mu$ measures the detuning from exact resonance. For an atom in a driven gravitational cavity at large initial energy this reduces to $T_\lambda = T_0 [1 - \frac{3}{8}(\lambda/E_r)^2/(1-\alpha)^4]$, so stronger modulation and lower initial energy both shorten the revival time. The paper reports that numerical simulations of cesium atoms in a modulated cavity confirm the quadratic dependence on $\lambda$ and the inverse-square dependence on $E_r$, matching the analytical curves.

Load-bearing premise

The load-bearing premise is that fast-oscillating terms in the coupled equations can be dropped so that only the resonant couplings $n = m \pm N$ survive, and that for the large quantum numbers involved the coupling matrix element $V_{m,m\pm N}$ can be replaced by a single constant $V$; if off-resonant transitions or the $m$-dependence of the coupling matter, the quasi-energy formula and the revival-time formula do not follow.

Editorial extensions

If this is right

For any one-dimensional system whose unperturbed spectrum and resonant coupling are known, Eq. (22) predicts the revival time under periodic driving without running a full simulation.
In the driven gravitational cavity, increasing $\lambda$ from 0 to 0.25 shortens the revival time by about 8% for $E_r = 104.1$ and by roughly 40% for $E_r = 70.28$, matching the numerical curves.
Because the leading correction is quadratic in $\lambda$, reversing the sign of the driving field leaves the revival-time shift unchanged at this order; the shift also grows as the initial energy falls, so low-energy wave packets are the most sensitive probes.
The formula's denominator $(\mu^2 - N^2/4)^3$ marks a breakdown boundary: as the detuning parameter approaches $N/2$, the perturbation expansion diverges, so near that boundary higher-order Mathieu terms are required.
For tight-binding potentials whose level spacing increases with energy, nonlinear resonances persist at stronger modulation, but the paper notes that higher-order terms in the Mathieu characteristic parameter must then be included.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A direct extension would sweep the driving frequency across the resonance and measure $T_\lambda$ as a function of the detuning $\mu$; Eq. (22) predicts a universal curve that diverges as $\mu \to N/2$, so the location and shape of that divergence would probe the resonance structure.
If the formula is right, the revival time becomes a precision diagnostic: a measured shift could determine the second derivative $E''_r$ of the unperturbed spectrum, a quantity normally obtained from spectroscopy.
The same expansion of the quasi-energy implies that fractional revivals, which are controlled by higher derivatives of $E_n$, will also be shifted by modulation; deriving their modified times is a natural next step.
Because the correction is even in the detuning, red and blue detuning should produce identical revival-time shifts at leading order, a symmetry an experiment could test by flipping the sign of the frequency offset.

Original abstract

We calculate the quantum revival time for a wave-packet initially well localized in a one-dimensional potential in the presence of an external periodic modulating field. The dependence of the revival time on various parameters of the driven system is shown analytically. As an example of application of our approach, we compare the analytically obtained values of the revival time for various modulation strengths with the numerically computed ones in the case of a driven gravitational cavity. We show that they are in very good agreement.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,548 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.06 · human participants 0.01 · controlled comparison 0.10 · randomized assignment 0.01 · states quantitative result 0.10 · compares with prior results 0.74 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We calculate the quantum revival time for a wave-packet initially well localized in a one-dimensional potential in the presence of an external periodic modulating field. The dependence of the revival time on various parameters of the driven system is shown analytically. As an example of application of our approach, we compare the analytically obtained values of the revival time for various modulation strengths with the numerically computed ones in the case of a driven gravitational cavity. We show that they are in very good agreement.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We calculate the quantum revival time for a wave-packet initially well localized in a one-dimensional potential in the presence of an external periodic modulating field.	contribution 0.98	0.95	0.07	asserted 0.97
s1	The dependence of the revival time on various parameters of the driven system is shown analytically.	contribution 0.52	0.91	0.12	asserted 1.00
s2	As an example of application of our approach, we compare the analytically obtained values of the revival time for various modulation strengths with the numerically computed ones in the case of a driven gravitational cavity.	contribution 0.50	0.95	0.08	asserted 0.91
s3	We show that they are in very good agreement.	result 0.98	0.91	0.38	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.95.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** A comparison between the numerically computed ratio

Figures reproduced from arXiv:quant-ph/0602236 by the authors.

References

Works cited (30)

[15]

F. Saif, Ph.D Thesis, University of Ulm (1998); Hyperthesis, 1, 2 (1999)

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[1]

W. H. F. Talbot, Philos. Mag. 4, 401 (1836)

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[2]

M. Born and E. Wolf, Principles of Optics , 5th Ed. (Pergamon Press, New York, 1975)

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[3]

M. Berry, I. Marzoli, and W.P. Schleich, Phys. World 14(6), 39 (2001)

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I. Sh. Averbukh and N. F. Perel’man, Phys. Lett. A 139, 449 (1989)

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C. Leichtle, I. Sh. Averbukh and W. P. Schleich, Phys. Rev. Lett ., 77, 3999 (1996); Phys. Rev. A 54, 5299 (1996). 11

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P. A. Braun and V. I. Savichev, J. Phys. B 29, L329 (1996)

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Wen-Yu Chen and G. J. Milburn, Phys. Rev. A 51, 2328 (1995)

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D. L. Aronstein and C. R. Stroud, Jr., Phys. Rev. A 62, 022102 (2000)

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F. Haake, Quantum Signatures of Chaos , (Springer, Berlin 1992)

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T. Hogg and B. A. Huberman, Phys. Rev. Lett. 48, 711 (1982)

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[12]

J. K. Breslin, C. A. Holmes, and G. J. Milburn, Phys. Rev. A 56, 3022 (1997)

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S. Tomsovic and J. Lefebvre, Phys. Rev. Lett. 79, 3629 (1997)

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[14]

F. Saif, Phys. Rev. E 62, 6308 (2000)

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[16]

F. Saif, G. Alber, V. Savichev, and W. P. Schleich, J. Opt. B 2, 668S (2000): F. Saif, J. of Phys. Soc. of Japan, 69(8) 2363L (2000)

View on Pith 2000
[17]

G. P. Berman and G. M. Zaslavsky, Phys. Lett. A 61, 295 (1977)

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[18]

M. E. Flatt´ e and M. Holthaus, Ann. Phys. NY 245, 113 (1996)

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[19]

L. E. Reichl, The Transition to Chaos in Conservative Classical Sys- tems: Quantum Manifestations (Springer-Verlag, Berlin, 1992). 12

View on Pith 1992
[20]

C. G. Aminoﬀ, A. M. Steane, P. Bouyer, P. Desbiolles, J. Dalibard , and C. Cohen-Tannoudji, Phys. Rev. Lett. 71, 3083 (1993)

View on Pith 1993
[21]

A. Steane, P. Szriftgiser, P. Desbiolles, and J. Dalibard, Phys. Rev. Lett. 74, 4972 (1995)

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[22]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, (Dover, New York, 1992)

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[23]

H. Wallis, J. Dalibard, and C. Cohen-Tannoudji, Appl. Phys. B, 54, 407 (1992)

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[24]

J. P. Dowling, and J. Gea-Banacloche, Adv. At. Mol. Opt. Phys. 37, 1 (1996)

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[25]

F. Saif, I. Bialynicki-Birula, M. Fortunato, and W. P. Schleich, Ph ys. Rev. A 58, 4779 (1998)

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[26]

F. Saif, Phys. Lett. A 274, 98 (2000)

View on Pith 2000
[27]

R. J. Cook and R. K. Hill, Opt. Commun. 43, 258 (1982)

View on Pith 1982
[28]

V. I. Balykin, V. S. Letokhov, Yu. B. Ovchinnikov, and A. I. Sido rov, Pis’ma Zh. Eksp. Teor. Fiz.45, 282 (1987) [JETP Lett. 45, 353 (1987)]; Phys. Rev. Lett. 60, 2137 (1988)

View on Pith 1987
[29]

M. A. Kasevich, D. S. Weiss, and S. Chu, Opt. Lett. 15, 607 (1990). 13 Figure 1: A comparison between the numerically computed ratio Tλ/T 0 for two diﬀerent initial conditions, Er = 104 . 1 (solid line with circles) and Er = 70. 28 (solid line with squares), and the corresponding [Eq. (25)] ana- lytical results (dashed lines) in the case of an atom bounci...

View on Pith 1990
[30]

Yu. B. Ovchinnikov, I. Manek, and R. Grimm, Phys. Rev. Lett. 79, 2225 (1997). 14 Τ 0 λΤ 1 λ 0.10 0.2 0.8 0.6

View on Pith 1997

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [15]

F. Saif, Ph.D Thesis, University of Ulm (1998); Hyperthesis, 1, 2 (1999)

View on Pith 1998

[2] [1]

W. H. F. Talbot, Philos. Mag. 4, 401 (1836)

View on Pith

[3] [2]

M. Born and E. Wolf, Principles of Optics , 5th Ed. (Pergamon Press, New York, 1975)

View on Pith 1975

[4] [3]

M. Berry, I. Marzoli, and W.P. Schleich, Phys. World 14(6), 39 (2001)

View on Pith 2001

[5] [4]

J. Parker and C. R. Stroud, Jr., Phys. Rev. Lett. 56, 716 (1986)

View on Pith 1986

[6] [5]

I. Sh. Averbukh and N. F. Perel’man, Phys. Lett. A 139, 449 (1989)

View on Pith 1989

[7] [6]

C. Leichtle, I. Sh. Averbukh and W. P. Schleich, Phys. Rev. Lett ., 77, 3999 (1996); Phys. Rev. A 54, 5299 (1996). 11

View on Pith 1996

[8] [7]

P. A. Braun and V. I. Savichev, J. Phys. B 29, L329 (1996)

View on Pith 1996

[9] [8]

Wen-Yu Chen and G. J. Milburn, Phys. Rev. A 51, 2328 (1995)

View on Pith 1995

[10] [9]

D. L. Aronstein and C. R. Stroud, Jr., Phys. Rev. A 62, 022102 (2000)

View on Pith 2000

[11] [10]

F. Haake, Quantum Signatures of Chaos , (Springer, Berlin 1992)

View on Pith 1992

[12] [11]

T. Hogg and B. A. Huberman, Phys. Rev. Lett. 48, 711 (1982)

View on Pith 1982

[13] [12]

J. K. Breslin, C. A. Holmes, and G. J. Milburn, Phys. Rev. A 56, 3022 (1997)

View on Pith 1997

[14] [13]

S. Tomsovic and J. Lefebvre, Phys. Rev. Lett. 79, 3629 (1997)

View on Pith 1997

[15] [14]

F. Saif, Phys. Rev. E 62, 6308 (2000)

View on Pith 2000

[16] [16]

F. Saif, G. Alber, V. Savichev, and W. P. Schleich, J. Opt. B 2, 668S (2000): F. Saif, J. of Phys. Soc. of Japan, 69(8) 2363L (2000)

View on Pith 2000

[17] [17]

G. P. Berman and G. M. Zaslavsky, Phys. Lett. A 61, 295 (1977)

View on Pith 1977

[18] [18]

M. E. Flatt´ e and M. Holthaus, Ann. Phys. NY 245, 113 (1996)

View on Pith 1996

[19] [19]

L. E. Reichl, The Transition to Chaos in Conservative Classical Sys- tems: Quantum Manifestations (Springer-Verlag, Berlin, 1992). 12

View on Pith 1992

[20] [20]

C. G. Aminoﬀ, A. M. Steane, P. Bouyer, P. Desbiolles, J. Dalibard , and C. Cohen-Tannoudji, Phys. Rev. Lett. 71, 3083 (1993)

View on Pith 1993

[21] [21]

A. Steane, P. Szriftgiser, P. Desbiolles, and J. Dalibard, Phys. Rev. Lett. 74, 4972 (1995)

View on Pith 1995

[22] [22]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, (Dover, New York, 1992)

View on Pith 1992

[23] [23]

H. Wallis, J. Dalibard, and C. Cohen-Tannoudji, Appl. Phys. B, 54, 407 (1992)

View on Pith 1992

[24] [24]

J. P. Dowling, and J. Gea-Banacloche, Adv. At. Mol. Opt. Phys. 37, 1 (1996)

View on Pith 1996

[25] [25]

F. Saif, I. Bialynicki-Birula, M. Fortunato, and W. P. Schleich, Ph ys. Rev. A 58, 4779 (1998)

View on Pith 1998

[26] [26]

F. Saif, Phys. Lett. A 274, 98 (2000)

View on Pith 2000

[27] [27]

R. J. Cook and R. K. Hill, Opt. Commun. 43, 258 (1982)

View on Pith 1982

[28] [28]

V. I. Balykin, V. S. Letokhov, Yu. B. Ovchinnikov, and A. I. Sido rov, Pis’ma Zh. Eksp. Teor. Fiz.45, 282 (1987) [JETP Lett. 45, 353 (1987)]; Phys. Rev. Lett. 60, 2137 (1988)

View on Pith 1987

[29] [29]

M. A. Kasevich, D. S. Weiss, and S. Chu, Opt. Lett. 15, 607 (1990). 13 Figure 1: A comparison between the numerically computed ratio Tλ/T 0 for two diﬀerent initial conditions, Er = 104 . 1 (solid line with circles) and Er = 70. 28 (solid line with squares), and the corresponding [Eq. (25)] ana- lytical results (dashed lines) in the case of an atom bounci...

View on Pith 1990

[30] [30]

Yu. B. Ovchinnikov, I. Manek, and R. Grimm, Phys. Rev. Lett. 79, 2225 (1997). 14 Τ 0 λΤ 1 λ 0.10 0.2 0.8 0.6

View on Pith 1997