Fractional revivals of quantum state in tight-binding chain

Bing Chen; C.P. Sun; Z. Song

REVIEW

Fractional revivals of quantum state in tight-binding chain

Bing Chen, Z. Song, C.P. Sun

Quantum Physics · Submitted 4 Mar 2006 · 29 references

Pith's verdict: Real and mostly sound fractional-revival formalism for a discrete chain, but Trev is a factor of 2 too short (wrong small-k dispersion) and the numerics appear to confirm the erroneous model rather than the exact chain.

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arXiv IDquant-ph/0603033 (version 2)

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CitationBing Chen, C.P. Sun, Z. Song. "Fractional revivals of quantum state in tight-binding chain." arXiv:quant-ph/0603033 (2006). https://pith.science/paper/2TXPZJDV

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Claim paper

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Editor's note

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You should know two things before spending time here. The revival time is off by a factor of two. Eq. (12) expands cos k ≈ 1 − k², but the small-k dispersion of the model in Eq. (2) is −2J + Jk². That factor propagates into ΔE, Trev, and every fractional revival instant in Eq. (21): the physical mirror revival time is (N+1)²/(Jπ), not (N+1)²/(2Jπ). The paper's own Sec. VI uses the correct value in its decoherence estimate, so the text contradicts itself. Second, the numerics appear to verify the wrong model. For a broad Gaussian packet in the exact chain at t=(N+1)²/(2Jπ), the even- and odd-n components interfere to give |F|² ≈ 1/2, not the 1 in Fig. 2a. I can't inspect the code, but the clean agreement in Figs. 2–6 is only explained if the simulations implemented the approximate Hamiltonian with the bad coefficient — which would make the claimed numerical confirmation circular.

What is genuinely good: taking the fractional-revival formalism from the infinite square well to a discrete tight-binding chain is a real extension, and the derivation is self-contained. The Gauss-sum structure, the clone positions N±r, and the vanishing-coefficient analyses for N₀ = N/3, N/4, and for superposed packets are done cleanly. The fractional fidelity Ff is a sensible measure for probabilistic transfer. And the paper is honest about its main validity limit — broad packets, Δ ≳ 24, its own Fig. 7. The phenomenon also survives the factor-of-two fix: at t = p(N+1)²/(qJπ) the same phase e^{-ipn²π/q} enters, so the clone structure and positions carry over unchanged; only the time label changes.

Softer spots, in proportion. Eq. (25), |b_r|² = 1/q, is stated too loosely: the b_r are Gauss sums and some vanish — the paper's own N₀ = N/3 case yields a single clone at T/3, not a q-way split. The QIT discussion overclaims 'perfect QIT': the transfer is probabilistic and no extraction/measurement step is specified; Ff = 1 only certifies the mirror component's shape. Minor: the parity stated in Eq. (10) is off by a sign from the sine eigenstates of Eq. (5), though it cancels in the fidelity. Citation pattern is unexceptionable; the self-citations are to the group's SPMC work, and the fanout idea is credited to Greentree et al.

For anyone working on spin-chain state transfer or quantum fanout, this is a useful contribution once the timescale is fixed. I wouldn't cite it as it stands, but I would send it to a serious referee rather than desk-reject: the error is large, localized, and fixable, and the formalism itself is reusable. The authors need to correct Eq. (12), recompute all revival times, and re-run the numerics against the exact Hamiltonian of Eq. (2).

Referee report

Significance

If the central claim is correct, the result is valuable: it shows that a simple uniform chain, without coupling engineering, can produce multiple well-separated copies of a wave packet and can serve as a quantum fanout or information splitter. The derivation from the model Hamiltonian via Fourier analysis and Gauss sums is transparent, and the proposed fractional fidelity is a useful figure of merit for partial state transfer. However, the quantitative predictions are currently undermined by a factor-of-two error in the small-k dispersion and by an imprecise probability statement for the clone amplitudes. These issues are load-bearing because the predicted revival times and the claimed numerical verification depend directly on them, but they are correctable within the manuscript's scope.

Major objections (3)

[Section III.A, Eqs. (10) and (12); Section IV; Section VI] The small-k expansion used in Eq. (12) has the wrong coefficient: for epsilon_k = -2J cos k, the expansion is cos k = 1 - k^2/2 + O(k^4), so the quadratic approximation is epsilon_k ≈ -2J + J k^2, not -2J + 2J k^2. Therefore Delta E in Eq. (10) should be J pi^2/(N+1)^2 and the revival time should be T_rev = (N+1)^2/(J pi), a factor of 2 larger than the value T_rev = (N+1)^2/(2 pi J) used in Section IV and in the time axes of Figs. 2-6. The paper itself contains both expressions: Section IV uses the smaller value, while Section VI quotes T_rev = (N+1)^2/(J pi), so the inconsistency is internal. Since Eq. (21) specifies all fractional revival instants as tau = p T_rev/q, the central quantitative claim depends on resolving this factor.
[Section III.A, Eqs. (19)-(25)] Equation (25) as stated, namely |b_r|^2 = 1/q for every r, is not correct because some of the generalized Gauss sums b_r vanish. For example, for p/q = 1/3 and l = 6, evaluating Eq. (19) gives b_2 = 0 (and also b_0 = b_4 = 0), so one cannot say that each of the l sub-GWPs in Eq. (21) carries probability 1/q. The correct statement is that only the nonvanishing coefficients obey |b_r|^2 = 1/q and that the number of nonvanishing clones is q, not l; this should be stated explicitly and used consistently in the fidelity expressions (23) and (33). As written, the l coefficients would sum to probability 2 for odd q, which further indicates that Eq. (25) cannot be read literally.
[Section IV, Figs. 2-6] The numerical verification needs to state explicitly which Hamiltonian was evolved. If the simulations used the exact Hamiltonian (2), then with the correct quadratic dispersion the first near-perfect mirror revival at low energy occurs at t = (N+1)^2/(J pi), which is t/T_rev = 2 in the units of Section IV. The reported unit-time peak |F(T_rev)|^2 = 1 in Fig. 2a is therefore incompatible with the exact model: at t = (N+1)^2/(2 pi J) the low-energy state is approximately a half-revival with mirror fidelity of order 1/2, not 1. If instead the simulations used the quadratic approximation with the erroneous coefficient, they verify the wrong model rather than the tight-binding chain. Either way, the numerical support for the central claim must be redone after correcting the dispersion coefficient.

Minor objections (4)

[Section III.A, Eq. (10)] The notation epsilon_k = n^2 Delta E omits the constant -2J and should be understood as the level difference relative to the band bottom; this is a presentation issue but is confusing in a paper whose central object is the level spacing.
[Fig. 2 caption] The caption says '2(b) is a part of 2(a) over one revival time' but it should say 'over one revival period', and the axis labels in Fig. 2(b) appear garbled, with entries such as '13/4 1/4 1/2'.
[Section V.C] The text contains the typo 'N0p = N - 50'; this should be 'Np0 = N + 1 - N0' or 'Np0 = N - 50' with consistent notation.
[Reference [26]] The arXiv identifier for Ref. [26], 'quant-ph/062209', appears to be a typo and should be checked and corrected.

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The paper

The paper claims

A uniform tight-binding chain splits a Gaussian wave packet into several shape-preserving copies at fractional revival times.

The reading

This paper claims that a Gaussian wave packet on a simple uniform hopping chain, when released at a low-energy site, does not just spread and decay: at times $\tau = p T_{\mathrm{rev}}/q$ it reassembles into several copies of itself, each with the original shape, located at mirror and shifted positions with probability $1/q$ per copy. If the claim holds, quantum information encoded in the packet (for instance its spin) can be routed to one or many receivers by time evolution alone, without engineering the hopping strengths. The result is derived by approximating the band dispersion as quadratic, which makes the lattice behave like the infinite square well, and is checked numerically on a 500-site chain with a packet width of about 24 sites. The paper also introduces a fractional fidelity that stays near 1 whenever the copy at the target site is faithful, even if only a fraction $1/q$ of the amplitude arrives there.

What carries the argument

The engine of the result is the finite Fourier decomposition of the quadratic phase factor, $e^{-ipn^2\pi/q} = \sum_{r=0}^{l-1} b_r e^{-2\pi i nr/l}$, applied to the wave-packet expansion coefficients; the period $l$ is $2q$ for odd $q$ and $q$ for even $q$, and the amplitudes obey $b_r=b_{l-r}$. Feeding this identity are two ingredients: the quadratic dispersion approximation $\varepsilon_k \approx -2J(1-k^2)$, which turns the uniform chain into a discrete version of the infinite square well, and the spectrum-parity matching condition, the condition that energy level spacings and eigenstate parities are both periodic functions of the standing-wave index $n$, which synchronises the phases of the sine standing waves so that the rephasing time is $T_{\mathrm{rev}}$ and the revived pattern appears at the mirror position. The machinery is measured by the mirror fidelity $F(\tau)$ and the fractional fidelity $F_f(\tau)$, the latter dividing out the clone amplitude at the target site.

Extended reading notes

Core claim

At the instant $\tau=pT_{\mathrm{rev}}/q$, with $T_{\mathrm{rev}}=(N+1)^2/(2\pi J)$, an initial Gaussian packet centred at $N_0$ evolves into $l$ sub-packets that closely reproduce the initial profile: one at the mirror position $N_{\mathrm{p}0}=N+1-N_0$, one at the original position $N_0$, and pairs at $N_0\pm 2(N+1)r/l$, where $l=2q$ for odd $q$ and $l=q$ for even $q$, each with probability $|b_r|^2=1/q$. This is the lattice analogue of fractional revivals in the infinite square well, and it follows because the low-energy dispersion is approximately quadratic. For special positions the pattern sharpens: a packet started at $N_0=N/3$ fully revives at its mirror site already at $T_{\mathrm{rev}}/3$, and a superposition of packets at $N/3$ and $2N/3$ revives at $T_{\mathrm{rev}}/24$. Numerically, for a 500-site chain and a packet width of 24 sites, the positions, shapes, and probabilities of the sub-packets match the formula, and the fractional fidelity at the mirror site is close to 1 at many rational times.

Load-bearing premise

The whole clone picture rests on treating the chain's energy spectrum as exactly quadratic for the packet, which is only true for low-energy, broad wave packets; the paper's own Figure 7 shows that when the packet width falls below roughly 24 lattice sites the fractional fidelity drops and the cloned sub-packets lose their shape.

Editorial extensions

If this is right

A uniform, unmodulated chain can transfer a qubit encoded in a Gaussian packet from its starting site to the mirror site with unit fidelity at the full revival time, and in one third of that time when the packet starts one third of the way along the chain.
Reading out only the clone at the target site gives perfect quantum information transfer at fractional revival times: the fractional fidelity is near 1 whenever that clone is well separated from its neighbours, even though the clone carries only a fraction of the total probability.
A superposition of two packets, placed one third and two thirds of the way along the chain, revives in one twenty-fourth of the revival time, giving a faster channel and a way to deliver entangled packet states.
Because the hopping Hamiltonian does not touch the spin degree of freedom, the same mechanism doubles as a flying qubit and as a passive quantum fanout, sending the same spin state to several spatial receivers at once.
For realistic quantum-dot parameters (hopping about 10 meV and a 500-site wire), the revival time is orders of magnitude shorter than typical decoherence times, so many revivals fit within the coherence window.

Reading between the lines

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

The same finite Fourier machinery is not special to Gaussian packets: any low-energy state expanded in sine standing waves should show the same clone structure, so testing a rectangular or chirped initial packet would show whether shape reproduction is generic.
The open-chain boundary conditions force the clones to form at mirror-symmetric positions; a ring geometry would change the standing-wave basis and likely produce translation clones, a design choice the paper does not explore.
Each clone carries probability 1/q, so the device acts as a passive beam splitter; detecting nothing at one clone position could herald the information at another, an entanglement-distillation-style use not considered in the paper.
The width threshold around 24 lattice sites is an experimentally accessible crossover: sweeping the initial packet width in a dot array should show fractional fidelity collapsing below that scale, which would map the validity region of the quadratic approximation.

Original abstract

We investigate the time evolution of Gaussian wave packet (GWP) in the tight-binding chain with uniform nearest neighbor (NN) hopping integral. Analytical analysis and numerical simulations show that the fractional revival of the quantum state occurs in such system, i.e., at appropriate time, a GWP can evolve into many copies of the initial state at different positions. The application of this quantum phenomenon to the scheme of quantum information transfer in solid-state system is discussed.

The compiled record

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The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We investigate the time evolution of Gaussian wave packet (GWP) in the tight-binding chain with uniform nearest neighbor (NN) hopping integral. Analytical analysis and numerical simulations show that the fractional revival of the quantum state occurs in such system, i.e., at appropriate time, a GWP can evolve into many copies of the initial state at different positions. The application of this quantum phenomenon to the scheme of quantum information transfer in solid-state system is discussed.

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 investigate the time evolution of Gaussian wave packet (GWP) in the tight-binding chain with uniform nearest neighbor (NN) hopping integral.	contribution 0.95	0.87	0.04	asserted 0.66
s1	Analytical analysis and numerical simulations show that the fractional revival of the quantum state occurs in such system, i.e., at appropriate time, a GWP can evolve into many copies of the initial state at different positions.	result 1.00	0.93	0.87	asserted 1.00
s2	The application of this quantum phenomenon to the scheme of quantum information transfer in solid-state system is discussed.	interpretation 0.85	0.49	0.09	unclear 0.59

What kind of paper the classifier reports this is: theoretical result 0.98 · evidence from proof or derivation 0.53.

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Figures

**Figure 1.** FIG. 1: (Color online) Schematic illustrations of quantum

**Figure 3.** FIG. 3: The illustrations of the fractional revival via the t

**Figure 5.** FIG. 5: (a) Plot of the square of fidelity

**Figure 6.** FIG. 6: (Color online) Plot of the square of the fractional

**Figure 7.** FIG. 7: (Color online) The square of fractional fidelity

Showing 5 of the paper's 7 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:quant-ph/0603033 by the authors.

References

Works cited (29)

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From the analytical results, at instants τ/T rev = 1/5, 1/4, 1 /3, 1 /2, and 1, the cloned sub-GWPs have the probabilities of 1 /5, 1/4, 1/3, 1/2, and 1, which result in the maximum values of the corresponding sub-GWPs to be 0 .089, 0.099, 0.114, 0.140 and 0 .198. The numeri- cal results, the number and the shapes of the sub-GWPs are in good accord with t...

View on Pith
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Alexandre G. M. Schmidt and M. G. E. da Luz, Phys. Rev. A 69, 052708 (2004)

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Peter Karbach and Joachim Stolze, Phys. Rev. A 72, 030301(R) (2005)

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M. Christandl, N. Datta, A. Ekert and A.J. Landahl, Phys. Rev. Lett. 92, 187902 (2004)

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M. Christandle, N. Datta T. C. Dorlas, A. Ekert, A. Kay and A.J. Landahl, Phys. Rev. A 71, 032312 (2005)

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C. Albanese, M.Christandle, N. Datta and A. Ekert, Phys. Rev. Lett. 93, 230502 (2004)

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Daniel Burgarth and Sougato Bose, New J. Phys. 7, 135 (2005)

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D. Burgarth and S. Bose, Physical Review A 71, 052315 (2005)

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[22]

T. Shi, B. Chen and Z. Song, Comm. Theor. Phys. 43, 795 (2005)

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[23]

T. Shi, Ying Li, Z. Song and C.P. Sun, Phys. Rev. A 71, 032309 (2005)

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[24]

S. Yang, Z. Song, and C.P. Sun, Phys. Rev. A 73, 022317 (2006)

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[25]

Y. Li, Z. Song, and C.P. Sun, quant-ph/0504175

View on Pith Pith review arXiv
[26]

Andrew D. Greentree, Simon J. Devitt, and Lloyd C. L. Hollenberg, Phys. Rev. A 73, 032319 (2006)

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[27]

S. Yang, Z. Song, and C.P. Sun, quant-ph/062209

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[28]

Guido Burkard and Daniel Loss, Phys. Rev. B 59, 2070 (1999); A.V.Onufriev and J.B.Marston, Phys. Rev. B 59, 12573 (1999); W Gvander Wiel, et al. New Journal of Physics 8, 28 (2006)

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[29]

Rogeriode Sousa and S.Das Sarma,cond-mat/0211567; Daniel Loss and David P. DiVincenzo, Phy. Rev. A 57, 120 (1998); Hans-Andreas Engel, et al. Quantum Infor- mation Processing, Vol. 3, 115 (2004)

View on Pith Pith review arXiv 1998

How the literature uses it

2 arXiv papers cite this in 2 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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] [1]

R.W. Robinett, Phys. Rep. 392 1 (2004)

View on Pith 2004

[2] [2]

I. Sh. Averbukh and N. F. Perelman, Phys. Lett. A 139, 449 (1989); Acta Phys. Pol. A 78, 33 (1990)

View on Pith 1989

[3] [3]

From the analytical results, at instants τ/T rev = 1/5, 1/4, 1 /3, 1 /2, and 1, the cloned sub-GWPs have the probabilities of 1 /5, 1/4, 1/3, 1/2, and 1, which result in the maximum values of the corresponding sub-GWPs to be 0 .089, 0.099, 0.114, 0.140 and 0 .198. The numeri- cal results, the number and the shapes of the sub-GWPs are in good accord with t...

View on Pith

[4] [4]

David L. Aronstein and C.R. Stroud, Phys. Rev. A 55, 4526 (1997)

View on Pith 1997

[5] [5]

G.S. Agarwal and J. Banerji, Phys. Rev. A 57, 3880 (1998)

View on Pith 1998

[6] [6]

G.A. Vugalter, A.K. Das and V.A. Sorokin, Phys. Rev. A 66, 012104 (2002)

View on Pith 2002

[7] [7]

O. Knospe and R. Schmidt, Phys. Rev. A 54, 1154 (1996)

View on Pith 1996

[8] [8]

M.J.J. Vrakking, D.M. Villeneuve and A. Stolow, Phys. Rev. A 54, R37 (1996)

View on Pith 1996

[9] [9]

Kevin F. Lee, D.M. Villeneuve, P.B. Corkum and E.A. Shapiro, Phys. Rev. Lett. 93, 233601 (2004)

View on Pith 2004

[10] [10]

Michael Spanner, E.A. Shapiro, and Misha Ivanov, Phys. Rev. Lett. 92, 093001 (2004)

View on Pith 2004

[11] [11]

S. Bose, Phys. Rev. lett. 91, 207901 (2003)

View on Pith 2003

[12] [12]

Y. Li, T. Shi, B. Chen, Z. Song, C.P. Sun, Phys. Rev. A 71, 032309 (2005)

View on Pith 2005

[13] [13]

Z. Song and C.P. Sun, Low Temperature Physics 31, 686 (2005)

View on Pith 2005

[14] [14]

Alexandre G. M. Schmidt and M. G. E. da Luz, Phys. Rev. A 69, 052708 (2004)

View on Pith 2004

[15] [15]

Peter Karbach and Joachim Stolze, Phys. Rev. A 72, 030301(R) (2005)

View on Pith 2005

[16] [16]

P. Jordan and E. Wigner, Z. Phys. 47, 631 (1928); E. Lieb, T. Schultz, and D. Mattis, Ann. Phys. 16, 407 (1961); S. Katsura, Phys. Rev. 127, 1508 (1962); N. Nagaosa, Quantum Field Theory in Strongly Correlated Electronic Systems, (Springer-Verlag, Berlin Heidelberg , Germany, 1999)

View on Pith 1928

[17] [17]

M. Christandl, N. Datta, A. Ekert and A.J. Landahl, Phys. Rev. Lett. 92, 187902 (2004)

View on Pith 2004

[18] [18]

M. Christandle, N. Datta T. C. Dorlas, A. Ekert, A. Kay and A.J. Landahl, Phys. Rev. A 71, 032312 (2005)

View on Pith 2005

[19] [19]

C. Albanese, M.Christandle, N. Datta and A. Ekert, Phys. Rev. Lett. 93, 230502 (2004)

View on Pith 2004

[20] [20]

Daniel Burgarth and Sougato Bose, New J. Phys. 7, 135 (2005)

View on Pith 2005

[21] [21]

D. Burgarth and S. Bose, Physical Review A 71, 052315 (2005)

View on Pith 2005

[22] [22]

T. Shi, B. Chen and Z. Song, Comm. Theor. Phys. 43, 795 (2005)

View on Pith 2005

[23] [23]

T. Shi, Ying Li, Z. Song and C.P. Sun, Phys. Rev. A 71, 032309 (2005)

View on Pith 2005

[24] [24]

S. Yang, Z. Song, and C.P. Sun, Phys. Rev. A 73, 022317 (2006)

View on Pith 2006

[25] [25]

Y. Li, Z. Song, and C.P. Sun, quant-ph/0504175

View on Pith Pith review arXiv

[26] [26]

Andrew D. Greentree, Simon J. Devitt, and Lloyd C. L. Hollenberg, Phys. Rev. A 73, 032319 (2006)

View on Pith 2006

[27] [27]

S. Yang, Z. Song, and C.P. Sun, quant-ph/062209

View on Pith

[28] [28]

Guido Burkard and Daniel Loss, Phys. Rev. B 59, 2070 (1999); A.V.Onufriev and J.B.Marston, Phys. Rev. B 59, 12573 (1999); W Gvander Wiel, et al. New Journal of Physics 8, 28 (2006)

View on Pith 1999

[29] [29]

Rogeriode Sousa and S.Das Sarma,cond-mat/0211567; Daniel Loss and David P. DiVincenzo, Phy. Rev. A 57, 120 (1998); Hans-Andreas Engel, et al. Quantum Infor- mation Processing, Vol. 3, 115 (2004)

View on Pith Pith review arXiv 1998