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REVIEW 4 major objections 4 minor 40 references

Quantum optimal control can shape Raman pulses that stay near-perfect under laser detuning and intensity errors, with no single optimizer winning outright.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-02 23:09 UTC pith:FDCVFNCY

load-bearing objection The abstract's three-optimizer benchmark is absent from the body; as submitted, the central claim is unsupported, but the Krotov part is competently done. the 4 major comments →

arxiv 2602.14494 v2 pith:FDCVFNCY submitted 2026-02-16 physics.atom-ph

Optimal Control Design of Robust Raman Pulses for High-Fidelity Cold-Atom Interferometry

Ziwen Song This is my paper
classification physics.atom-ph PACS 37.25.+k
keywords quantum optimal controlKrotov algorithmRaman pulsesatom interferometryrobust pulse shapingfringe contrasttwo-level systemgravimetry
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper tackles a practical bottleneck in cold-atom interferometry: laser frequency drift and intensity fluctuations degrade the Raman pulses that split, reflect, and recombine atomic wave packets, compressing the final interference fringe contrast. The author's central claim is that quantum optimal control—specifically the Krotov algorithm—can design amplitude- and phase-modulated Raman mirror pulses whose transition probability forms a broad plateau over detuning and coupling errors, instead of the narrow peak of standard rectangular or Gaussian pulses. In a simulated Mach-Zehnder sequence, this robustness translates into markedly higher fringe contrast under a fixed systematic detuning. The abstract further reports a three-way comparison of Krotov, GRAPE, and CRAB under a common peak-amplitude limit, finding that GRAPE attained the lowest terminal ensemble error and highest contrast, Krotov a slightly larger high-fidelity grid fraction, and CRAB the worst, while emphasizing trade-offs rather than universal superiority. The body text itself focuses exclusively on the Krotov optimization and does not present the GRAPE or CRAB details.

Core claim

The paper's own claim, on its own terms, is that a Krotov-optimized Raman pulse for the effective two-level atom can actively cancel detuning-induced phase errors by tracing a carefully shaped three-dimensional path on the Bloch sphere. The optimized pulse, seeded from a Gaussian and updated with a Blackman window and step-size λ=0.5, produces a top-hat response curve: atomic transition probability stays near unity across a wide range of two-photon detuning and coupling strength, in contrast to the narrow, bull's-eye region of standard pulses. When this pulse is used as the mirror π pulse in a simulated Mach-Zehnder interferometer under a fixed laser detuning, the final fringe retains a much

What carries the argument

The central object is the effective two-level atom Hamiltonian H(t) = (ℏ/2)(δσz + Ω_eff(t)[cos φ_L(t) σx + sin φ_L(t) σy]), where δ is the two-photon detuning and the time-dependent amplitude Ω_eff(t) and phase φ_L(t) are the controls. The Krotov algorithm—an iterative quantum optimal-control method that monotonically decreases a cost functional—updates the control field according to ∆ε(t) ∝ S(t)/λ Im⟨χ|∂H/∂ε|ψ⟩, with a Blackman shape function S(t) enforcing smooth turn-on and turn-off. The workhorse of the argument is the propagator-based calculation of fringe contrast from the full π/2–π–π/2 sequence, which avoids relying on pulse-sequence symmetry and lets the robustness of an asymmetric,

Load-bearing premise

The advertised comparison assumes that the 25-member detuning–amplitude ensemble and the dense out-of-sample grid used to evaluate Krotov, GRAPE, and CRAB are identical, well-defined, and genuinely independent of the training data, with the same peak-amplitude limit of 3.0—but the body neither defines the ensemble, the grid, the normalization convention, nor the GRAPE/CRAB protocols, so the reported numbers could reflect properties of the training set rather than transferable

What would settle it

Re-run the Krotov, GRAPE, and CRAB optimizations using the paper's stated 25-member ensemble and dense out-of-sample grid with identical normalization and amplitude limits; if GRAPE's terminal-error or contrast advantage reverses, or if the Krotov pulse's transition-probability plateau (P_e ≥ 0.9) does not appear across the claimed detuning–coupling range, the central robustness claim fails.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Krotov-optimized Raman mirror pulses maintain high atomic-manipulation fidelity over a much broader range of laser detuning and intensity fluctuations than rectangular, Gaussian, or super-Gaussian pulses.
  • Using such a pulse as the mirror in a simulated Mach-Zehnder interferometer preserves fringe contrast under a systematic detuning, directly improving the signal-to-noise ratio for precision gravimetry.
  • The Krotov step-size parameter λ=0.5 balances convergence speed and numerical stability, converging in roughly 1000 iterations to a waveform similar to the λ=1.0 result but with less computation.
  • Robustness optimization deliberately trades a small loss of fidelity at the ideal operating point for a large gain across the error range, a feature visible in the early rise of the unperturbed cost.
  • The abstract's three-optimizer comparison indicates no universal winner: GRAPE leads in terminal ensemble error and contrast, Krotov in robust grid fraction, and CRAB trails, so practical choice depends on the target metric.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the robustness plateau holds in a full three-dimensional treatment including atomic velocity spread, extending the optimization from π pulses to π/2 beam-splitter pulses could compound the contrast gain and further relax laser-stability requirements in field-deployable gravimeters—an extension the paper leaves for future work.
  • The top-hat response suggests these pulses could allow cheaper, less-stabilized laser systems without sacrificing measurement precision, a practical consequence the author does not spell out.
  • A testable extension would be to run the same 25-member ensemble and dense out-of-sample grid with GRAPE and CRAB using exactly the same normalization and pulse-duration conventions as the Krotov runs; if the reported ordering persists, the trade-off conclusion is robust, and if not, the comparison becomes a statement about training-set properties rather than transferable performance.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The manuscript (as posted under arXiv:2602.14494) advertises a reproducible framework that compares Krotov, GRAPE, and CRAB optimizers for robust Raman pulse design under a common normalized peak-amplitude limit of 3.0, reporting specific terminal ensemble errors, P_e>=0.9 grid fractions, and fringe contrasts. The body provided, however, describes only a Krotov-based optimization (Sections II.E and III.A) and compares the resulting KR1/KR2/KR3 pulses against rectangular, Gaussian, and super-Gaussian pulses (Section III.B) and in an interferometer simulation (Section III.C). The GRAPE/CRAB comparison, the 25-member ensemble, the out-of-sample grid, and all quantitative results attributed to the abstract are absent from the body. The Krotov results are qualitatively plausible, but the central advertised multi-optimizer benchmark is not available for assessment.

Significance. Robust Raman pulse shaping is a relevant and active direction for improving cold-atom interferometer contrast and systematic-error robustness. The Krotov optimization described in the body is a legitimate technique, and the qualitative demonstration of broadened fidelity plateaus and improved fringe amplitude is encouraging. However, the advertised contribution is a reproducible, fixed-budget comparison of Krotov, GRAPE, and CRAB. That contribution is not present: no code or data are supplied, the ensemble and out-of-sample grid are not defined, and no GRAPE/CRAB results appear anywhere in the body. The specific numbers in the abstract are therefore unfalsifiable from the submitted material. If the missing comparison and definitions were added, the work could be of practical value; as submitted, the significance is not established.

major comments (4)
  1. [Abstract vs. Sections II–III] The abstract reports concrete numerical results: GRAPE terminal ensemble error 1.224e-2, projected Krotov 1.243e-2, CRAB 3.081e-2, P_e>=0.9 grid fractions 0.177 versus 0.170, and contrasts 0.582±0.019, 0.454±0.019, 0.439±0.023. The body contains no GRAPE or CRAB implementation, no definition of a 25-member detuning–amplitude ensemble, no out-of-sample grid, and no fixed-budget protocol. Section II.E only describes the Krotov algorithm, and Section III.B compares Krotov pulses with standard rectangular, Gaussian, and super-Gaussian pulses. The central comparison claimed in the abstract is therefore unsupported by the manuscript's content.
  2. [§II.E, Eqs. (20)–(25); §III.A] The ensemble entering the Krotov cost functional is never specified. Equation (20) defines J = J_T + g_a + g_b and Fig. 3 shows averages over 'different perturbation parameters,' but the manuscript does not state the number of ensemble members, the detuning and Rabi-coupling values used, their distribution, or the pulse duration T and temporal discretization. Without this information, the convergence curves in Fig. 3 and the robustness calculations in Figs. 5 and 6 are not reproducible, and the abstract's '25-member ensemble' is not substantiated.
  3. [§III.B, Figs. 5 and 6] The reported P_e>=0.9 grid fraction (0.177 for Krotov versus 0.170 for GRAPE) is not defined or derived in the body. The text at one point mentions a threshold of 0.80 for the 'high-fidelity robustness area,' while the abstract uses 0.90. The heat maps in Fig. 6 have no stated grid limits, resolution, or declaration of whether the grid is the training ensemble or an independent out-of-sample set. The grid fraction is thus a claim that cannot be verified from the submitted material.
  4. [§III.C, Fig. 8] The fringe-contrast values in the abstract (0.582±0.019 for GRAPE, 0.454±0.019 for Krotov, 0.439±0.023 for CRAB) do not appear in the body. Section III.C and Fig. 8 only state qualitatively that the KR2 pulse gives the largest oscillation amplitude under a fixed detuning; no contrast numbers, uncertainties, or statistical procedure are provided. The abstract's contrast comparison is therefore unsubstantiated.
minor comments (4)
  1. [Title] The PDF title ('Design of Robust Raman Pulses for Cold Atom Interferometers Based on the Krotov Algorithm') differs from the arXiv metadata title ('Optimal Control Design of Robust Raman Pulses for High-Fidelity Cold-Atom Interferometry'). The title should match the actual content.
  2. [Eq. (19) and nearby text] There is an empty citation marker immediately after 'the system Hamiltonian becomes' in Section II.D, and the reference for the final Hamiltonian is missing. Please add the intended citation.
  3. [Normalization convention] The abstract states a 'normalized peak-amplitude limit of 3.0,' and Fig. 4 shows an amplitude axis reaching 3. The normalization convention (e.g., relative to the nominal π-pulse Rabi frequency) is never stated explicitly, which impedes independent reproduction.
  4. [Threshold definitions] The high-fidelity threshold is described as 'e.g., 0.80' in Section III.B but the abstract uses a P_e>=0.9 threshold for the grid fraction. These should be aligned and precisely defined.

Circularity Check

1 steps flagged

Krotov robustness plateau is the optimized objective; abstract's out-of-sample grid comparison is absent from the body.

specific steps
  1. fitted input called prediction [Section III.B (Fig. 6) vs Section II.E (Eq. 21 and Fig. 3 caption)]
    "[III.B] 'To conduct a more comprehensive evaluation, we have plotted 2D heat maps of the atomic transition probability in a two-parameter error space of detuning and coupling strength, as shown in Fig. 6.' [Fig. 3 caption] 'the black curves represent the average cost functional, J_T(avg), and the colored curves represent the cost functionals for different perturbation parameters.' [Eq. 21] 'J_T = 1 − |⟨ψ_tgt|ψ(T)⟩|^2.'"

    The Krotov pulse is obtained by minimizing J_T, the infidelity, over an ensemble of perturbation parameters (the 'average cost functional' with 'different perturbation parameters' in Fig. 3). The body's robustness 'evaluation' in Fig. 6 then plots transition probability over the same detuning–coupling error space, so the high-fidelity 'robustness island' is the trained objective, not an independent out-of-sample test. The abstract's 'dense out-of-sample grid' is nowhere defined in the body, so the reported P_e ≥ 0.9 grid fraction and robustness comparison cannot be shown to be out-of-sample; if the dense grid is merely a dense version of the training domain, the claimed robustness reduces by construction to the fitted cost.

full rationale

The body's derivation chain from the two-level Hamiltonian, through the propagator and Krotov update, to the interferometer fringe-contrast simulation is internally self-contained: no load-bearing self-citation chain, uniqueness theorem, or ansatz-smuggling is present. The clearest circularity is the presentation of the Krotov-optimized pulse's fidelity plateau over detuning and coupling strength as an 'evaluation' when that plateau is precisely the objective minimized over the perturbation ensemble. The abstract goes further and asserts a 25-member ensemble, a dense out-of-sample grid, and a three-optimizer comparison with specific numeric values, none of which are defined or reported in the body; that is a missing-support/omitted-proof problem rather than an additional circularity, but it prevents independent verification of the advertised comparison. Because the fringe-contrast simulation in Section III.C is a downstream consequence not identical to the training objective, the circularity is partial, not total.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

The paper introduces no new physical entities. Its central claims rest on standard two-level Raman physics, the numerical validity of time-slicing, and—most importantly—an ensemble/grid design that is quoted in the abstract but never specified in the body.

free parameters (3)
  • Krotov step-size parameter lambda = 0.5
    Chosen by hand after 'numerous numerical experiments' (Section II.E and III.A). It controls the update magnitude and convergence/stability trade-off, and all Krotov results depend on this choice.
  • 25-member detuning–amplitude ensemble = unspecified
    The abstract says a 25-member ensemble is used, but the body gives no ranges, weights, or member values. Its composition determines the robustness plateau, so it is a free design choice affecting every reported number.
  • Out-of-sample grid density and range = unspecified
    The abstract claims a dense out-of-sample grid, but the body never defines it. Whether this grid is truly independent of the training ensemble is load-bearing for the robustness claims.
axioms (5)
  • domain assumption Stimulated Raman transition can be modeled as an effective two-level system with Hamiltonian in Eq. (2).
    Used throughout Section II; the intermediate excited state is adiabatically eliminated. This is standard but restricts validity to far-detuned Raman pairs.
  • standard math Arbitrary pulse shapes can be simulated as a product of short rectangular propagators via the Magnus expansion (Eq. 8).
    Invoked in Section II.A to justify time-slicing; requires sufficiently small dt and convergence of the Magnus series.
  • domain assumption The Krotov update in Eq. (25) guarantees monotonic convergence in discrete time with the chosen lambda.
    The text notes monotonicity is strictly guaranteed only in the continuous limit; discrete-time performance is tuned via lambda=0.5.
  • ad hoc to paper The detuning–amplitude ensemble used in the optimization is representative of experimental noise.
    The body does not define the ensemble or justify its match to experimental laser noise; the robustness conclusions depend on this representativeness.
  • ad hoc to paper The dense out-of-sample evaluation grid used in the abstract is independent of the optimization ensemble.
    Abstract asserts it; body never demonstrates it. Without independence, robustness scores may reflect training rather than prediction.

pith-pipeline@v1.3.0-alltime-deepseek · 3669 in / 4373 out tokens · 125538 ms · 2026-08-02T23:09:53.006131+00:00 · methodology

0 comments
read the original abstract

The performance of high-precision cold-atom interferometers is often limited by imperfections in the Raman laser fields. We present a reproducible framework for robust Raman mirror-pulse design and compare Krotov, GRAPE, and CRAB under a common normalized peak-amplitude limit of 3.0. The effective two-level model uses a 25-member detuning--amplitude ensemble and a dense out-of-sample grid. In the fixed-budget study, GRAPE attained a terminal ensemble error of $1.224\times10^{-2}$ and projected Krotov attained $1.243\times10^{-2}$; Krotov occupied a slightly larger $P_e\ge0.9$ grid fraction (0.177 versus 0.170). The best of five CRAB seeds gave $3.081\times10^{-2}$. In the interferometer calculation, GRAPE produced the largest contrast, $0.582\pm0.019$, while Krotov and the selected CRAB pulse gave $0.454\pm0.019$ and $0.439\pm0.023$, respectively. These results establish a reproducible comparison of trade-offs, rather than universal superiority of any one optimizer.

Figures

Figures reproduced from arXiv: 2602.14494 by Ziwen Song.

Figure 1
Figure 1. Figure 1: FIG. 1. Energy level diagram for a two-photon stimulated [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Convergence process of the Krotov algorithm for [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Final pulse waveforms obtained from optimizations with different step-size parameters [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Comparison of the robustness of different pulses [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Simulated final interference fringes for the atom [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

40 extracted references · 10 canonical work pages

  1. [1]

    Flechtner, C

    F. Flechtner, C. Reigber, R. Rummel, and G. Balmino, Surveys in Geophysics42, 1029–1074 (2021)

    2021

  2. [2]

    T. E. Jensen, B. Dale, A. Stokholm, R. Forsberg, A. Bres- son, N. Zahzam, A. Bonnin, and Y. Bidel, Earth System Science Data17, 1667–1684 (2025)

    2025

  3. [3]

    G. Rosi, F. Sorrentino, L. Cacciapuoti, M. Prevedelli, and G. M. Tino, Nature510, 518–521 (2014)

    2014

  4. [4]

    Lamporesi, A

    G. Lamporesi, A. Bertoldi, L. Cacciapuoti, M. Prevedelli, and G. M. Tino, Physical Review Letters100, 10.1103/physrevlett.100.050801 (2008)

    doi:10.1103/physrevlett.100.050801 2008

  5. [5]

    Canuel, F

    B. Canuel, F. Leduc, D. Holleville, A. Gauguet, J. Fils, A. Virdis, A. Clairon, N. Dimarcq, C. J. Bord´ e, A. Lan- dragin, and P. Bouyer, Physical Review Letters97, 10.1103/physrevlett.97.010402 (2006)

    doi:10.1103/physrevlett.97.010402 2006

  6. [6]

    Cheiney, L

    P. Cheiney, L. Fouch´ e, S. Templier, F. Napolitano, B. Battelier, P. Bouyer, and B. Barrett, Physical Review Applied10, 10.1103/physrevapplied.10.034030 (2018)

    work page doi:10.1103/physrevapplied.10.034030 2018

  7. [7]

    S. M. Dickerson, J. M. Hogan, A. Sugarbaker, D. M. S. Johnson, and M. A. Kasevich, Physical Review Letters 111, 10.1103/physrevlett.111.083001 (2013)

    doi:10.1103/physrevlett.111.083001 2013

  8. [8]

    Kasevich and S

    M. Kasevich and S. Chu, Physical Review Letters67, 181–184 (1991)

    1991

  9. [9]

    C. R. Ekstrom, J. Schmiedmayer, M. S. Chapman, T. D. Hammond, and D. E. Pritchard, Physical Review A51, 3883–3888 (1995)

    1995

  10. [10]

    Bouchendira, P

    R. Bouchendira, P. Clad´ e, S. Guellati-Kh´ elifa, F. Nez, and F. Biraben, Physical Review Letters 106, 10.1103/physrevlett.106.080801 (2011)

    work page doi:10.1103/physrevlett.106.080801 2011

  11. [11]

    Bonnin, N

    A. Bonnin, N. Zahzam, Y. Bidel, and A. Bresson, Phys- ical Review A92, 10.1103/physreva.92.023626 (2015)

    work page doi:10.1103/physreva.92.023626 2015

  12. [13]

    X. Wu, Z. Pagel, B. S. Malek, T. H. Nguyen, F. Zi, D. S. Scheirer, and H. M¨ uller, Science Advances5, 10.1126/sci- adv.aax0800 (2019)

    doi:10.1126/sci- 2019

  13. [14]

    S. S. Szigeti, J. E. Debs, J. J. Hope, N. P. Robins, and J. D. Close, New Journal of Physics14, 023009 (2012)

    2012

  14. [15]

    Le Gou¨ et, T

    J. Le Gou¨ et, T. Mehlst¨ aubler, J. Kim, S. Merlet, A. Cla- iron, A. Landragin, and F. Pereira Dos Santos, Applied Physics B92, 133–144 (2008)

    2008

  15. [18]

    van Frank, A

    S. van Frank, A. Negretti, T. Berrada, R. B¨ ucker, S. Montangero, J.-F. Schaff, T. Schumm, T. Calarco, and J. Schmiedmayer, Nature Communications5, 10.1038/ncomms5009 (2014)

    work page doi:10.1038/ncomms5009 2014

  16. [19]

    D. M. Reich, M. Ndong, and C. P. Koch, The Journal of Chemical Physics136, 10.1063/1.3691827 (2012)

    doi:10.1063/1.3691827 2012

  17. [20]

    S. J. Glaser, U. Boscain, T. Calarco, C. P. Koch, W. K¨ ockenberger, R. Kosloff, I. Kuprov, B. Luy, S. Schirmer, T. Schulte-Herbr¨ uggen, D. Sugny, and F. K. Wilhelm, The European Physical Journal D69, 10.1140/epjd/e2015-60464-1 (2015)

    doi:10.1140/epjd/e2015-60464-1 2015

  18. [21]

    J. C. Saywell, M. S. Carey, P. S. Light, S. S. Szigeti, A. R. Milne, K. S. Gill, M. L. Goh, V. S. Perunicic, N. M. Wilson, C. D. Macrae, A. Rischka, P. J. Everitt, N. P. Robins, R. P. Anderson, M. R. Hush, and M. J. Biercuk, Nature Communications14, 10.1038/s41467-023-43374- 0 (2023)

    work page doi:10.1038/s41467-023-43374- 2023

  19. [22]

    B. W. Shore,Manipulating Quantum Structures Using Laser Pulses(Cambridge University Press, 2011)

    2011

  20. [23]

    P. R. Berman,Atom Interferometry(Cambridge Univer- sity Press, 1997)

    1997

  21. [24]

    J. C. Saywell, I. Kuprov, D. Goodwin, M. Carey, and T. Freegarde, Physical Review A98, 10.1103/phys- reva.98.023625 (2018)

    doi:10.1103/phys- 2018

  22. [25]

    Dunning, R

    A. Dunning, R. Gregory, J. Bateman, N. Cooper, M. Himsworth, J. A. Jones, and T. Freegarde, Physical Review A90, 10.1103/physreva.90.033608 (2014)

    work page doi:10.1103/physreva.90.033608 2014

  23. [26]

    L´ opez-Monjaraz, H

    C. L´ opez-Monjaraz, H. Pe˜ na Vega, K. Jim´ enez-Garc ´ ıa, J. M. L´ opez Romero, and N. V. Corzo, Physica Scripta 99, 125414 (2024)

    2024

  24. [27]

    Magnus, Communications on Pure and Applied Mathematics7, 649–673 (1954)

    W. Magnus, Communications on Pure and Applied Mathematics7, 649–673 (1954)

    1954

  25. [28]

    Miao, Physics Letters A271, 296–302 (2000)

    X. Miao, Physics Letters A271, 296–302 (2000)

    2000

  26. [29]

    Y. Luo, S. Yan, Q. Hu, A. Jia, C. Wei, and J. Yang, The European Physical Journal D70, 10.1140/epjd/e2016- 70428-6 (2016)

    work page doi:10.1140/epjd/e2016- 2016

  27. [30]

    T. M. Brzozowski, M. Maczynska, M. Zawada, J. Za- chorowski, and W. Gawlik, Journal of Optics B: Quan- tum and Semiclassical Optics4, 62–66 (2002)

    2002

  28. [31]

    Stoner, D

    R. Stoner, D. Butts, J. Kinast, and B. Timmons, Journal of the Optical Society of America B28, 2418 (2011)

    2011

  29. [32]

    D. L. Butts, J. M. Kinast, B. P. Timmons, and R. E. Stoner, Journal of the Optical Society of America B28, 416 (2011)

    2011

  30. [33]

    Y. Wang, J. Cheng, Y. Liu, and T. Lin, Physical Review A109, 10.1103/physreva.109.053501 (2024)

    work page doi:10.1103/physreva.109.053501 2024

  31. [34]

    V. F. Krotov and H. Kong (1993)

    1993

  32. [35]

    Morzhin and A

    O. Morzhin and A. Pechen (arXiv, 2018)

    2018

  33. [36]

    Saywell, M

    J. Saywell, M. Carey, M. Belal, I. Kuprov, and T. Free- garde, Journal of Physics B: Atomic, Molecular and Op- tical Physics53, 085006 (2020)

    2020

  34. [37]

    J. P. Palao and R. Kosloff, Physical Review A68, 10.1103/physreva.68.062308 (2003)

    doi:10.1103/physreva.68.062308 2003

  35. [38]

    J. P. Palao, R. Kosloff, and C. P. Koch, Physical Review A77, 10.1103/physreva.77.063412 (2008)

    work page doi:10.1103/physreva.77.063412 2008

  36. [39]

    Serban, J

    I. Serban, J. Werschnik, and E. K. U. Gross, Physical Review A71, 10.1103/physreva.71.053810 (2005)

    work page doi:10.1103/physreva.71.053810 2005

  37. [40]

    Kaiser and V

    A. Kaiser and V. May, The Journal of Chemical Physics 121, 2528–2535 (2004)

    2004

  38. [41]

    S. E. Sklarz and D. J. Tannor, Physical Review A66, 10.1103/physreva.66.053619 (2002)

    doi:10.1103/physreva.66.053619 2002

  39. [42]

    Kobzar, T

    K. Kobzar, T. E. Skinner, N. Khaneja, S. J. Glaser, and B. Luy, Journal of Magnetic Resonance194, 58–66 11 (2008)

    2008

  40. [43]

    Kobzar, T

    K. Kobzar, T. E. Skinner, N. Khaneja, S. J. Glaser, and B. Luy, Journal of Magnetic Resonance170, 236–243 (2004)

    2004