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Theory of composite Ramsey sequences of radiofrequency pulses beyond the rotating wave approximation

T0 review · 3 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read The paper argues that the rotating wave approximation, which is standard for optical transitions, fails for composite Ramsey sequences of radiofrequency pulses once the sequence contains several tens of pulses.

desk verdict A credible numerical demonstration that the rotating wave approximation fails for long rf composite Ramsey sequences, with the key threshold condition stated but not derived. read the letter →

arxiv 2502.13973 v1 pith:QAQSJCBA submitted 2025-02-04 physics.atom-ph

V. I. Yudin , O. N. Prudnikov , A. V. Taichenachev , M. Yu. Basalaev , V. G. Pal'chikov , S. N. Bagayev This is my paper · ORCID
classification physics.atom-ph
keywords compositeRamseysequencesrotatingwaveapproximationradiofrequencypulseslocalLorentzinvarianceZeemanstructureFourieranalysisBloch-Siegertshifttensor
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that the rotating wave approximation, which is standard for optical transitions, fails for composite Ramsey sequences of radiofrequency pulses once the sequence contains several tens of pulses. The non-resonant parameter for rf fields is only about $10^{-2}$ to $10^{-3}$, so each pulse leaves a small correction that accumulates over many pulses. For the specific sequence $\{(\pi/2,-\pi/2)^n,\pi\}$ proposed to search for Lorentz-invariance-violating tensor shifts $\kappa \hat{J}_z^2$, the exact dynamics differ radically from the resonant approximation for large $n$. Moreover, when condition (10), $\hbar\kappa/\Omega_{\rm rf} \ll (\tau/\tau_R)(\Omega_{\rm rf}/\Omega_L)$, holds, the small tensor shift has practically no observable effect, so the sequence cannot reliably detect an LLI-violating shift even if it exists. The paper therefore leaves the viability of such rf Ramsey tests open and calls for analysis beyond the rotating wave approximation.

What carries the argument

The central object is the composite Ramsey sequence $\{(\varphi_1,\varphi_2,\dots,\varphi_N)^n,\varphi_{\rm fin}\}$, made of an initial $\pi/2$ pulse, $n$ identical boxes of $N$ $\pi$ pulses, and a final $\pi/2$ pulse, with free evolution times $\tau_R$ between pulses. The carrying mechanism is Fourier analysis of the time-dependent Schr\"odinger equation in the rotating frame: after the unitary transformation $a_m(t)=\tilde a_m(t)e^{-im\nu t}$, the Hamiltonian splits into resonant terms and rapidly oscillating counter-rotating terms proportional to $e^{\pm 2i\nu t}$. The resonant approximation keeps only the former; the full treatment expands the wave function in harmonics and keeps all of them. This reveals a per-pulse non-resonant error of order $\Omega_{\rm rf}/\Omega_L$, which is the mechanism behind the breakdown. The small tensor shift $\kappa \hat{J}_z^2$ enters through condition (10), which sets the scale below which the shift is hidden by the non-resonant dynamics.

What would settle it

For parameters satisfying condition (10), compute the exact final population $P_{m'}$ versus $n$ for the sequence $\{(\pi/2,-\pi/2)^n,\pi\}$ with a known tensor shift just above the bound; the paper predicts the curve is indistinguishable from $\kappa=0$ at large $n$, whereas the resonant Hamiltonian predicts a growing separation. If the exact curves separate visibly, the criterion is wrong.

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Extended reading notes

Core claim

On its own terms, the paper's central discovery is that the dynamics of a multilevel atom driven by a long composite Ramsey sequence of rf pulses are governed by non-resonant contributions that the rotating wave approximation discards, and these contributions change the outcome qualitatively once the number of pulses exceeds several tens. Concretely, for the sequence $\{(\pi/2,-\pi/2)^n,\pi\}$ with $J=7/2$ and parameters $\Omega_L/\Omega_{\rm rf}=100$, the exact solution of the full Hamiltonian (3) (computed by Fourier expansion over harmonics up to $F=50$) differs drastically from the resonant Hamiltonian (9) for $n\gg 1$, whereas the two agree for small $n$. The accumulated error is of order $\Omega_{\rm rf}/\Omega_L$ per pulse, so with $\Omega_{\rm rf}/\Omega_L\sim 10^{-2}$–$10^{-3}$, breakdown appears after tens to hundreds of pulses. Under condition (10), the tensor shift $\kappa \hat{J}_z^2$ has essentially no visible effect in the exact dynamics, in contrast to the resonant prediction where it grows with $n$. The paper concludes that the rf Ramsey method of Ref. [9] cannot reliably detect an LLI-violating tensor shift, and that the general effectiveness of dynamical decoupling for rf pulses needs separate justification beyond the rotating wave approximation.

Load-bearing premise

The paper's key assumption is that condition (10) correctly identifies when the tensor shift becomes unobservable; this criterion is supported by numerical examples and a rough per-pulse error estimate rather than a rigorous proof.

Editorial extensions

If this is right

  • If the paper is right, the rotating wave approximation cannot be used to design or predict composite rf Ramsey sequences with more than a few tens of pulses; exact or beyond-RWA propagation is required.
  • The specific sequence $\{(\pi/2,-\pi/2)^n,\varphi_{\rm fin}\}$ cannot serve as a reliable detector of an LLI-induced tensor shift when condition (10) holds, because the shift's effect is masked by non-resonant dynamics.
  • Agreement between resonant theory and experiment for the $^{88}$Sr$^+$ data with $n$ up to 55 does not validate the method for LLI searches, because that agreement arises only when the controlled tensor shift is large enough to violate condition (10).
  • Dynamical decoupling's ability to suppress magnetic-field noise in the rf regime is not guaranteed by the two-level rotating-wave justification and must be re-examined.
  • For rf transitions, unlike optical ones, the non-resonant parameter cannot be made arbitrarily small: increasing $\Omega_L$ enlarges the second-order Zeeman shift of the same tensor form, and decreasing $\Omega_{\rm rf}$ increases relative field fluctuations.

Reading between the lines

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

  • A testable extension is to scan the ratio $\Omega_{\rm rf}/\Omega_L$ and the number of boxes $n$ to locate the crossover where exact and RWA dynamics diverge; the paper's qualitative argument predicts the crossover near $n\sim \Omega_L/\Omega_{\rm rf}$.
  • Condition (10), if correct, implies a sensitivity floor for this class of rf composite-pulse LLI searches: any tensor shift below $\hbar\Omega_{\rm rf}^2 \tau/(\Omega_L \tau_R)$ is invisible, which is worse than the sensitivity predicted by the resonant Hamiltonian.
  • The breakdown mechanism is generic for any multilevel system driven by many pulses with a small but finite counter-rotating field, so the same Fourier-harmonic method could be applied to other rf metrology schemes, such as magic rf dressing in optical clocks.
  • Because the paper's conclusion about condition (10) rests on numerical examples and a qualitative error-accumulation argument, a rigorous analytic estimate of the accumulated non-resonant error as a function of $n$ would either confirm the condition or reveal a different scaling; such a derivation is the natural next step.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper develops a beyond-rotating-wave-approximation (RWA) theory of composite Ramsey sequences of radiofrequency pulses acting on a Zeeman manifold with angular momentum J. Using the full Hamiltonian of Eq. (3) [or Eq. (7)] and a Fourier expansion over harmonics, the authors numerically compare the dynamics generated by the full Hamiltonian with the RWA Hamiltonian of Eq. (9) for the sequence {(π/2, −π/2)^n, π} proposed in Ref. [9]. They find that for n exceeding several tens the RWA becomes inadequate, that the small tensor shift κJ_z^2 has practically no observable effect when condition (10) holds, and that the modified resonant Hamiltonian of Eq. (11), including the Bloch–Siegert shift, still does not reproduce the exact dynamics. The paper concludes that the effectiveness of such rf composite Ramsey sequences for detecting local Lorentz invariance (LLI) violation is not fully determined and requires further study.

Significance. If the numerical result is correct, this is an important cautionary result for precision tests of local Lorentz invariance with rf composite Ramsey spectroscopy. The paper's strength is that the full and RWA Hamiltonians are independently defined and no parameters are fitted to the target conclusion; the comparison is a direct numerical test. The explicit identification of the Bloch–Siegert correction in Eq. (11) and the demonstration that this first-order correction is insufficient for long sequences are useful and falsifiable. The main quantitative condition, inequality (10), is, however, asserted from numerical observation rather than derived, and the paper's practical conclusion about undetectability of LLI tensor shifts rests on that condition. If the scaling in (10) is not correct, the negative conclusion about the sequence of Ref. [9] would not follow even though the RWA breakdown itself would remain real.

major comments (3)
  1. [§3, Eq. (10)] The criterion ℏκ/Ω_rf ≪ (τ/τ_R)(Ω_rf/Ω_L) is introduced after Fig. 2 as a numerically observed condition, but it is load-bearing for the paper's practical conclusion that the sequence {(π/2, −π/2)^n, π} cannot reliably detect an LLI tensor shift. No derivation from Eq. (7) is supplied, and the factor τ/τ_R is not justified: the tensor shift κJ_z^2 acts during free-evolution intervals of duration τ_R, whereas the leading non-resonant correction in Eq. (11) acts only during pulses of duration τ, so accumulated phases proportional to κτ_R and Ω_rf^2 τ/Ω_L are the more natural quantities to compare. A Magnus or Floquet expansion of Eq. (7) retaining the leading κ-dependent term is needed to establish the scaling. Without such a derivation, the detectability conclusion is conditional on a plausible but undemonstrated estimate, and this is exactly the concern raised by the stress-test note; I agree that it lands.
  2. [§3, Fourier truncation (p. 2)] The numerical method truncates the Fourier expansion at F = 50, and the text states that F = 50 'was enough with a good margin' for the curves in Figs. 2–4. No convergence test or error estimate is shown, despite the fact that the central claim is a quantitative discrepancy for n ≫ 1. The paper should display the convergence of P_{m'} as a function of F for at least one long-sequence case and for the parameter set of Fig. 3, where the dynamics are governed by a different regime of condition (10).
  3. [§3, Figs. 2–3 and subsequent paragraph] The text asserts that 'similar radical discrepancies' are observed for other values of m′ and J, including J = 1/2, 3/2, 5/2, and that calculations for J = 5/2 using experimental data from Ref. [9] confirm good agreement with the resonant Hamiltonian. None of these cases is displayed or described quantitatively. Because the practical conclusion is meant to apply beyond J = 7/2, the paper should include at least a summary figure or table for the J = 5/2 comparison with the experimental parameters, and should specify which m′ values were scanned. Without this evidence, the generality claim is asserted rather than demonstrated.
minor comments (4)
  1. [Title] The title contains a typo ('pul ses' instead of 'pulses'); it should be corrected.
  2. [Abstract and Conclusion] The abstract states that the effectiveness of the sequences 'has not yet been fully determined', while the text below Eq. (10) makes the stronger claim that the sequence 'will not allow to reliably detect this shift'. These statements should be reconciled, because the stronger claim depends on the undemonstrated condition (10).
  3. [Conclusion, final paragraph] The sentence 'the effectiveness of the dynamical decoupling technique ... is also not guaranteed' is vague; the paper should specify which property is not guaranteed (e.g., robustness to magnetic-field fluctuations) and what concrete test would establish it.
  4. [Reference [9] comparison] The J = 5/2 comparison against data from the Supplemental material of Ref. [9] is mentioned but not located precisely; the authors should identify the specific dataset and parameters used.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the full and RWA Hamiltonians are independently defined, and the central claim is supported by direct numerical solution rather than by a fitted or self-referential construction.

full rationale

The paper's derivation chain is self-contained in the relevant sense. The starting Hamiltonian (3) is defined from the physical field geometry and the Zeeman Hamiltonian, and the transformation (5) leading to Eq. (7) is exact. The rotating-wave-approximation Hamiltonian (9) is obtained by dropping the explicitly oscillatory terms under condition (8), and the modified Hamiltonian (11) is derived as the first non-resonant correction. The comparison of the full and RWA dynamics in Figs. 2-4 is a numerical experiment, not a fit: no parameter appearing in the claimed result is extracted from the same observable that is then 'predicted'. Condition (10) is asserted as a scaling threshold and supported by numerical examples and a qualitative error-accumulation argument; it may be insufficiently derived, but it is not circular, because the inequality is not defined as 'the condition under which the tensor shift has no effect' by construction, and the numerics are independent of that assertion. References to Ref. [9] are citations to external experimental and theoretical work, not load-bearing self-citations, and no uniqueness theorem or prior result of the present authors is invoked to force the conclusion. The main limitation—that condition (10) is not rigorously proven—is a correctness or rigor concern, not a circularity concern.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim rests primarily on the Hamiltonian model from Ref. [9] and on the numerical convergence of the Fourier expansion. No free parameters are fitted, and no new entities are introduced.

assumptions (3)
  • domain assumption The Hamiltonian in Eq. (3), with tensor shift κJz^2, correctly represents LLI violation and relevant systematic shifts in trapped-ion rf spectroscopy.
    This model is taken from Ref. [9]; the paper does not derive or independently justify it.
  • domain assumption The Fourier expansion in harmonics from -F to F with F=50 converges for the pulse sequences studied.
    The authors state this truncation is sufficient 'with a good margin' but provide no convergence study; the accuracy of the exact calculation is load-bearing for the RWA breakdown claim.
  • domain assumption Decoherence, magnetic field noise, and pulse imperfections are neglected; unitary ideal-pulse dynamics are assumed.
    The comparison between full and resonant Hamiltonians is made for ideal pulses without noise; this is standard in the cited works as well but is an unstated idealization.

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Cite this review

Pith. "Pith review of Theory of composite Ramsey sequences of radiofrequency pulses beyond the rotating wave approximation." pith.science (2026). https://pith.science/paper/QAQSJCBA

@misc{pith2026250213973,
  author       = {Pith},
  title        = {Pith review of: Theory of composite Ramsey sequences of radiofrequency pulses beyond the rotating wave approximation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QAQSJCBA}},
  note         = {Machine review of arXiv:2502.13973}
}
read the original abstract

We develop a theory of composite Ramsey sequences of rf pulses interacting with the Zeeman structure at the long-lived atomic level, beyond the rotating wave approximation. Such sequences are proposed in experiments to detect the violation of local Lorentz invariance [R. Shaniv, et al., Phys. Rev. Lett. 120, 103202 (2018)]. Based on Fourier analysis, we have shown that taking into account non-resonant contributions leads to a radical change in the dynamics of the quantum system (with respect to the rotating wave approximation) in the case when the number of Ramsey pulses exceeds several tens. As a result, the effectiveness of using such rf pulses sequences to test local Lorentz invariance has not yet been fully determined and requires additional research.

Figures

Figures reproduced from arXiv: 2502.13973 by the authors.

Figure 1
Figure 1. FIG. 1. General scheme of a composite Ramsey sequence [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Calculations of the value [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 2
Figure 2. (a)]. However, as can be seen from a comparison of Figs. 2(a) and (b), the rotating wave approximation very unsatisfactorily describes the atomic dynamics for n ≫ 1. Moreover, our calculations for the full Hamiltonian (3) [or (7)] show that the small tensor contribution (κJˆ2 z ) has practically no noticeable effect if the condition is met: ~κ Ωrf ≪ τ τR Ωrf ΩL . (10) This fact is also clearly confirmed by Figs. 2(c… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: , even the use of the modified resonant approxima￾tion based on the Hamiltonian (11) does not lead to an adequate description of the atomic dynamics for compos￾ite sequence of rf pulses with n ≫ 1. In conclusion, we have considered the composite Ram￾sey sequences of rf…

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Reference graph

Works this paper leans on

28 extracted references · 27 canonical work pages

  1. [9]

    [like the green curve in Fig. 2(a)]. However, as can be seen from a comparison of Figs. 2(a) and (b), the rotating wave approximation very unsatisfactorily describes the atomic dynamics for n ≫ 1. Moreover, our calculations for the full Hamiltonian ( 3) [or (7)] show that the small tensor contribution ( κ ˆJ 2 z ) has practically no noticeable effect if th...

    work page

  2. [1]

    Ω rf τ / 2 = π/ 2) has phase φ in = 0, while the fi- nal π/ 2-pulse has a phase φ fin

    The initial π/ 2-pulse with duration τ / 2 (i.e. Ω rf τ / 2 = π/ 2) has phase φ in = 0, while the fi- nal π/ 2-pulse has a phase φ fin . Between these two π/ 2- pulses on the time scale, there are n of identical N -boxes, each of which consists of N number of π -pulses with duration τ (Ω rf τ = π ) and with corresponding phases (φ 1, φ 2, ..., φ N ). Thus, ...

    work page

  3. [2]

    Pospelov and Y

    M. Pospelov and Y. Shang, Lorentz violation in Horava- Lifshitz-type theories, Phys. Rev. D 85, 105001 (2012)

    work page 2012

  4. [3]

    In this paper, we will consider composite Ramsey se- quences of rf pulses using the dynamic decoupling tech- nique

    [or ( 7)] and compare them with cal- culations for the effective Hamiltonian ( 9). In this paper, we will consider composite Ramsey se- quences of rf pulses using the dynamic decoupling tech- nique. The general scheme of such sequences is pre- sented in Fig

    work page

  5. [4]

    Pruttivarasin, M

    T. Pruttivarasin, M. Ramm, S. G. Porsev, I. I. Tupitsyn, M. S. Safronova, M. A. Hohensee, and H. H¨ affner, Michel- son–Morley analogue for electrons using trapped ions to test Lorentz symmetry, Nature 517, 592–595 (2015)

    work page 2015

  6. [5]

    is explained by the fact that the controlled tensorial shift (quadrupole shift of the linear Paul trap) is large enough and the condition (

    work page

  7. [6]

    M. S. Safronova, D. Budker, D. DeMille, Derek F. Jack- son Kimball, A. Derevianko, and Charles W. Clark, Search for new physics with atoms and molecules, Rev. Mod. Phys. 90, 025008 (2018)

    work page 2018

  8. [7]

    and us- ing only the effective (reduced) resonant Hamiltonian ˆHres = −δ ˆJz + κ ˆJ 2 z + η(t) Ω rf 2 { eiφ (t) ˆJ− + e−iφ (t) ˆJ+ } = − δ ˆJz + κ ˆJ 2 z + η(t)Ω rf { cos φ(t) ˆJx + sin φ(t) ˆJy } . (9) .... ....... /c116 /c116/c47/c50 /c116 /c116 /c116 /c116/c47/c50 /c116R/c116R /c472 /c116R /c116R/2 /c112/c47/c50 /c112 /c112 /c47/c50/c112 /c112 /c112 /c1...

    work page

Show all 28 references
  1. [8]

    Thus, if the tensorial shift κ ˆJ 2 z is mainly determined by LLI violation, then the use of rf Ramsey sequence {(π/ 2, −π/ 2)n, φ fin }, considered in Ref

    for real experiments with n ≫ 1. Thus, if the tensorial shift κ ˆJ 2 z is mainly determined by LLI violation, then the use of rf Ramsey sequence {(π/ 2, −π/ 2)n, φ fin }, considered in Ref. [9], will not al- low to reliably detect this shift, even if it exists in reality. The e...

  2. [10]

    Indeed, our calculations for J = 5/ 2, using experimental data from the Supplemental materials to Ref

    is not satisfied. Indeed, our calculations for J = 5/ 2, using experimental data from the Supplemental materials to Ref. [9], con- firm a fairly good agreement between the rotating wave approximation and exact calculations. However, the con- tribution due to the hypothetical LLI...

  3. [11]

    parameter of non-resonant con- tributions

    is satisfied] is for- mulated at a qualitative level as follows. For example, let the ratio (Ω rf / Ω L) ∼ 0. 01 holds. Then, as follows from the Fourier analysis of the Schr¨ odinger equation ( 6) with the Hamiltonian ( 7), the contribution of higher har- monics to the wave fu...

  4. [12]

    parameter of non-resonant contributions

    under the conditions (8), it can be easily shown that taking into account the first non-resonant contributions leads to the modified res- onant Hamiltonian ˆH (mod) res = ˆHres + η(t) Ω 2 rf 4Ω L ˆJz, (11) where we used η2(t) = η(t). Thus, the expression for ˆH (mod) res differs ...

  5. [13]

    Horava, Quantum gravity at a Lifshitz point, Phys

    P. Horava, Quantum gravity at a Lifshitz point, Phys. Rev. D 79, 084008 (2009)

    2009

  6. [14]

    Cognola, R

    G. Cognola, R. Myrzakulov, L. Sebastiani, S. Vagnozzi, and S. Zerbini, Covariant Horrava-like and mimetic Horndeski gravity: cosmological solutions and perturba- tions, Class. Quantum Gravity 33, 225014 (2016)

    2016

  7. [15]

    V. A. Dzuba, V. V. Flambaum, M. S. Safronova, S. G. Porsev, T. Pruttivarasin, M. A. Hohensee, and H. 5 H¨ affner, Strongly enhanced effects of Lorentz symme- try violation in entangled Yb + ions, Nature Phys. 12, 465–468 (2016)

    2016

  8. [16]

    Megidish, J

    E. Megidish, J. Broz, N. Greene, and H. H¨ affner, Im- proved test of local Lorentz invariance from a determin- istic preparation of entangled states, Phys. Rev. Lett. 122, 123605 (2019)

    2019

  9. [17]

    Sanner, N

    C. Sanner, N. Huntemann, R. Lange, C. Tamm, E. Peik, M. S. Safronova, and S. G. Porsev, Optical clock compar- ison for Lorentz symmetry testing, Nature 567, 204–208 (2019)

    2019

  10. [18]

    Shaniv, R

    R. Shaniv, R. Ozeri, M. S. Safronova, S. G. Porsev, V. A. Dzuba, V. V. Flambaum, and H. H¨ affner, New Meth- ods for Testing Lorentz Invariance with Atomic Systems, Phys. Rev. Lett. 120, 103202 (2018)

    2018

  11. [19]

    L. S. Dreissen, Ch.-H. Yeh, H. A. F¨ urst, K. C. Grense- mann, and T. E. Mehlst¨ aubler, Improved bounds on Lorentz violation from composite pulse Ramsey spec- troscopy in a trapped ion, Nat. Comm. 13, 7314 (2022)

    2022

  12. [20]

    Viola, E

    L. Viola, E. Knill, and S. Lloyd, Dynamical Decoupling of Open Quantum Systems, Phys. Rev. Lett. 82, 2417 (1999)

    1999

  13. [21]

    Khodjasteh and D

    K. Khodjasteh and D. A. Lidar, Fault-Tolerant Quan- tum Dynamical Decoupling, Phys. Rev. Lett. 95, 180501 (2005)

    2005

  14. [22]

    Khodjasteh and D

    K. Khodjasteh and D. A. Lidar, Performance of deter- ministic dynamical decoupling schemes: Concatenated and periodic pulse sequences, Phys. Rev. A 75, 062310 (2007)

    2007

  15. [23]

    Khodjasteh, J

    K. Khodjasteh, J. Sastrawan, D. Hayes, T. J. Green, M. J. Biercuk, and L. Viola, Designing a practical high- fidelity long-time quantum memory, Nat. Comm. 4, 2045 (2013)

    2013

  16. [24]

    D. A. Lidar and T. A. Brun, Quantum Error Correc- tion (Cambridge University Press, Cambridge, England, 2013)

    2013

  17. [25]

    G. T. Genov, D. Schraft, N. V. Vitanov, and T. Half- mann, Arbitrarily Accurate Pulse Sequences for Robust Dynamical Decoupling, Phys. Rev. Lett. 118, 133202 (2017)

    2017

  18. [26]

    C.-H. Yeh, K. C. Grensemann, L. S. Dreissen, H. A. F¨ urst, and T. E. Mehlst¨ aubler, Robust and scalable rf spectroscopy in first-order magnetic sensitive states at second-long coherence time, New J. Phys. 25, 093054 (2023)

    2023

  19. [27]

    Bloch and A

    F. Bloch and A. Siegert, Magnetic Resonance for Nonro- tating Fields, Phys. Rev. 57, 522 (1940)

    1940

  20. [28]

    Zanon-Willette, E

    T. Zanon-Willette, E. de Clercq, and E. Arimondo, Magic Radio-Frequency Dressing of Nuclear Spins in High- Accuracy Optical Clocks, Phys. Rev. Lett. 109, 223003 (2012)

    2012

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