the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 11 Sep 2026
Floquet Theory and Average Hamiltonian Theory Revisited: Equivalence, Convergence and Applications to NMR
Abstract. An accurate theoretical treatment of periodically driven quantum systems is crucial for various fields in the exact sciences, for instance Nuclear Magnetic Resonance (NMR) spectroscopy. Conventionally, either average Hamiltonian theory or Floquet theory is used to predict or describe experimental outcomes, such as the time evolution or the spectra yielding the information of the sample under study. A detailed analysis of the equivalence of these two approaches with an emphasis on applications in NMR will help to improve the theoretical understanding of NMR experiments.
In this work, we identify the Floquet–Magnus expansion as essential to prove the mathematical equivalence of Floquet theory and average Hamiltonian theory. We advocate a calculation scheme which is less prone to algebraic mistakes because explicit integration is avoided. On this basis, we provide the first four orders of both theories. We further examine their applicability to some experiments in NMR. As examples, we investigate the Bloch–Siegert shift and dipolar coupled spin systems under magic-angle spinning.
Based on our analysis, we recommend the use of the Floquet–Van Vleck approach including both the effective Hamiltonian and the kick operator. The consistent separation of secular and non-secular contributions appears to be especially advantageous for numerical robustness. Its accuracy is about three times better than the one provided by average Hamiltonian theory despite their formal equivalence.
Our findings provide important insights into the theoretical background of Floquet theory and average Hamiltonian theory. This includes the extent of their algebraic and perturbative equivalence, with an emphasis on how these findings are of relevance in the analysis of solid-state NMR experiments.
Competing interests: Matthias Ernst, who is one of the (co-)authors, is a member of the editorial board of Magnetic Resonance.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors. Views expressed in the text are those of the authors and do not necessarily reflect the views of the publisher.- Preprint
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CC1: 'Comment on mr-2026-10', Tom Barbara, 24 Sep 2026
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CC2: 'Reply on CC1', Matthias Ernst, 24 Sep 2026
Dear Tom,
thank you for the positive words about our manuscript.
We hope that the analytical equations up to fourth order are helpful in situations where such higher orders are needed.
I think the reason the Bloch-Siegert shift is often not noticed stems from the fact that all lines in the spectrum shift the same amount and we typically observe only relative frequencies. And if you always decouple, you have no reference to notice the shift. However, there have been recent publications where the heteronuclear Bloch-Siegert shift was observed and even utilized to calibrate the rf-field amplitude of low-γ nuclei (I. Hung, P. Gor’kov, Z. Gan, Using the heteronuclear Bloch-Siegert shift of protons for B1 calibration of insensitive nuclei not present in the sample, J. Magn. Reson. 310 (2020) 106636–5. https://doi.org/10.1016/j.jmr.2019.106636. and E. Nehra, V. Agarwal, Y. Nishiyama, Low-power 14N decoupling at fast MAS of 70 kHz, Solid State Nucl. Magn. Reson. 137 (2025) 102006. https://doi.org/10.1016/j.ssnmr.2025.102006.). The Bloch-Siegert shift is much stronger if you irradiate (decouple) the low-γ nucleus while observing the high-γ nucleus than the other way round. In this situation, even a field of 40 kHz nutation frequency on 14N can lead to a Bloch-Siegert shift of 500 Hz on the protons. When I was last sitting at the spectrometer myself and measuring 15N-1H correlation spectra, I actually noticed for the first time the significant shift of protons when decoupling 15N during the acquisition from undecoupled spectra.
I agree that Shirley's thesis is worthwhile reading. It has a lot more context than the Phys.Rev. article from 1965.
Best regards,
Matthias
Citation: https://doi.org/10.5194/mr-2026-10-CC2 -
CC3: 'Reply on CC2', Tom Barbara, 24 Sep 2026
HI Matthias,
Thanks for mentioning those heteronuclear decoupling papers. This reminded me of how confusing things can get with the use of the terminology. Of course, once transverse magnetization is created by a resonant pulse, a very weak far off resonance irradiation can produce a phase or frequency shift. This method is used in MRI to map B1 fields and those researchers that pursue such matters always refer this as a "Bloch-Siegert" shift. I always found this aspect confusing because to me Bloch-Siegert was about the "counter rotating component", which has a very different symmetry under rotations. I can recall this confusion at the very start from the papers by Bodenhausen and Emsley. Not to long ago I became motivated to write up a short note about this because some coworkers were interested in mapping B1 of a head coil for out 7T whole body scanner and wanted to know how it worked. If you are interested, I have added my note as a supplement. The calculation appears to give the same result as Hung, Gorkov and Gan who mentioned in the introduction the "AC stark effect" and give the expression w1^2/2*(delta).
Thanks again for an interesting discussion and the references.
-
CC3: 'Reply on CC2', Tom Barbara, 24 Sep 2026
-
CC2: 'Reply on CC1', Matthias Ernst, 24 Sep 2026
-
RC1: 'Comment on mr-2026-10', Anonymous Referee #1, 27 Oct 2026
In solid-state NMR the spin Hamiltonian is frequently periodic in time because of magic-angle spinning (MAS), radio-frequency (rf) irradiation or repeated pulse sequences. Two perturbative approaches are used to obtain a simple, time-independent description of such systems: AHT (Magnus expansion over one cycle; “coherent averaging”) and Floquet theory combined with Van Vleck perturbation theory (FVV; “secular averaging”). Practitioners often find that the two appear to give different expressions at second and higher order, and it is not always clear why, or which is preferable. The authors set out to (a) show precisely how the two approaches are related, (b) provide reliable expressions up to fourth order, and (c) compare their accuracy on representative NMR problems. The question that the manuscript addresses: how Average Hamiltonian Theory (AHT) and Floquet theory are related, and which one to use in practice for magnetic resonance experiments. This manuscript also provides a single reference that collects the relations and the higher-order expressions in a consistent notation and should be useful to many readers. Overall I would rate the manuscript highly but I do have a few concerns and comments that should be potentially addressed by the authors before publication.
Comments:
- The manuscript states at different places: The abstract says that FVV’s “accuracy is about three times better than the one provided by average Hamiltonian theory despite their formal equivalence”; Section 3.3 reports an average ratio ΔAHT/ΔFVV ≈ 3.1, and the conclusions describe FVV as “more robust, reliable and efficient”.
This factor is obtained from random test Hamiltonians (uniform coefficients, cut-off C = 5 harmonics, Appendix C) and two specific error measures. The authors attribute the difference to FVV retaining the kick operator as an exact rotation while AHT truncates it. This explains why the two approaches differ and makes FVV’s advantage plausible, but it does not predict the size of the factor, it’s in/dependence on t₀ and on the test ensemble, or that FVV is always the more accurate. As phrased in the abstract, the statement nevertheless reads like a general property of the two methods. It’s not clear if this factor of 3 is so or a result of the test case under consideration.
Fig. 3 itself shows how variable is the ΔAHT/ΔFVV factor. The plateau heights of ΔAHT/ΔFVV range from about 1.5 to about 14-16 (2-dim eigenvalues, third order), with no consistent trend with order or dimension, and the spread is largest for the smallest matrices. Why should the error scale or specifically in this scenario reduce with the dimension of the matrix is not so clear?
Is there a possible regime (possibly larger dimension matrix) in which the error would eventually collapse to ~1, if so should be discussed.
I think most of the x-range (ω/ρ ≈ 10²–10⁶) lies far beyond experimental conditions; for the MAS examples, ω/ρ ≈ 2–10, i.e. the left edge of the plots, Could the authors provide some context here, on the relevant regimes for NMR.
Figure 3: The manuscript does not state how many random Hamiltonians were sampled per dimension (Fig. 2 suggests a single realisation), which t₀ was used, or how many points were averaged to generate Figure 3. The authors could provide all details to exactly reproduce figure 3.
- In NMR, AHT is very often applied with the cycle chosen time-symmetric about its midpoint, H(t₀ + τ) = H(t₀ + T − τ). In such a scenario all terms with odd powers of 1/ω, in the authors’ notation F⁽²⁾, F⁽⁴⁾, …, then vanish identically. Thus FVV’s ability to exploit symmetry without a special choice of t₀ is a genuine advantage. I am wondering if it possible to include ΔAHT/ΔFVV error calculations systematically for time-symmetric sequences to highlight the errors, the behaviour of the kick operator and potential implications of the errors if they are significant?
- Figure 4: It’s very difficult to see the intensity and position changes for the single crystal in two different orientation. Perhaps zoom in plots might help giving the reader a better idea about change. Is the reported error only change in intensity or also change in position. It appears that the position of the lines also shift, how large are the shifts.
- In Appendix D the authors state that the small effect of the kick operator on stroboscopic MAS spectra “cannot be understood based on the analytical framework”, and they list it as an open question. For powder samples, I believe it follows from a basic property of MAS.
Under MAS, a crystallite rotated by the angle γ about the rotor axis is physically the same as the γ = 0 crystallite with a different to; as γ enters only through the factor e^{ik(ωᵣt − γ)}. Averaging over γ is therefore the same as averaging over the starting time t₀. In the FVV picture the effective Hamiltonian F does not depend on t₀, so all these crystallites have the same line positions and differ only through the kick Λ(t₀). By construction, the FVV kick averages to zero over one rotor period. Since the powder average samples γ uniformly the leading effect of the kick cancels in the powder spectrum, leaving only a much smaller second-order remainder. Does this sound reasonable or maybe it’s more complicated and I missed something fundamental ?
5. The conclusions recommends FVV as “the best generic choice for solid-state NMR”, The equations in of Section 3 assume that the modulation is much faster than every interaction, including H₀; all denominators are of the form nω. Many of the most important uses of Floquet theory in solid-state NMR are the opposite case: experiments are generally at a resonance condition, such as rotational resonance (Δωiso = nωᵣ) or rf/MAS recoupling sequences. There, two frequencies nearly match, denominators like (nωᵣ − Δω) approach zero, and the expansions presented here break down. Those are exactly the situations in which the choice between AHT and Floquet theory matters most in NMR, and they would require a multiple-frequency Floquet treatment that the paper does not attempt. I feel adding a short discussion of the different order terms at resonance conditions, maybe just rotational resonance, should help in establishing the relevance of the conclusions in the manuscript.
- Line 535: “Based on our exemplary calculation……” I agree the manuscript is exemplary but I think the calculations are really for model systems.
- Reference: Haeberlen and Waugh (1968a) and (1968b) are the same paper;
- Reference: the ISBN given for Mehring (1983) is identical to that of Haeberlen (1976).
Citation: https://doi.org/10.5194/mr-2026-10-RC1 -
RC2: 'Comment on mr-2026-10', Anonymous Referee #2, 27 Oct 2026
The comment was uploaded in the form of a supplement: https://mr.copernicus.org/preprints/mr-2026-10/mr-2026-10-RC2-supplement.pdf
Status: closed (peer review stopped)
-
CC1: 'Comment on mr-2026-10', Tom Barbara, 24 Sep 2026
This is a very useful effort to have available, especially the comparison with Average Hamiltonian Theory. The calculation out to 3rd order for Bloch Siegert is very revealing. Since I worked on micro-coil NMR and had read about some groups that enjoyed 1 MHz Rabi frequencies, I often wondered why no one observed such shifts. On a historical note, this inspired me to look up J.H. Shirley's Cal Tech thesis, which I borrowed via inter library loan when I was a post doc at UCSD. Now of course, you can get it right away via the Cal Tech web site!
Citation: https://doi.org/10.5194/mr-2026-10-CC1 -
CC2: 'Reply on CC1', Matthias Ernst, 24 Sep 2026
Dear Tom,
thank you for the positive words about our manuscript.
We hope that the analytical equations up to fourth order are helpful in situations where such higher orders are needed.
I think the reason the Bloch-Siegert shift is often not noticed stems from the fact that all lines in the spectrum shift the same amount and we typically observe only relative frequencies. And if you always decouple, you have no reference to notice the shift. However, there have been recent publications where the heteronuclear Bloch-Siegert shift was observed and even utilized to calibrate the rf-field amplitude of low-γ nuclei (I. Hung, P. Gor’kov, Z. Gan, Using the heteronuclear Bloch-Siegert shift of protons for B1 calibration of insensitive nuclei not present in the sample, J. Magn. Reson. 310 (2020) 106636–5. https://doi.org/10.1016/j.jmr.2019.106636. and E. Nehra, V. Agarwal, Y. Nishiyama, Low-power 14N decoupling at fast MAS of 70 kHz, Solid State Nucl. Magn. Reson. 137 (2025) 102006. https://doi.org/10.1016/j.ssnmr.2025.102006.). The Bloch-Siegert shift is much stronger if you irradiate (decouple) the low-γ nucleus while observing the high-γ nucleus than the other way round. In this situation, even a field of 40 kHz nutation frequency on 14N can lead to a Bloch-Siegert shift of 500 Hz on the protons. When I was last sitting at the spectrometer myself and measuring 15N-1H correlation spectra, I actually noticed for the first time the significant shift of protons when decoupling 15N during the acquisition from undecoupled spectra.
I agree that Shirley's thesis is worthwhile reading. It has a lot more context than the Phys.Rev. article from 1965.
Best regards,
Matthias
Citation: https://doi.org/10.5194/mr-2026-10-CC2 -
CC3: 'Reply on CC2', Tom Barbara, 24 Sep 2026
HI Matthias,
Thanks for mentioning those heteronuclear decoupling papers. This reminded me of how confusing things can get with the use of the terminology. Of course, once transverse magnetization is created by a resonant pulse, a very weak far off resonance irradiation can produce a phase or frequency shift. This method is used in MRI to map B1 fields and those researchers that pursue such matters always refer this as a "Bloch-Siegert" shift. I always found this aspect confusing because to me Bloch-Siegert was about the "counter rotating component", which has a very different symmetry under rotations. I can recall this confusion at the very start from the papers by Bodenhausen and Emsley. Not to long ago I became motivated to write up a short note about this because some coworkers were interested in mapping B1 of a head coil for out 7T whole body scanner and wanted to know how it worked. If you are interested, I have added my note as a supplement. The calculation appears to give the same result as Hung, Gorkov and Gan who mentioned in the introduction the "AC stark effect" and give the expression w1^2/2*(delta).
Thanks again for an interesting discussion and the references.
-
CC3: 'Reply on CC2', Tom Barbara, 24 Sep 2026
-
CC2: 'Reply on CC1', Matthias Ernst, 24 Sep 2026
-
RC1: 'Comment on mr-2026-10', Anonymous Referee #1, 27 Oct 2026
In solid-state NMR the spin Hamiltonian is frequently periodic in time because of magic-angle spinning (MAS), radio-frequency (rf) irradiation or repeated pulse sequences. Two perturbative approaches are used to obtain a simple, time-independent description of such systems: AHT (Magnus expansion over one cycle; “coherent averaging”) and Floquet theory combined with Van Vleck perturbation theory (FVV; “secular averaging”). Practitioners often find that the two appear to give different expressions at second and higher order, and it is not always clear why, or which is preferable. The authors set out to (a) show precisely how the two approaches are related, (b) provide reliable expressions up to fourth order, and (c) compare their accuracy on representative NMR problems. The question that the manuscript addresses: how Average Hamiltonian Theory (AHT) and Floquet theory are related, and which one to use in practice for magnetic resonance experiments. This manuscript also provides a single reference that collects the relations and the higher-order expressions in a consistent notation and should be useful to many readers. Overall I would rate the manuscript highly but I do have a few concerns and comments that should be potentially addressed by the authors before publication.
Comments:
- The manuscript states at different places: The abstract says that FVV’s “accuracy is about three times better than the one provided by average Hamiltonian theory despite their formal equivalence”; Section 3.3 reports an average ratio ΔAHT/ΔFVV ≈ 3.1, and the conclusions describe FVV as “more robust, reliable and efficient”.
This factor is obtained from random test Hamiltonians (uniform coefficients, cut-off C = 5 harmonics, Appendix C) and two specific error measures. The authors attribute the difference to FVV retaining the kick operator as an exact rotation while AHT truncates it. This explains why the two approaches differ and makes FVV’s advantage plausible, but it does not predict the size of the factor, it’s in/dependence on t₀ and on the test ensemble, or that FVV is always the more accurate. As phrased in the abstract, the statement nevertheless reads like a general property of the two methods. It’s not clear if this factor of 3 is so or a result of the test case under consideration.
Fig. 3 itself shows how variable is the ΔAHT/ΔFVV factor. The plateau heights of ΔAHT/ΔFVV range from about 1.5 to about 14-16 (2-dim eigenvalues, third order), with no consistent trend with order or dimension, and the spread is largest for the smallest matrices. Why should the error scale or specifically in this scenario reduce with the dimension of the matrix is not so clear?
Is there a possible regime (possibly larger dimension matrix) in which the error would eventually collapse to ~1, if so should be discussed.
I think most of the x-range (ω/ρ ≈ 10²–10⁶) lies far beyond experimental conditions; for the MAS examples, ω/ρ ≈ 2–10, i.e. the left edge of the plots, Could the authors provide some context here, on the relevant regimes for NMR.
Figure 3: The manuscript does not state how many random Hamiltonians were sampled per dimension (Fig. 2 suggests a single realisation), which t₀ was used, or how many points were averaged to generate Figure 3. The authors could provide all details to exactly reproduce figure 3.
- In NMR, AHT is very often applied with the cycle chosen time-symmetric about its midpoint, H(t₀ + τ) = H(t₀ + T − τ). In such a scenario all terms with odd powers of 1/ω, in the authors’ notation F⁽²⁾, F⁽⁴⁾, …, then vanish identically. Thus FVV’s ability to exploit symmetry without a special choice of t₀ is a genuine advantage. I am wondering if it possible to include ΔAHT/ΔFVV error calculations systematically for time-symmetric sequences to highlight the errors, the behaviour of the kick operator and potential implications of the errors if they are significant?
- Figure 4: It’s very difficult to see the intensity and position changes for the single crystal in two different orientation. Perhaps zoom in plots might help giving the reader a better idea about change. Is the reported error only change in intensity or also change in position. It appears that the position of the lines also shift, how large are the shifts.
- In Appendix D the authors state that the small effect of the kick operator on stroboscopic MAS spectra “cannot be understood based on the analytical framework”, and they list it as an open question. For powder samples, I believe it follows from a basic property of MAS.
Under MAS, a crystallite rotated by the angle γ about the rotor axis is physically the same as the γ = 0 crystallite with a different to; as γ enters only through the factor e^{ik(ωᵣt − γ)}. Averaging over γ is therefore the same as averaging over the starting time t₀. In the FVV picture the effective Hamiltonian F does not depend on t₀, so all these crystallites have the same line positions and differ only through the kick Λ(t₀). By construction, the FVV kick averages to zero over one rotor period. Since the powder average samples γ uniformly the leading effect of the kick cancels in the powder spectrum, leaving only a much smaller second-order remainder. Does this sound reasonable or maybe it’s more complicated and I missed something fundamental ?
5. The conclusions recommends FVV as “the best generic choice for solid-state NMR”, The equations in of Section 3 assume that the modulation is much faster than every interaction, including H₀; all denominators are of the form nω. Many of the most important uses of Floquet theory in solid-state NMR are the opposite case: experiments are generally at a resonance condition, such as rotational resonance (Δωiso = nωᵣ) or rf/MAS recoupling sequences. There, two frequencies nearly match, denominators like (nωᵣ − Δω) approach zero, and the expansions presented here break down. Those are exactly the situations in which the choice between AHT and Floquet theory matters most in NMR, and they would require a multiple-frequency Floquet treatment that the paper does not attempt. I feel adding a short discussion of the different order terms at resonance conditions, maybe just rotational resonance, should help in establishing the relevance of the conclusions in the manuscript.
- Line 535: “Based on our exemplary calculation……” I agree the manuscript is exemplary but I think the calculations are really for model systems.
- Reference: Haeberlen and Waugh (1968a) and (1968b) are the same paper;
- Reference: the ISBN given for Mehring (1983) is identical to that of Haeberlen (1976).
Citation: https://doi.org/10.5194/mr-2026-10-RC1 -
RC2: 'Comment on mr-2026-10', Anonymous Referee #2, 27 Oct 2026
The comment was uploaded in the form of a supplement: https://mr.copernicus.org/preprints/mr-2026-10/mr-2026-10-RC2-supplement.pdf
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This is a very useful effort to have available, especially the comparison with Average Hamiltonian Theory. The calculation out to 3rd order for Bloch Siegert is very revealing. Since I worked on micro-coil NMR and had read about some groups that enjoyed 1 MHz Rabi frequencies, I often wondered why no one observed such shifts. On a historical note, this inspired me to look up J.H. Shirley's Cal Tech thesis, which I borrowed via inter library loan when I was a post doc at UCSD. Now of course, you can get it right away via the Cal Tech web site!