the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 02 Apr 2026
Optimally controlled NMR in electrochemistry: Larmor and nutation frequency selective spin excitation for locally selective NMR experiments
Abstract. Spectroelectrochemical nuclear magnetic resonance (NMR) experiments are faced with numerous challenges originating from shielding effects and susceptibility gradients in samples, leading to inhomogeneous static magnetic fields B0 and radio frequency (rf) fields B1. Moreover, magnetic feedback caused by eddy currents in conductors can obstruct precise measurements. Previous works have shown that these eddy current induced magnetic field distortions can be accurately predicted by finite element method (FEM) simulations. In this work, we present a workflow combining FEM predictions with quantum optimal control (QOC) to tailor custom NMR pulses that exploit specific magnetic field distortions for selective excitation of affected sample regions. The desired selectivity was achieved using pattern pulses optimized for a particular B1 or Larmor frequency ν0. Experimental validation was performed on a heterogeneous phantom consisting of two cavities filled with two spectroscopically distinguishable liquids, one between copper disks to mimic an electrochemical cell, and one between polymer disks as reference. An over 30-fold suppression of the reference resonance in between polymer compared to the resonance in between copper disks was achieved, demonstrating how QOC-tailored pulses can selectively address FEM-predicted B1 distortions to achieve spatial selectivity. It was also demonstrated how QOC-tailored pulses can selectively excite specific ν0 despite of B0 distortions, which implies that difficulties with conventional solvent suppression techniques in electrochemical setups can be mitigated using the adjustable robustness of QOC-tailored pulses. The presented approach sets the stage for gradient-free, localized in operando NMR in electrochemistry and material sciences, with the prospect of surface selectivity down to the detection limit of the setup.
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RC1: 'Comment on mr-2026-7', Anonymous Referee #1, 17 Apr 2026
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AC1: 'Reply on RC1', Simone Koecher, 12 Jun 2026
We thank the reviewer for their concise summary of our work and in particular for their extensive review of the
draft as well as their detailed and productive comments. We worked actively to improve the presentation, to make the text
easier to read, and to highlight the key messages more clearly (see following responses for details).Please refer to the attached pdf for consistent formatting and figures of the response. As advised by MR, we are not attaching the revised manuscript, which will be released at a later date.
1. I am not an expert on OC pulses so I cannot judge how demanding the generation of such pulses is. My impression is that
this is quite standard and that the application to electrochemical problems is the new part. My expectation from the title and the
first reading och Chapter 2.1, I expected the pulses to be simultaneously selective in frequency and nutation space. The title says
"Larmor and nutation frequency selective pulses". After a more careful reading, I think now that these are frequency selective
pulses that are tolerant to deviations in the nutation frequency and nutation selective pulses that are insensitive to changes in
the resonance frequency. If this is correct, I think this should be made clearer in phrasing of the title and the text. Later on,
the paper always talks about B0 selective or nutation-frequency selective pulses. I just realized that the phrasing in the SI is
much clearer: B1-robust selective excitation within ±500 Hz in a ±2000 Hz suppression band" and "Larmor frequency-robust
selective excitation for 0% artificial nutation frequency increase". It would be nice if the phrasing in the main text and title
would reflect that precision of the SI. By the way, I think the use of the term "B0 selective" pulses is not a good choice since B0
by definition is the same everywhere. I think it would be good to use a different term here like resonance frequency selective
pulses.Response:
The reviewer is correct. Neither ν0 -, nor ν1 -tailored pulses, nor the combination are new in the NMR optimal control
community as we detailed in the Introduction. So far, B0 and B1 field distortions due to metallic components have mostly
been considered in order to minimize and compensate for them. The new aspect presented in our work, is exploiting the
B1 distortions near electric conductors to achieve spatial and ultimately surface selectivity for in operando electrochemical
catalysis applications while accounting for B0 distortions, too.
We agree that the phrasing in places was not clear about which frequency (or field inhomogeneity) the respective pulses are
robust or sensitive to. We revised the text (and title) with respect to this point. In particular, we included an additional paragraph
at the beginning of the Results and Discussion section and rephrased the subtitles.
We consistently replaced B0 -selective/-robust with ν0 -selective/-robust, and analogously for B1 and ν1 , to characterize the
pulses.
Action:
We revised the title and sections of the text.Revised Paragraph:
New title: Optimally controlled NMR in electrochemistry: Larmor and versus nutation frequency selective spin excitation
for locally selective NMR experiments
Results and Discussion
QOC pattern pulse design is capable of providing either selectivity or robustness with respect to both ν0 and ν1 independently.
Each variation is useful for certain applications. Selectivity with respect to the Larmor frequency allows for selective measurement
of specific spin species, which enables solvent suppression or the suppression of other dominating bulk signals to increase
sensitivity with respect to minority spin species. In contrast, ν0 robustness facilitates quantitative, phase-stable measurements
despite B0 inhomogeneities. Robustness with respect to the nutation frequency ν1 allows for the uniform, quantitative excitation
within an electrochemical cell despite B1 -distorting metallic components. On the other hand, ν1 selectivity potentially promises
to enable spatial selectivity to B1 -distorting metallic surfaces.
With electrochemical applications in mind, we discuss in the following ν1 -robust, ν0 -selective pulses, which will enable
efficient solvent suppression even in the presence of B1 -distorting electrical conductors. Secondly, we demonstrate ν0 -robust,
ν1 -selective excitation in order to achieve spatial selectivity in proximity to electrical conductors independent of their B0 -
distorting effects.2. I was wondering how much the improvement of these pulses are compared to "standard" frequency band selective pulses
like BURP pulses (H. Geen, R. Freeman, J. Magn. Reson. 93 (1991) 93–141) or nutation-frequency selective pulses in the
rotating frame (K. Aebischer, N. Wili, Z. Tosner, M. Ernst, Magn. Reson. 1 (2020) 187–195) that also have some inherent
nutation or frequency frequency bandwidth. Can the authors comment, please, how important the simultaneous optimization
for both parameters is for this application?Response:
The results based on GRAPE-optimized QOC pulses were compared against E-BURP pulses on a freshly constructed model
setup. The superior performance of the QOC approach is illustrated in a comparison of two waterfall spectra (analogously to
Figure 3 and 4), detailing the application of QOC (a) versus E-BURP (b) pulses. Figure (a) shows again an efficient excitation
within the selective bandwidth of the QOC pulse and a suppression of resonances outside of this bandwidth. While the E-BURP
pulses in Figure (b) also manage to provide decently selective excitation of n-dodecane, its suppression when selectively
exciting H2O is less efficient. Additionally whenever selectively exciting n-dodecane, its peak shape seems to be affected
uncontrollably, while E-BURP pulses never affect the peak shape of H2O. Finally, the application of E-BURP pulses heavily
distorts the baseline for some resonance frequency offsets. The most important individual spectra of the experiments (on-
resonance pulses for n-dodecane and H2O) were summarized in Figure S28 and added to the SI.We assume that the poorer performance of E-BURP lies indeed in the lack of robustness against nutation frequency deviations,
as strong residuals appear predominantly when attempting suppression of the signal associated with the Cu cavity. The BURP
excitation pulses as originally published by Geen and Freeman (1991), therefore, seem to be only suited for Larmor-frequency
selective measurements with unperturbed B1 , since suppression of the signal associated with the PEEK cavity was still
satisfactory. In summary, the robustness against nutation frequency deviations enabled with QOC pulses provides a drastic
improvement in selectivity for measurements in the vicinity of conductive surfaces.Further reasons for using GRAPE over BURP regard pulse length. Pulses optimized in a Fourier basis tend to require
longer pulse durations τ due to less flexibility in the pulse shape. For example, an excitation band of ±500 Hz around the
irradiation frequency, as used in our paper, corresponds to a BURP-1 pulse length of 4 ms according to the BURP-1 design
rule 0 ≤ νexc ≤ 2/τ . Since GRAPE can optimize pulses with arbitrary shape within the piecewise-constant treatment, a shorter
pulse duration of 1 ms was sufficient, while providing robustness against nutation frequency deviations on top. Shorter pulse
durations are crucial for future applications in the electrochemical context. In particular, in operando applications and their
temporal resolution depend critically on the duration of the individual pulse sequence as well as the selectivity and efficiency
of the excitation (and suppression), as the latter one impacts how many repetitions have to be conducted to achieve a reasonable
signal-to-noise. Furthermore, relaxation or dephasing may be enhanced in the vicinity of electrodes, which leads to broad
4lines. Having excitation pulses that are as short as possible to achieve a required selectivity is crucial to minimize the T2 or
T2⋆ weighting effects. In addition, GRAPE-optimized pulses can have multiple excitation bands, which might find applications
for CO2 reduction producing various different solvated products in low concentrations, such as CO, methanol, ethanol, formic
acid, etc.
Nutation-frequency selective inversion pulses in the spin-lock frame as published by Aebischer et al. (2020) can be adapted
for excitation experiments. Again, we chose GRAPE due to its flexibility. For example, the nutation frequency band targeted
by the pulses in the spin-lock frame is defined by the pulse length. In GRAPE, the parameters can be tuned more freely.
Furthermore, GRAPE-optimized pattern pulses can excite multiple nutation frequency bands, which may be required in future
experiments, as both increases and decreases in B1 intensity have been observed in the proximity of conductive surfaces.
Simultaneous optimization for both Larmor frequency and nutation frequency might become relevant for future spectroelectrochemical
experiments with more complex sample mixtures, as well as more sophisticated electrochemical cells or conductive interface
properties. Combinability of both parameters has been shown in the original work on pattern pulses (Kobzar et al., 2005).Action:
Testing of E-BURP pulses and additions in introduction, conclusion as well as in the SI.
Revised Paragraph:
Introduction
The rf modulation impedes the effectiveness of established selective pulse sequences such as BURP, which are not optimized
for systems with inherently distorted B1 fields Geen and Freeman (1991).
Conclusion
To support this claim, a comparison of QOC excitation pulses and E-BURP pulses on the herein presented model setup
was undertaken (SI Fig. S28). The comparison evidently visualized, that when using literature-known ν0 -selective pulses
without ν1 robustness for electrochemical setups, baseline distortions and unsatisfactory selectivity may occur due to strong
B1 field distortions near conductive cell components. Furthermore, the B1 field distortions near conductive electrochemical
cell components in the model setup were accurately predicted by FEM and integrated into a QOC workflow to tailor pattern
pulses which exploit the simulated sharp B1 enhancement near conductive interfaces.3. A second more general point is the use of the term "quantum optimal control" for the generation of the pulses. What is
"quantum" about the optimal control methods used to generate pulses on a classical computer? I realize that "quantum" is
very trendy and sells well but I really think that "optimal control" would be sufficient here.Response:
The term “quantum optimal control” indicates that the equations of motion which are integrated during the
optimization process are quantum mechanical. The method of “optimal control” is not bound to quantum mechanical problems,
but is also applied in engineering and economics. Thus, the specification of “quantum optimal control” is not trivial and the
term has been in use in the quantum control scientific community since at least the 90s (Zhu and Rabitz, 1999). It is true that
within the magnetic resonance literature mainly the term “optimal control” has been used. But we argue that even within the
field of magnetic resonance, this distinction is warranted, considering that optimal control can be applied to both the classical
Bloch equations and quantum master equations.4. The presentation of the results in Figs. 3 and 4 using double color coding of the spectra and the background is not easy
to understand and interpret. Would it not be much simpler to plot the theoretical excitation profile over the spectra with a little
arrow or dot indicating the irradiation frequency. What is especially grating is the fact that the color coding on the right goes
to -1 but is in reality never smaller than 0.Response:
We replaced the double color coding with an additional y-axis specifying the accurate frequency offset. Furthermore, a
green dot and arrow now mark the irradiation frequency of each measurement. We decided against removing or changing
the color gradient for the excitation profile. First, it depicts clearly that the transition from excitation to suppression is not
instantaneous but continuous in behavior, despite the distinct transition from excitation to suppression specified for the QOC
optimization. Second, we did not change the limits of the color bar. The light blue coloring of certain areas show, that the
theoretical x-magnetization does go into the negative range (about −0.1). Furthermore, the physical limits of the theoretical
x-magnetization are given by [−1, 1] defining the limits of our color bar.Revised Paragraph:
Results and Discussion
The revised Figure 3 and the corresponding text excerpt are shown as an example. The other figures and excerpts were
revised accordingly.
Here, ∆ν0 was varied with a step size of 200 Hz between 2975 Hz and 775 Hz (marked with a green dot and arrow), with
the excitation centered on either the resonance of H2O at ∆ν0 = 2375 Hz or on the resonance of n-dodecane at ∆ν0 = 1375 Hz.5. Fig. 2 and text: Why use ppm for the frequency axis if the ppm are meaningless. I think here a Hz scale would be much
more meaningful since also the selectivity of the pulses was specified in Hz. (applies also to Figs. 3,4 and 7)Response:
We added a second x-axis, too, for simultaneous ppm and Hz labeling. The unit ppm is the standard, relative unit for NMR
spectra and Hz is more in focus here, since the pulses are optimized for specific absolute frequencies in Hz.6. Fig. 2: Add a z scale to the drawing on the left to make the mapping from the drawing to the plot of the spectra clearer.
Maybe even scale the drawing and the spectral plot such that the scale is the same.Response:
Thanks for pointing out the shortcomings of the figure. We have revised it.
It becomes obvious when comparing the z-axis of the phantom setup and the spatially encoded chemical shift imaging,
that the signals of n-dodecane and H2 O are not restricted to the 0.1 mm corresponding to the copper and PEEK cavities
respectively. We hypothesize, that the presence of the solvent signals outside of the z-coordinates associated with the cavities
might originate from one of the following reasons. First, the solvents could obviously leak out of their cavities despite their
immiscibility. Second, the magnetic field distortions due to the copper do not only impact B0 and B1, but naturally also the
applied gradients for spatial resolution, hence corrupting the mapping of resonance frequency to spatial coordinate.7. Maybe Figures 1 and 2 could be combined in a single figure that contains all the details.
Response:
We agree with the reviewer that Figure 1 and 2 could be combined. We merged the figures and added some details that help
with the clarity of the figure.
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AC1: 'Reply on RC1', Simone Koecher, 12 Jun 2026
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RC2: 'Comment on mr-2026-7', Anonymous Referee #2, 07 May 2026
In this article, the authors propose use of specially prepared RF pulses to selectively excite signals from two model liquids (water and n-dodecane) placed in a specially designed phantom made of plastic PEEK and copper disks. According to the authors, this phantom simulates an electrochemical cell for potentially conducting NMR measurements during electrochemical reactions directly within the NMR sprobe. Indeed, such experiments, for example, electrolysis or charging of chemical energy storage devices, are very interesting from an NMR perspective. However, as presented in the introduction, such measurements in complex heterogeneous structures using conductive metals, as well as other materials with magnetic susceptibilities different from those of the solvents, dramatically disrupt the homogeneity of the magnetic fields B0 and B1. Research aimed at finding solutions to these problems is extremely relevant.
The authors propose using the finite element method in conjunction with quantum optimal control to calculate specially prepared RF pulses for a specific phantom geometry or future electrochemical cell. This allows for selective signal excitation, for example, near electrodes or within the cell volume.
As a result of their work, the authors demonstrated the feasibility of their approach, specifically, they demonstrated that special pulses can selectively excite water protons in a PEEK cell and dodecane protons in a copper cell. They also demonstrated that signals from these liquids can be not only excited but also selectively suppressed in such a heterogeneous phantom. Convincing spectra and numerical parameters of the selectivity of these pulses are presented.
This work appears to be based on careful research and relates to the highly specialized field of NMR resonance. It demonstrates proof-of-concept of this approach rather than a ready-made algorithm for general use. Obviously, the pulses shown on Figures in the ESI will only work for the phantom presented in the article. I believe the work is worth publishing in the journal Magnetic Resonance Copernicus after minor revisions.
As I read, I had some questions about the work:
- Figure 1 shows a diagram and a 3D view of the phantom. It needs to be improved. Why did you choose a huge font for the dimensions? Please make the font appropriate. The overall design isn't immediately clear; I think it would be helpful to show a three-dimensional cutout in the 3D view, like a slice of a pie, so the cell structure is clear. I would highlight the different liquids by different colors and label them in this Figure. Also, it would be helpful to show the location of the RF coil to scale compared to the phantom.
- How were the liquids added to the cells? Were there any problems with air gaps? Could air gaps or bubbles also pose problems with B0 inhomogeneity?
- Line 152 describes the nutation experiments. It's not entirely clear what RF pulses were used for the nutation experiments. Specially optimized ones or simple hard rectangular ones. This should be specified.
- Experiments with B0 field selectivity. They're generally clear, but my personal opinion is that before using special pulses, it would be worth demonstrating how standard selective pulses work for this phantom, such as a Gaussian or E-burb pulse, or others available in Topspin. Then it would be clearer to what extent special pulses are needed.
- Figure 5 is about B1 selectivity. This graph is completely unclear. What is shown on the Y ordinate axis? How this graph or spectrum was obtained is unclear. The nutation frequency is shown in kHz at the top, and the B1 field in a.u. units at the bottom. I believe that if the nutation frequency for protons is known, then the B1 magnetic field magnitude can be expressed in microtesla.
- Figure C19 in the supplementary materials shows water on a gray background, although it's most likely represented by blue disks. It would be more accurate to indicate it with arrows or another method. At first glance, it appears that the entire gray area around it consists of water.
- A general question regarding images c1-c4. It's clear that the amplitude of B1 over time has some high-frequency noise. How important is this noise? What determines the amplitude of this noise? What would happen if, instead of an amplitude pulse with high-frequency noise, we applied a pulse without this noise, but only with an envelope, after applying a low-pass filter?
- Line 51. The abbreviation PEM is not revealed.
Citation: https://doi.org/10.5194/mr-2026-7-RC2 -
AC2: 'Reply on RC2', Simone Koecher, 12 Jun 2026
We thank the reviewer for their thorough summary of our work and their detailed, productive comments.
The reviewer is correct, that this work presents a proof-of-concept for a rather specific FEM-QOC application for a particular,
well defined phantom setup. However, the methodology and workflow can be readily adopted to different cell geometries,
solvents, field strengths, and applications.Please refer to the attached pdf for consistent formatting and figures of the response. As advised by MR, we are not attaching the revised manuscript, which will be released at a later date.
1. Figure 1 shows a diagram and a 3D view of the phantom. It needs to be improved. Why did you choose a huge font for
the dimensions? Please make the font appropriate. The overall design isn’t immediately clear; I think it would be helpful to
show a three-dimensional cutout in the 3D view, like a slice of a pie, so the cell structure is clear. I would highlight the different
liquids by different colors and label them in this Figure. Also, it would be helpful to show the location of the RF coil to scale
compared to the phantom.Response:
Thanks for the feedback on Fig. 1. Due to the cylindrical symmetry of the phantom, we do not see any benefit in three-
dimensional cutout presentation suggested. We agree, that Fig. 1 is badly proportioned in particular with respect to the size of
the labels. Instead of the RF coil, we depicted the homogenously NMR-sensitive volume which is centered between the lower
copper and upper PEEK coin of the model setup and extends ±4 mm in z-direction along the added z-axis.2. How were the liquids added to the cells? Were there any problems with air gaps? Could air gaps or bubbles also pose
problems with B0 inhomogeneity?Response:
The individual cavities were assembled separately. The components were immersed and assembled in the respective liquids,
i.e. copper coins and PEEK separator in n-dodecane and PEEK coins and PEEK separator in H2 O. The sandwiched coins filled
and coated with the respective solvent were removed from the liquid reservoir and combined within the NMR tube. Due to
the surface tension of the liquids to the respective material, the coins adhered to the liquid and thus to another. Also, due to
the small volume of the sandwiched liquid, any bubble or air gap would be very small on an absolute scale (max. 100 µm
diameter), but simultaneously also counteract the adhesion between the coins, thus making them immediately noticeable in the
cell construction. In conclusion, the formation of air gaps or bubbles cannot be rigorously eliminated, but their presence in the
setup is highly unlikely.
Nonetheless, we would like to highlight that air bubbles potentially impact the B0 homogeneity as has been demonstrated
by Schatz et al. (2024) for air bubbles trapped underneath copper. We attempted to test the impact of air bubbles on a setup
just consisting of PEEK coins and water. However, the susceptibility gradients across the microscopic liquid filled volume
surrounded by polymer dominated the linewidth of the water signal (15 Hz full width at half maximum, 400-600 Hz of peak
base width for a sample without air gaps), making the assessment of the air bubble impact on B0 homogeneity difficult.3. Line 152 describes the nutation experiments. It’s not entirely clear what RF pulses were used for the nutation experiments.
Specially optimized ones or simple hard rectangular ones. This should be specified.Response:
Thank you for pointing out this missing detail.
Action:
In the methods section, we added the specification that the nutation experiments were recorded using rectangular
pulses. Furthermore, the original Bruker files for the experiment are included in the dataset (file acqus in
NMR_data/Exp_nut_spectrum.rar) that is to be published alongside the paper.Revised Paragraph:
Nutation experiments were performed with rectangular pulses of varying pulse length and a constant
pulse power of 1.5 W. In total, 100 pulse lengths were screened with a step size
of 30 µs.4. Experiments with B0 field selectivity. They’re generally clear, but my personal opinion is that before using special pulses,
it would be worth demonstrating how standard selective pulses work for this phantom, such as a Gaussian or E-burb pulse, or
others available in Topspin. Then it would be clearer to what extent special pulses are needed.Response:
The results based on GRAPE-optimized QOC pulses were compared against E-BURP pulses on a freshly constructed model
setup. The superior performance of the QOC approach is illustrated in a comparison of two waterfall spectra (analogously to
Figure 3 and 4), detailing the application of QOC (a) versus E-BURP (b) pulses. Figure (a) shows again an efficient excitation
within the selective bandwidth of the QOC pulse and a suppression of resonances outside of this bandwidth. While the E-
BURP pulses in Figure (b) also manage to provide decently selective excitation of n-dodecane, the selective excitation of
H2 O is imperfect and leads to higher residuals of n-dodecane signal compared to the use of QOC pulses. Additionally, the
application of E-BURP pulses heavily distorts the baseline for some resonance frequency offsets and affects the shape of the
n-dodecane peak when selectively exciting it. The most important individual spectra of the experiments (on-resonance pulses
for n-dodecane and H2 O) were summarized in Figure S28 and added to the SI.
We assume that the poorer performance of E-BURP lies indeed in the lack of robustness against nutation frequency deviations,
as strong residuals appear predominantly when attempting suppression of the signal associated with the Cu cavity. The BURP
excitation pulses as originally published by Geen and Freeman (1991), therefore, seem to be only suited for Larmor-frequency
selective measurements with unperturbed B1 , since suppression of the signal associated with the PEEK cavity was still
satisfactory. In summary, the robustness against nutation frequency deviations enabled with QOC pulses provides a drastic
improvement in selectivity for measurements in the vicinity of conductive surfaces.
Further reasons for using GRAPE over BURP regard pulse length. Pulses optimized in a Fourier basis tend to require
longer pulse durations τ due to less flexibility in the pulse shape. For example, an excitation band of ±500 Hz around the
irradiation frequency, as used in our paper, corresponds to a BURP-1 pulse length of 4 ms according to the BURP-1 design
rule 0 ≤ νexc ≤ 2/τ . Since GRAPE can optimize pulses with arbitrary shape within the piecewise-constant treatment, a shorter
pulse duration of 1 ms was sufficient, while providing robustness against nutation frequency deviations on top. Shorter pulse
durations are crucial for future applications in the electrochemical context. In particular, in operando applications and their
temporal resolution depend critically on the duration of the individual pulse sequence as well as the selectivity and efficiency
of the excitation (and suppression), as the latter one impacts how many repetitions have to be conducted to achieve a reasonable
signal-to-noise. Furthermore, relaxation or dephasing may be enhanced in the vicinity of electrodes, which leads to broad
lines. Having excitation pulses that are as short as possible to achieve a required selectivity is crucial to minimize the T2 or
T2⋆ weighting effects. In addition, GRAPE-optimized pulses can have multiple excitation bands, which might find applications
for CO2 reduction producing various different solvated products in low concentrations, such as CO, methanol, ethanol, formic
acid, etc.Action:
Testing of E-BURP pulses and additions in conclusion as well as in the SI.
Revised Paragraph:
New figure:
Figure S28: 1 H spectra recorded utilizing a ν0 -selective QOC excitation pulse (1 ms) versus an E-BURP-1 pulse (4 ms) with
a selective excitation range of 2.5 ppm (±500 Hz). The top spectrum depicts the reference 1 H spectrum recorded using a hard
pulse. Hereby, the resonance at approx. -3.3 ppm is assigned to n-dodecane and the resonance at approx. -0.8 ppm to H2 O. The
pulses were applied on-resonance for either H2 O (a) or n-dodecane (b). The selective excitation of n-dodecane is achieved to
a similar degree with both pulses, while the selective excitation of water is less efficient when using the E-BURP-1 pulse. The
E-BURP-1 pulse is observed to distort the spectrum baseline and excites the n-dodecane resonance to a significant degree. The
shift in frequencies with respect to previous measurements (other figures) originates from a severe drift of the magnet in the meantime.Introduction
The rf modulation impedes the effectiveness of established selective pulse sequences such as BURP, which are not optimized
for systems with inherently distorted B1 fields Geen and Freeman (1991).Conclusion
To support this claim, a comparison of QOC excitation pulses and E-BURP pulses on the herein presented model setup
was undertaken (SI Fig. S28). The comparison evidently visualized, that when using literature-known ν0 -selective pulses
without ν1 robustness for electrochemical setups, baseline distortions and unsatisfactory selectivity may occur due to strong
B1 field distortions near conductive cell components. Furthermore, the B1 field distortions
in the model setup were accurately predicted by FEM and integrated into a QOC workflow to tailor pattern
pulses which exploit the simulated sharp B1 enhancement near conductive interfaces.5. Figure 5 is about B1 selectivity. This graph is completely unclear. What is shown on the Y ordinate axis? How this graph
or spectrum was obtained is unclear. The nutation frequency is shown in kHz at the top, and the B1 field in a.u. units at the
bottom. I believe that if the nutation frequency for protons is known, then the B1 magnetic field magnitude can be expressed in
microtesla.Response:
Indeed, the dual x-axis structure can be disorienting and we thank the reviewer for pointing out this potential risk. We made
adjustments to the figure and caption for better clarification. The unit of the bottom x-axis was changed to µT, leading to slight
changes due to rounding (25.7% instead of 26.1%). Detailed information on how the nutation spectrum was measured was
already given in the Methods section.Action:
The plot was updated such that the x-axis of the FEM-simulation results are now given in µT. Furthermore, the caption now
starts by emphasizing that Fig. 5 is a dual x-axis plot. It contains a brief description on how the nutation spectrum was recorded
and how the FEM-histogram was obtained. We added a color-coded y-axis labels for more clarity. Scales on the y-axes were
omitted as the absolute histogram counts and the absolute intensity are not relevant for the interpretation of the plot.6. Figure C19 in the supplementary materials shows water on a gray background, although it’s most likely represented by
blue disks. It would be more accurate to indicate it with arrows or another method. At first glance, it appears that the entire
gray area around it consists of water.Response:
We agree with the reviewer, that parts of the figure were ambiguous. The gray background was omitted and the
positioning of the text adjusted.7. A general question regarding images c1-c4. It’s clear that the amplitude of B1 over time has some high-frequency noise.
How important is this noise? What determines the amplitude of this noise? What would happen if, instead of an amplitude
pulse with high-frequency noise, we applied a pulse without this noise, but only with an envelope, after applying a low-pass
filter?Response:
The “noise” in the pulse shape is not noise. It is a byproduct of the numerical optimization process of GRAPE. GRAPE-
pulses with complex tasks such as pattern pulses often exhibit unintuitive shapes (Kobzar et al., 2005). But this is not an
annoyance. It is part of the solution determined by the GRAPE algorithm. Recently published phase-modulated ultrabroadband
pulses also exhibit shapes resembling noise, but this “noise” is an integral part of the solution (see SI of (Woordes et al., 2026)).
Thus, “smoothening” these pulse can potentially change the quantum dynamics and the pulse outcome. For example, applying
a Butterworth low-pass filter to the nutation frequency selective pulses with a cutoff frequency of 50 kHz lead to a quality
function decline of 1.1% in the case of ΓB1 = 1.25. In the cases ΓB1 = 1.0 and 1.8, the noise filter had negligible effect on
the quality function. For the remaining pulses, the decline was between 0% and 1%. Although 1% does not feel like much,
in ensemble optimal control, where many quality functions are averaged, a few percent can make a strong difference in pulse
performance. Luckily, modern spectrometers are well capable of faithfully implementing pulses with non-smooth shapes in a
time resolution of sub-microseconds.8. Line 51. The abbreviation PEM is not revealed.
Response:
We thank the reviewer for spotting this unassigned abbreviation, which escaped our proofreading.Revised Paragraph:
FEM simulations have also been utilized to validate and optimize uniform B1 distribution within in operando cell setups to
study proton exchange membrane (PEM) fuel cells (Zhang et al., 2011), as well as battery applications, (Aguilera et al., 2021;
Sanders et al., 2022) up to commercial coin cell scales (Walder et al., 2021).
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The paper describes the use of OC generated selective pulses in frequency and nutation space for use in electrochemical cells. It demonstrates the performance of such pulses on a phantom sample containing two different liquids between copper plates and PEEK plates. The two lines are clearly shifted in frequency space and also in the nutation space. I think the paper is suitable for MR but the presentation is sometimes confusing and should be improved.
I am not an expert on OC pulses so I cannot judge how demanding the generation of such pulses is. My impression is that this is quite standard and that the application to electrochemical problems is the new part. My expectation from the title and the first reading och Chapter 2.1, I expected the pulses to be simultaneously selective in frequency and nutation space. The title says "Larmor and nutation frequency selective pulses". After a more carefull reading, I think now that these are frequency selective pulses that are tolerant to deviations in the nutation frequency and nutation selective pulses that are insensitive to changes in the resonance frequency. If this is correct, I think this should be made clearer in phrasing of the title and the text. Later on, the paper always talks about B_0 selective or nutation-frequency selective pulses. I just realized that the phrasing in the SI is much clearer: "B1-robust selective excitation within ±500 Hz in a ±2000 Hz suppression band" and "Larmor frequency-robust selective excitation for 0 % artificial nutation frequency increase". It woould be nice if the phrasing in the main text and title would reflect that precision of the SI. By the way, I think the use of the term "B_0 selective" pulses is not a good choice since B_0 by definition is the same everywhere. I think it would be good to use a different term here like resonance frequency selective pulses.
I was wondering how much the improvement of these pulses are compared to "standard" frequency band selective pulses like BURP pulses (H. Geen, R. Freeman, J. Magn. Reson. 93 (1991) 93–141) or nutation-frequency selective pulses in the rotating frame (K. Aebischer, N. Wili, Z. Tosner, M. Ernst, Magn. Reson. 1 (2020) 187–195) that also have some inherent nutation or frequency frequency bandwidth. Can the authors comment, please, how important the simultaneous optimization for both parameters is for this application?
A second more general point is the use of the term "quantum optimal control" for the generation of the pulses. What is "quantum" about the optimal control methods used to generate pulses on a classical computer? I realize that "quantum" is very trendy and sells well but I really think that "optimal control" would be sufficient here.
The presentation of the results in Figs. 3 and 4 using double color coding of the spectra and the background is not easy to understand and interpret. Would it not be much simpler to plot the theoretical excitation profile over the spectra with a little arrow or dot indicating the irradiation frequency. What is especially grating is the fact that the color coding on the right goes to -1 but is in reality never smaller than 0.
minor points:
Fig. 2 and text: Why use ppm for the frequency axis if the ppm are meaningless. I think here a Hz scale would be much more meaningful since also the selectivity of the pulses was specified in Hz. (applies also to Figs. 3,4 and 7)
Fig. 2: Add a z scale to the drawing on the left to make the mapping from the drawing to the plot of the spectra clearer. Maybe even scale the drawing and the spectral plot such that the scale is the same.
Maybe Figures 1 and 2 could be combined in a singel figure that contains all the details.