the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Visualization of dynamics in coupled multispin systems
Jingyan Xu
Dmitry Budker
Download
 Final revised paper (published on 09 Aug 2022)
 Supplement to the final revised paper
 Preprint (discussion started on 16 May 2022)
Interactive discussion
Status: closed

CC1: 'Comment on mr20229', Andrey Pravdivtsev, 20 May 2022
Dear Jingyan, Dmitry and Danila
Thanks for advancing MR field!
I read your preprint and have a few comments and questions regarding your work.
in the intro you write, “Most NMR textbooks visualize the motion of a single spin1/2 (or an ensemble of spins) using the Bloch vector.”
I understand this because it gives us a simple rule to explain and describe spin evolution. Moreover, there are modified rules for the week coupling case that use product operator formalism. How do we know what they are useful? Because they give us an excellent tool to predict an effect on paper before measuring it.
In conclusion, you write: “AMP/AMC surfaces conveniently represent symmetries of density matrices and allow spotting their presence (orientation, alignment, etc.) or absence even when direct analysis of density matrices is not obvious. <..> The presented AMP/AMC surface approach allows visualizing complex dynamics in multispin systems and may find applications for describing hyperpolarization experiments”
I miss new insights into spin evolution. I do not understand how this approach can help to explain spin evolution simpler or predict something new.
Some other comments:
Comment 1. I can guess Figure 1c, need corrections. Something like Fk,mF>Fk,mj, Fl,mF>Fl,mi, for mi,mj<Fl, rhoFlFk_mF,mF>rhoFlFk_mi,mj and then in the caption only mF=mi=mj were used for visualization.
Comment 2. L 135 “Lastly, Fig. 5E, which measures the outofphase coherence reveals extra information not covered by Fig. 5B, i.e., the singlettriplet coherence, ˆI1y ˆI2x−ˆI1x ˆ 135 I2y, is transiently formed during the experiment.”
 It was your free choice not to plot oop coherence on B, which you can add, and then again no new information I can gain from the visualization. Or I’m I missing something?
Comment 3. Fig 6. T1 and T2 are the unfortunate choices because they have some specific mining in NMR. Maybe small tau instead?
Comment 4. Line 150: “Dynamics of the AMC surfaces shown in Fig 6DF is more intricate. Their motion is a superimposed oscillation of shapes with a period of T2 = 1/J and a slow precession about the xaxis with the abovementioned period of T1. From the visualization, one may conclude that inphase and outofphase zero quantum coherences give rise to the doublet shown in Fig 6B”
 What is your observable? If something evolves with the same frequency it does not mean that it is the reason for the signal (i'm not saying that it is the case here).
 You need to explain how the evolution of ZQC results in the observation of the ULF spectrum.
Comment 5 : “Code availability. The software code for the graphics shown in this paper is available from the authors upon request”
I hope this statement will be not acceptable for any journals and MR will have a strict policy on publishing all scripts together with the paper or EUrepositories with doi like zenodo.org
With kind regards
Andrey Pravdivtsev

CC2: 'Reply on CC1', Danila Barskiy, 27 May 2022
Andrey, thank you for valuable comments. The most important message we want to emphasize is that all information about the spin system is contained in the density matrix (DM), thus, no "new insights into spin evolution" are necessarily expected from AMP/AMCs. However, such visualization approach allows seeing DM symmetries in a more convenient way  thus, new insights may potentially be generated by an inspired reader. Imagine a state (I1zI2z) globally rotated by 33 degrees about xaxis. Good luck understanding what state it is by looking at DM directly (not that it is impossible): all elements will be populated when DM is written in the original basis. Simple visualization algorithm presented in this work produces the surface from which the symmetry (alignment) is immediately obvious. In addition, Appendix C shows that DM and the visualized surfaces form onetoone map. We plan to elaborate on this subject more in a revised version of the manuscript and shift the focus from "new insights" into convenience of observing symmetries.
Comment 1: We plan to modify Figure 1 in the revised manuscript to enhance explanation and reduce confusion.
Comment 2: See above comment about information content in the visualizations. In order to keep onetoone correspondence between the DM and the visualized AMP/AMC surfaces, we present the outofphase coherence for completeness.
Comment 3: We modify the labeling in the revised version.
Comment 4: Very good comments, for the AMP/AMC surfaces – it is probability that is important, not necessarily the measured property. However, in a particular case of inphase zeroquantum coherence (I1zI2z) represented by AMC surface in Figure 6DE, the intersection of this surface with positive zaxis represents directly a measured signal in ZULF NMR experiments. We will elaborate on this subject in more detail in the revised manuscript.
Comment 5: The code will be provided although we followed several published examples in the MR journal where such statements about the code availability were provided.

CC2: 'Reply on CC1', Danila Barskiy, 27 May 2022

RC1: 'Comment on mr20229', Anonymous Referee #1, 25 May 2022
The authors provide a new way of visualizing density matrix components during pulse sequences in multispin system using angular momentum probability surfaces. The approach appears to be superior to the previously described DROPS method. One component of the approach involves coupling angular momenta, but it seems that the actual visualization is not very intuitive to follow. Either way, it seems the authors provide some useful examples. I am wondering whether the authors could identify cases where this approach can produce some level of intuition that would exceed the level of intuition one would get from examining the spin components directly.

AC1: 'Reply on RC1', Danila Barskiy, 27 May 2022
We thank the referee for assesing our approach as superior to the previously described methods. One particlar case where the presented approach may produce an enhanced level of intuition is the presented nearzerofield NMR experiment (Figure 6). Additional discussion is added to the revised version of the manuscript.
Initial magnetization of two coupled spins (with gyromagnetic ratios g1 and g2) in high magnetic field is (g1*I1z + g2*I2z). This state can be presented as a sum of symmetric and antisymmetric components: 0.5*(g1+g2)*(I1z+I2z) and 0.5*(g1g2)*(I1zI2z). Remarkably, both of these components can be detected by a magnetometer in ZULF NMR experiments. The first symmetric component corresponds to the orientation of a collective magnetic moment (I1z+I2z) and its visualization is related to an AMP surface presented by the Fig.6C (however, the surface represents probability and not exactly the measured property). This component typically corresponds to static magnetization at zero field. The experiment shown in Fig. 6C makes this component precessing with Larmor frequency 0.5*(g1+g2)*Bx upon application of the magnetic field Bx. The second component of decomposition represents a zeroquantum coherence (I1zI2z) which is directly detectable by the magnetometer. Intersection of the plotted AMC surface with any axis is a direct representation of a measured signal by a magnetometer along that axis.

AC1: 'Reply on RC1', Danila Barskiy, 27 May 2022

RC2: 'Comment on mr20229', Malcolm Levitt, 26 May 2022
The paper describes a visualization tool for the representation of density operators in multiplespin systems. The approach is inspired by the DROPS software of Glaser and coworkers, and shares with that work most strengths and weaknesses. The strength is a graphical representation which might possibly lead to a helpful visualization of complex spin dynamics, sidestepping the need for complex mathematics, and possibly help inspire new procedures or give new insights. The main weakness is that although the tool allows a pretty graphical representation of the mathematics, it does not replace the mathematics, at least not as far as I can see. So the result seems to be pretty graphics (which I am all for) but not with evident real utility, in contrast to the Bloch vector picture, from which many NMR effects and experiments have been derived.
So I am not yet convinced of the utility of this representation. Furthermore I cannot see exactly how it works, and the authors do not help since they choose a mathematically dense exposition which is very hard to follow, right from the beginning. Despite my best efforts I cannot understand equation 1 and the following equations. It may be that the terms used by the authors are selfevident to the atomic physics community but I suspect that most readers of this journal will, like myself, struggle greatly with it. For this to work the authors must make far greater efforts to express themselves in language comprehensible to us mere magnetic resonance mortals. What on earth is the "block (Fk, Fk)"? Scientists on the same level of mathematical physics as myself will need to be led far more slowly through this material, using helpful simple examples on the way.
In addition the authors introduce the term "angular momentum coherence (AMC)". I suspect that the term coherence is used here to mean something very different from its standard usage in magnetic resonance, as defined by Ernst and coworkers (an offdiagonal density operator term, when written in the Hamiltonian eigenbasis). I am not sure though since I cannot follow the authors' meaning. In general I do not think a redefinition or loose usage of this fundamental term is advisable.
In summary I am sympathetic to the aims of this paper but find the presentation impossible to follow. In addition, I am far from convinced of its usefulness, but recognise that it could be of value if explained well enough and made sufficiently accessible.
On the issue of accessibility, I agree with another referee that it is not longer acceptable, for work of this kind, to say that the code is available on request.
 AC2: 'Reply on RC2', Danila Barskiy, 04 Jun 2022

RC3: 'Comment on mr20229', Malcolm Levitt, 29 May 2022
A further comment: The article emphasizes the total spin angular momentum quantum number (denoted F, I believe). Offdiagonal density operator elements spanning states with different values of F appear to be called "angular momentum coherences" (AMCs). This nomenclature and analysis might be appropriate for atomic physics, where the Hamiltonian has isotropic, or nearisotropic symmetry. However this situation is rarely encountered in magnetic resonance of bulk matter, since we hardly ever deal with isotropic systems. Trivially, the application of a strong magnetic field breaks isotropic symmetry (leading, amongst other things, to the Zeeman effect, upon which most magnetic resonance is based). Even the solution NMR of isotropic liquids does not involve an isotropic Hamiltonian. Very often, chemical shift differences and other interactions break the symmetry further. In most cases these are not small perturbations but conpletely break the isotropy of the spin Hamiltonian. There are rare exceptions, such as zerofield NMR.
Since highfield NMR almost always uses Hamiltonian eigenstates that do not have welldefined values of F, I do wonder what utility this diagrammatic approach might have. Furthermore, although the concept of AMC's "angular momentum coherences" might possibly mean something in atomic physics, I suspect that it has no, or little, relevance to the vast majority of magnetic resonance experiments, and probably conflicts with the conventional use of the term coherence in magnetic resonance  namely a coherent superposition of Hamiltonian eigenstates.
I think the article will not be appropriate for the magnetic resonance community unless these sharp differences between the atomic spectroscopy and bulk magnetic resonance contexts are highlighted more clearly.

AC4: 'Reply on RC3', Danila Barskiy, 07 Jul 2022
Dear Malcolm, we thank you for the critique. The paper was substantially rewritten and now the visualization approach is based on the use of generalized measurementrelated operators. We note that this approach provide us with valuable information which is not easy to grasp through direct observation of numerical values of the densitymatrix elements. First, the action of global rotations applied to the density operator is directly reflected by the rotations of the plotted surfaces (see new Fig. 3). Second, there is a close relationship between the density matrix coherences and the symmetries of the plotted surfaces (see new Fig. 4). Third, our visualization allows one to quantitively understand the measured ZULF NMR spectrum by looking at the intersection of the surface with an axis representing sensitive direction of a magnetometer. As an example, the ZULF NMR spectrum of an AX system is now pictorially explained through our visualization. Specifically, we explained the small asymmetry of the doublet centered at Jfrequency which is not easy to grasp through the product operator formalism.
We also improved the text by simplifying some mathematical notations and by using terminology which is more familiar to the NMR community. In simple words, our visualization is now performed by plotting measurements with zeroquantum Hermitian operators rotated along various directions.
Despite coherences (visualized via AMC surafes) were indeed offdiagonal elements of the density operator written in the total angular momentum basis (i.e., Hamiltonian eigenbasis at zero magnetic field), we agree that excessive use of the “coherence” terminology did not add additional value to the paper. For this reason, we abandoned the term "AMC surfaces" in the updated version of the paper and now refer to the visualized surfaces directly via the measurement operators.
Regarding the software availability. The statement “The software code for the graphics shown in this paper is available from the authors on reasonable request” was a direct copy (except deletion of the word reasonable) from the following article published in Magnetic Resonance (see https://mr.copernicus.org/articles/2/395/2021/). Nonetheless, we gladly provide the code in the revised version of the manuscript.
Taken together, thanks to your comments, we believe the paper is significantly improved an, hopefully, it is now more accessible for the magnetic resonance community.

AC4: 'Reply on RC3', Danila Barskiy, 07 Jul 2022

CC4: 'Comment on mr20229', Tom Barbara, 31 May 2022
Without being more current on aspects of this topic as outlined by Steffen Glaser (CC3), I was initially reminded of the efforts put forward in the late 1970’s on the topic of coherences and their visualization. As I remember it, it was Vega and Pines who would point out the analogies between spin coherences and angular momentum “orbitals”. This did connect college chemistry to spin coherences, but never had much to add in efforts to understand dynamics. Rather, the dynamics would be visualized in some subspace that had favorable rotationlike properties, so that Bloch Sphere type pictures could be drawn. There were lots of ways of slicing the problem and, just as an example, I have an entire binder of notes on two spin ½ mapped onto spin 3/2 looking for dynamical groupings that offered some visual picture. Alas they were (almost) always limited in utility even though the few exceptions were of some interest for spin 1 and spin 3/2 or two coupled spin ½.
I sense that a similar situation holds for the pictures offered here. Furthermore, I agree with Malcolm’s comments that the mathematical exposition is obtuse. Given that hurdle, who is going to master the language if it actually adds only an incremental insight?
 CC5: 'Reply on CC4', Tom Barbara, 01 Jun 2022
 AC3: 'Reply on CC4', Danila Barskiy, 04 Jun 2022

RC4: 'Comment on mr20229', Steffen Glaser, 01 Jun 2022
The stated aim of the manuscript is to extend the concept of angular momentum probability (“AMP”) surfaces (which have been shown to be useful in the understanding of atomic physics experiments) by socalled angular momentum coherence (“AMC”) surfaces. The authors show that this makes it possible to create threedimensional graphical representations of the density operator of coupled spins, which is illustrated for three concrete NMR pulse sequences. The authors also show that the suggested visualization is complete in the sense that the density operator can be reconstructed from a full set of “AMP” and “AMC” surfaces. The mathematical basis of the visualization approach appears to be solid (but I agree with the comments of other reviewers that the presentation of the derivations and proofs should probably be adapted to the readers of Magnetic Resonance).
In my view, the most important weak point of the current manuscript is a thorough discussion of how the presented approach is related to similar visualization techniques that have been introduced before for the visualization of coupled spin/angular momentum dynamics. In particular, the readers (as well as the referees and the editor) will be interested to see what are truly novel aspects in terms of the visualization approach or novel applications and to give a proper account of closely related previous work. (Before I discuss these aspects in more detail below, I would like to point out that even if the visualization should be closely related (or even be essentially identical) to previously published approaches, I would still be in favor to publish a revised manuscript in which these points are considered, because as far as I see, the proposed visualization variant has not been applied to concrete NMR settings yet and it should be interesting and useful for the readers to see whether or not it could have advantages compared to other visualizations approaches.)
Point #1:
Before addressing the main point (the potential novelty of the “AMC” surface representation), I would like to briefly discuss the established “AMP” surface representation for uncoupled spins or atoms or molecules with arbitrary angular momentum (called F in the nomenclature used in the manuscript or J in other settings). I think the proper context in which this work should be placed is the general field of phase space representations, at least by referring to the books by W. P. Schleich, Quantum Optics in Phase Space (Wiley, New York, 2001), C. K. Zachos, D. B. Fairlie, and T. L. Curtright, Quantum Mechanics in Phase Space: An Overview with Selected Papers (World Scientific, Singapore, 2005). F. E. Schroeck, Jr., Quantum Mechanics on Phase Space (Springer, New York, 2013). T. L. Curtright, D. B. Fairlie, and C. K. Zachos, A Con cise Treatise on Quantum Mechanics in Phase Space (World Scientific, Singapore, 2014 and the recent review by R. P. Rundle and M. J. Everitt in Adv. Quantum Technol. 2100016 (2021). For a general discussion and comparison of different families of phasespace representations, I refer in particular to the recent paper (B. Koczor, R. Zeier, S. J. Glaser, Continuous PhaseSpace Representations for FiniteDimensional Quantum States and their Tomography", Phys. Rev. A 101, 022318, 2020) and references therein.
In (Koczor et al., 2020), the focus is the family of socalled sparametrized phasespace functions of which the widelyused Glauber P function (with s=1), the Wigner W function (with s=0) and the Husimi Q function (with s=1) are special cases. The paper gives an overview how the plethora of different finite and infinitedimensional phase space representations are related and can be mapped to each other. In particular, see Fig. 2 of (Koczor et al., 2020) for a graphical overview, section III on phasespace representations for spins and Ref. 76 (Stratonovich, 1956), Ref. 58 (Argawal, 1981), Ref. 77 (Varilly et al., 1989), and Ref. 57 (Brif et al, 1999).
As far as I can see (but please correct me if I am wrong), the definition of the “AMP” representation appears to be identical with the finitedimensional version of the Husimi Q function, which to my knowledge was first defined (under different names) in the early 1980s (Argawal, 1981). Surprisingly, the close connection (if not identity) of the “AMP” surfaces and the Husimi Q function for finitedimensional quantum systems seems to have gone unnoticed – or at least has apparently not been pointed out in the previous literature (and I need to apologize that before reviewing the current manuscript, I was not aware of the “AMP” representation and therefore we had not explicitly mentioned it in (Koczor et al., 2020).
So far, I have not received through the library the book (M. Auzinsh, D. Budker, S. Rochester, 2010) cited in the manuscript by Xu et al. and could not check if more details are given there on the relation between “AMP” surfaces and other phasespace representations. However, I found Simon Rochester’s thesis from 2010 online, in which he points out that Chapters 25 of his thesis are largely identical to sections in the book, of which he is an author. In chapter 2 of his thesis (section 2.3.3), it is pointed out that the expansion of “AMP” functions in terms of spherical harmonics in Eq. 2.39 is “quite similar” to the Wigner function for angular momentum states given in Eq. 2.40 (from Dowling et al., 1994). I assume that this is also pointed out in the book (M. Auzinsh, D. Budker, S. Rochester, 2010). In this section of the thesis is also correctly stated that the essential difference between the expansion of the Wigner function and the “AMP” function is “that the contributions of polarization moments of various ranks are weighted differently”, resulting in the fact that “AMP” functions are nonnegative. However, this is exactly the property of the Husimi Q function, which has the same weighting factors as the “AMP” function and the same physical interpretation, see also (Koczor et al., 2020), where the sdependent weighting factors are explicitly given for the family of sparametrized phasespace functions (see Eq. 5 and Fig. 3).
To summarize point #1, I think it would be very beneficial to point out in the paper that the “AMP” functions are in fact identical with the Husimi Q function and to add corresponding original references and Ref. (Koczor et al., 2020), in order to avoid potential confusion that can arise if different names are used for the same concept in closely related scientific communities and to clearly define the relation between Husimi Q/”AMP” functions and other members of the family of sparametrized phasespace functions. It would also be interesting to state in the manuscript whether or not my impression is true that the definition of (Argawal, 1981) predates the (identical) definition of what is called the “AMP” function.
Point #2:
This point concerns the question about the degree of novelty of the “AMC” surface representation. In the introduction, of the manuscript, the authors mention the “DROPS” representation introduced by (Garon et al., 2015), but state that “while the approach conveniently reflects the dynamics and symmetry of individual spins …, it is challenging to extract information from the drops representing systems of equivalent (or nearly equivalent) spins” and that the approach based on “AMP” and “AMC” surfaces is introduced “to address these limitations”. This section raises several issues:
Point 2.1: To avoid any confusion of nomenclature, let me first address a minor point:
In the introduction of the manuscript, it is implied that “spin drops” is equivalent to the “DROPS” representation, which is not the case and the two concepts should be clearly distinguished:
“DROPS” is a general approach to visualize operators of general spin systems (coupled or not) that was introduced in the paper (Garon et al., 2015). In addition, the implementation of the “DROPS” representation and visualization (based both on the “LISA basis and the multipole basis”) in a Mathematica package is publicly available by downloading the file “DROPS_1.0.zip” at https://www.ch.nat.tum.de/en/ocnmr/mediareports/downloads.
“SpinDrops” is the name of an interactive software package that is freely available for the community (see www.spindrops.org), which (in addition to other approaches) also provides the option to visualize the density operator (as well as Hamiltonians, propagators etc.) using a “DROPS” representation.
Point 2.2:
The “DROPS” representation is a general mapping between operators and a set of spherical functions (socalled droplets). As pointed out in (Garon et al., 2015), it is a Wignertype (generalized) phasespace representations, which is applicable for arbitrary spin systems. In (Garon et al., 2015), the detailed mapping (which is based on symmetryadapted spherical tensors) was explicitly presented for systems consisting of up to three coupled spins ½. More recently, explicit symmetryadapted spherical tensors were constructed based on which systems consisting of up to six spins ½ as well as for coupled spins > ½, based on which arbitrary operators in such systems can be represented and visualized (see Leiner, Zeier, Glaser, "SymmetryAdapted Decomposition of Tensor Operators and the Visualization of Coupled Spin Systems", J. Phys. A: Math. Theor. 53, 495301, 2020).
Both in (Garon et al., 2015) and in (Leiner et al, 2020), we focused on the “DROPS” representation based on the socalled “LISA” basis of spherical tensor operators (with defined linearity, subsystem, and auxiliary criteria, such as permutation symmetry), which is specifically constructed to visualize the individual spin contributions, which are relevant in most highfield NMR experiments. (In fact, due to the symmetryadapted “LISA” basis, also magnetically equivalent spins can be efficiently represented).
However, in addition to the “LISA” basis, in section VII and VIII, appendix F, Tables II, VI and VII, Figures 6 and 12 of (Garon et al., 2015), we also explicitly and extensively discussed “DROPS” representations based on socalled multipole spherical tensor basis operators, generalizing a visualization introduced by Merkel et al. in 2008 (Ref. 18 in Garon et al., 2015). I did not have a chance to make an indepth comparison yet, but I appears that this “DROPS” representation based on the multipole basis is essentially identical (or at least very closely related) to the “AMC” functions. Note that in particular the form of the “multipole operators” corresponding to transitions between blocks with different total angular momentum quantum numbers defined in Eq. (F1) in appendix F of (Garon et al., 2015) appears to be identical to the corresponding definition of the spherical tensor operators in Eq. (A1) in appendix A of the manuscript by Xu, Budker and Barskiy on which the definition of the “AMC” surface functions is based. In fact, both the “multipole operators defined by (Garon et al., 2015) and the spherical tensor operators in Eq. (A1) of the manuscript by Xu et al. differ from the tensor operators in the “LISA” basis in not having a defined particle number (i.e., “linearity”) and inducing a different grouping into droplets. One (apparently trivial) difference seems to be that the droplets corresponding to transitions from “F_l” to “F_k” and from “F_k” to “F_l” are separately displayed in droplets in of (Garon et al., 2018) and (Merkel et al., 2008), whereas they are merged in the manuscript by Xu et al, whereas droplets corresponding to zeroquantum phase phi=0 and phi=pi/2 are displayed separately. Another small difference is that (at least the droplets corresponding to the diagonal blocks in Fig. 1 are nonnegative (corresponding to a Husimi Q function representing angular momentum pointing probabilities), whereas in the “DROPS” representation based on multipole operators, the droplets corresponding to the diagonal blocks can have negative values (corresponding to a Wigner function, representing the expectation values of socalled axial tensor operators). However, as discussed above in point #1 (and as indicated in the thesis of Simon Rochester and also discussed in (Koczor et al., 2020)), it is straightforward to transform between a Husimi Q and a Wigner W representation by simply changing the rankdependent weighting factors of the polarization moments.
To summarize point 2: It would be very helpful for the readers to clearly state how closely the combined “AMP” and “AMC” surface representation is related to the “DROPS” representation based on multipole operators and to point out potential differences and whether or not the differences are significant. In particular, the following statements need to be corrected:
Line 30 “… it is challenging to extract information from the drops representing systems of equivalent (or nearly equivalent) spins.” This is clearly not the case, in particular if the DROPS representation based on multipole operators is applied.
Line 34: “… To address these limitations, we introduce …”. As pointed out above, the DROPS representation based on multipole operators does not have the stated limitations
For the presented NMR examples, comparison of the “AMP/AMC” with the standard DROPS representation based on the “LISA” basis and/or with the DROPS representation based on multipole operators is not mandatory, but could be quite useful (at least for one of the presented NMR examples), to make it possible for the readers to judge (potential) advantages or disadvantages of the different visualization approaches.

AC5: 'Reply on RC4', Danila Barskiy, 07 Jul 2022
Dear Steffen, we highly appreciate your constructive criticism and comments. The manuscript is now improved by citing the relevant literature you mentioned. In addition, the equivalence between the AMPS and the Husimi Qfunction was indeed established and the proof is now presented in the Appendix G.
The manuscript was updated in multiple places. First, the terminology of AMC (angular momentum coherence) surfaces has been abandoned and now the visualization is based upon using general zeroquantum Hermitian operators for plotting the surfaces. This makes the surfaces represent measurable properties as now discussed in the section 3.2. The manuscript was updated in multiple places to use the correct terminology (DROPS representation).
Regarding differences and similarities of the presented approach and the DROPS representation. Decomposition of the density matrix into blocks corresponding to the total angular momentum quantum numbers F is the same as in (Garon et al., 2015, when using multipole tensor operator basis), however, the visualization procedure is different. DROPS approach is a Wignertype representation (visualization is complete but the composed surfaces do not directly represent measurable properties of the density matrix), while our approach is measurementbased representation (visualization is complete and the radius of the surface directly represents a measurable property of the density matrix).
We performed additional simulations and now present a direct comparison between our visualization approach and the DROPS approach (see Section 3.5) when applied to the ZULF NMR experiment. While DROPS approach allows to fully describe the spin dynamics, it is less straightforward to convert color into the measurable property. We also point out that DROPS representation may be discriminatory with respect to colorblindness of some groups of the population while the presented approach avoids the use of the color map (since the plotted surfaces correspond to measurable properties which are real numbers).
We hope the updated version of the paper addresses all criticism and we thank you once again for expremely valuable comments.

AC5: 'Reply on RC4', Danila Barskiy, 07 Jul 2022

CC6: 'Comment on mr20229', Balint Koczor, 06 Jun 2022
Thank you for the interesting paper! I completely agree with two prior comments. First, I also think the surface obtained in Eq (1) as the representation in the maximal F block spanned by F_k,F_k> should be the standard Q function and the connection to literature on phasespaces should certainly be discussed. Second, I think the paper has presentational as well as more technical issues: The main selling point of the paper is that the representation is based on eigenblocks of the total angular momentum operator as illustrated in Fig1, but the following central quantities are nowhere mathematically defined
  F, m > are never defined. They're presumably eigenvectors of a certain total angular momentum operator. A comment in parentheses above Eq (C4) attempts to define these as `defined with the projection mF along the zdirection)' which still gives no detail as to what these are
 The total angular momentum operator is nowhere defined and, moreover, not even a symbol is introduced for it!
Furthermore, the authors present some derivations in appendix A. These again give no detail where the involved mathematical objects come from. Most of these proofs are actually quite standard and can be found in the literature, yet the authors present these without citing any literature at all. At least some information should be given how these derivations connect to existing literature and the involved objects, e.g., spherical tensor operators, ClebschGordan coefficients etc. should be defined or references should be cited.If these comments are properly addressed I think the paper could be an interesting addition to literature. AC7: 'Reply on CC6', Danila Barskiy, 19 Jul 2022

EC1: 'Comment on mr20229', Geoffrey Bodenhausen, 11 Jun 2022
This paper introduces a graphical convention for the representation of various forms of spin order in coupled systems.
Personally, I find figures such as Fig. 4B perfectly adequate, and I fail to see any advantages in ‘AMC’ pictures such as those in Fig. 4E and 4F. True, the ‘AMP’ pictures of Fig. 4 C and 4D are even less informative. Likewise, in Fig. 5B it would have been easy to add a line to show the timedependence of the singlettriplet coherence. I would prefer to call it a longlived coherence or LLC.
The problem with many graphical representations is that, while their beauty and elegance may be obvious to their inventors, they may remain opaque for many readers.
The beauty of some nontrivial trajectories on Bloch’s sphere is one of the very few exceptions that confirm the rule. Their success is mostly due to the unmatched talents of the regretted Ray Freeman and some of his students like Gareth Morris who illustrated DANTE’s spiraling descent to hell, and Malcolm Levitt’s composite pulses.
I myself have made several illfated attempts to become famous by introducing various graphical conventions. Thus I failed to convince Ray that phase corrections in 2D spectra should be represented on the surface of a taurus rather than on a sphere (Ray, who would never make any disparaging comments, kindly referred to ’Geoffrey’s doughnuts’.) A few years later, I tried to represent a doublequantum coherence in deuterium NMR by a ‘d’ orbital with two positive and two negative lobes in the equatorial plane. Bob and Gitte Vold, with whom I was working in those days, wisely discouraged me. On the occasion of Xmas 1978, I offered them some ‘doublequantum cookies’ by adapting the shape of a Swiss cake mold. Those cookies were not intended to gain celebrity status.
While at ETH with Richard Ernst, we invested a fair amount of time in graphically representing transformations of product operators by socalled ‘windmills’ that show triads of noncommuting operators. Nobody ever uses these. Likewise, we chose to represent coherences by squiggly lines drawn onto energy level diagrams, using dashed lines for imaginary terms, and solid lines for real operators. Again, nobody ever cared about these carefully designed pictures. On the other hand, our representations of populations of eigenstates in terms of positive and negative deviations from the demagnetized state were often misunderstood and frequently misused. We affectionately referred to our blackandwhite ‘pingpong balls’, but Richard disapprovingly called them ‘Boppeli’, using a disparaging SwissGerman diminutive that did not bode well. How right he was.
The only graphical convention from this period that had some sort of success (cited 1420 times in Google’s beauty contest) appears to be our ‘coherence transfer pathways’. The graphical conventions that show the path through different levels of zero, single and pquantum coherences are commonly used to clarify what suchandsuch a pulse sequence is supposed to achieve. However, it seems that these pictures are rarely used to derived phase cycles, and few appear to appreciate the implications of selecting ‘mirror’ pathways to obtain pure phase signals.
With Norbert Müller, we developed an elaborate graphical convention (no doubt too elaborate!) that keeps track separately of the coherence orders ‘p’ and tensor ranks ‘l’ for three methyl protons M and a neighboring proton A in AM3 systems. Figure 7 in ‘CrossCorrelation of Chemical Shift Anisotropy and Dipolar Interactions in Methyl Protons Investigated by Selective Nuclear Magnetic Resonance Spectroscopy’ by N. Müller and G. Bodenhausen, J. Chem. Phys. 98, 60626069 (1993), may serve as a warning for future generations against oversophistication. I intend to upload this figure on ‘MR discussions’ in a separate pdf file for the amusement of today’s readers. If one can trust Google, the paper was cited a whopping 16 times in 29 years.
With Urs Eggenberger, we tried to develop a graphical scheme that allows one to sketch transformations of product operators under shifts and couplings without the need to write out bulky trigonometric expressions in "Modern NMR Pulse Experiments: A Graphical Description of the Evolution of Spin Systems” by U. Eggenberger and G. Bodenhausen, Angew. Chem. Int. Ed. Engl. 29, 374383 (1990). The objective of this hapless paper is actually more ambitious than the aim of the paper submitted by Budker et al. Yet, despite our best efforts, and despite the reputation of Angewandte Chemie, our paper was a complete flop, since it was only cited 24 times in 32 years. (I firmly reject the widespread idea that high citation numbers correlate with originality, but concede that low numbers indicate poor impact.)
Yet, to our surprise, our ‘coherence transfer pathways’ turned out to be most successful in solidstate NMR, in particular for quadrupolar nuclei. Phil Grandinetti invented socalled ’transition pathways’, which show successive transformations of singletransition operators representing coherences, or of polarization operators representing populations (J. Am. Chem. Soc.,130, 10858–10859, 2008.) This idea evolved into a paper entitled ‘Symmetry pathways in solidstate NMR’, by P. J. Grandinetti, J. T. Ash, and N. M. Trease, Progress in Nuclear Magnetic Resonance Spectroscopy, 59, 121196, 2011. According to Google Scholar, it was cited 47 times in 11 years, hardly a resounding success.
So I do not expect the paper submitted by Budker and coworkers to become a ‘citation classic’. Fortunately, ‘Magnetic Resonance’ is not striving to optimize its impact factor. We cannot exclude that some younger scientists may find inspiration in their graphs. It would therefore be counterproductive to reject this paper.
As pointed out be one of the reviewers, it is not acceptable to write ‘The software code for the graphics shown in this paper is available from the authors upon request.’ It should be downloadable form a server.
As editor of Magnetic Resonance, I strongly object to the authors’ use of the word ‘coherence’ in their abbreviation ‘angular momentum coherence (AMC)’. Whatever witty name they may choose for their graphs, our scientific community should not allow them to get away with such a misleading abbreviation that can only create confusion. Perhaps ‘Local angular momentum probability (LAMP)’ would convey the authors’ ideas? Some readers might find this abbreviation enlightening.

AC6: 'Reply on EC1', Danila Barskiy, 16 Jul 2022
Dear Geoffrey,
Many thanks for considering our manuscript and your comments. We well realize that we will not get famous by proposing the generalized measurementbased visualization. Luckily, that is not our goal. The story is that that we found AMPS exceptionally useful for describing spin evolution in many situations in atomic physics (especially, things like alignmenttoorientation conversion) and for NQR [see D. Budker, D. F. Kimball, S. M. Rochester, J. T. Urban, Alignmenttoorientation conversion and nuclear quadrupole resonance, Chem. Phys. Lett. 378 (34), 440448 (2003), https://doi.org/10.1016/S00092614(03)013277]. The point is that AMPS immediately reveal the symmetries of the density matrix, helping predict and optimize signals.
Our current manuscript addresses (in our opinion, successfully) the painful deficiencywe did not have an analogue of AMPS for coupled spin systems which are of primary importance, for instance, in ZULF NMR.
You are 100% correct, however, that the usefulness of visual representations is not universalwhat helps some people does not necessarily help others, especially, if the latter had already developed their own “crutches.” I remember being elated at what I thought was a clear exposition of the mechanism of NQR in powdered samples presented in the paper quoted above that I tried to “sell” to the great Erwin Hahn, whose office at Berkeley was just a few doors down from mine. “Finally, you can see how alignment evolves in each crystallite, even though this has nothing to do with spin precession!” said I. “Not so fast,” said Erwin, “Think of alignment as two spin vectors pointing in opposite directions,” said Erwin, “the quadrupole splitting pattern is equivalent to these two vectors rotating in opposite directions, which,” he raised his arms and started waving them in circles, “immediately shows you how alignment can convert to orientation, resulting in observable signal in NQR!” While Erwin’s picture may, indeed, be useful, the additional strength of AMPS over his handwaving is that AMPS have precise onetoone connection to the density matrix and allows visualizing arbitrary polarization moments, for which Erwin would not have enough hands.
Thank you so much for your strong advice against the AMC terminology; indeed, we have completely eradicated it from the manuscript.
Sincerely,
Dmitry Budker

AC6: 'Reply on EC1', Danila Barskiy, 16 Jul 2022

EC2: 'Comment on mr20229', Geoffrey Bodenhausen, 11 Jun 2022
The comment was uploaded in the form of a supplement: https://mr.copernicus.org/preprints/mr20229/mr20229EC2supplement.pdf