23 Sep 2021
23 Sep 2021
Spin relaxation: under the sun, anything new?
 ^{1}Laboratoire des biomolécules, LBM, Département de chimie, Ecole normale supérieure, PSL University, Sorbonne Université, CNRS, 75005 Paris, France
 ^{2}International Tomography Center, Siberian Branch of the Russian Academy of Science, Novosibirsk 630090, Russia
 ^{3}Novosibirsk State University, Novosibirsk 630090, Russia
 ^{1}Laboratoire des biomolécules, LBM, Département de chimie, Ecole normale supérieure, PSL University, Sorbonne Université, CNRS, 75005 Paris, France
 ^{2}International Tomography Center, Siberian Branch of the Russian Academy of Science, Novosibirsk 630090, Russia
 ^{3}Novosibirsk State University, Novosibirsk 630090, Russia
Abstract. Spin relaxation has been at the core of many studies since the early days of NMR, and the undelying theory worked out by its founding fathers. However, this theory has been recently questioned (Bengs and Levitt (2020)) in the light of Linblad theory of quantum Markovian master equations. In this article, we review the conventional approach of quantum mechanical theory of NMR relaxation and show that under the usual assumptions, it is equivalent to the Linblad formulation. We also comment on the debate over semiclassical versus quantum versions of spectral density functions involved in relaxation.
Bogdan A. Rodin and Daniel Abergel
Status: closed

CC1: 'Comment on mr202162', Tom Barbara, 23 Sep 2021
This effort gives a modern and more accessable account of many details in the theories of Bloch and Hubbard that were not covered in the exposition I wrote up in MR26892021 on the comparison between these and the Lindbladian master equation. A derivation that does not wrangle through the same path taken by Bloch and Hubbard (and Redfield) can be very helpful to readers interested in learning the fundamentals, especially the starting point of iteration of the density matrix evolution as I mentioned in my summary section of MR26892021. A good emphasis is also made in the discussion of "left" and "right" spectral densities and their thermal symmetrization, which allows one to approach the classical limit, as was done by Hubbard with his SECH scale factor.

AC1: 'Reply on CC1', Daniel Abergel, 24 Sep 2021
Thanks for the careful reading of our manuscript. This work has been the opportunity to realize that the seminal work by Bloch not only presents a comprehensive and fairly general approach to relaxation that extends beyond spin relaxation.

CC13: 'Reply on AC1', Tom Barbara, 28 Sep 2021
Yes it is true that one cannot underestimate the significance of Bloch. For that reason I was surprised that modern terminology ascribes equations like your #21 as "Redfield" in kind. Redfield never wrote down this general form in his IBM article. The closest he came was with his Eq.2.30 in that paper and only then in the extreem narrowing limit. I noticed this "new termiinology" first when I read Manzano's paper. From what I read in the literature, equations like Eq.2.30 were first written down by Abragam in his monograph. From that equation, which is very general, the different treatments follow depending on how one interprets the Hamiltonians and the density operators, which is the difference between Bloch and the semiclassical Redfield treatments. So I believe it is misleading to title your section 3 as "The Redfield equation is equivalent to the Lindblad form..." Even the semiclassical Redfield equation is equivalent to Lindblad, but in that case the identity operator is the steady state and this has to be corrected ad hoc. I believe it is not easy to capture these details in pithy eponymous wordings and that some care is required.

CC13: 'Reply on AC1', Tom Barbara, 28 Sep 2021

AC1: 'Reply on CC1', Daniel Abergel, 24 Sep 2021

CC2: 'Comment on mr202162', Malcolm Levitt, 23 Sep 2021
In this submission Rodin and Abergel comment extensively on the 2020 paper by Christian Bengs and myself entitled "A master equation for spin systems far from equilibrium" (doi.org/10.1016/j.jmr.2019.106645). From the wording used by Rodin and Abergel, many readers will assume that (1) we stated in our paper that the "founding fathers" (Bloch, Redfield, Abragam) got the relaxation theory wrong, and (2) we introduced a new theory based on the Lindbladian equations that give the correct result. If Rodin and Abergel deliberately intended that impression to be conveyed, they would be guilty of a bad misrepresentation of our work. In fact, in our paper we state explicitly, many times, that the "founding fathers" were well aware, and mentioned explicitly many times over, the limitations of the semiclassical relaxation theory to a welldefined regime; and (2) the "founding fathers" were well aware that the crude "fix" that leads to the inhomogeneous master equation is invalid outside the assumptions of weak spin order and high spin temperature. In our paper we even cite extensively Redfield's phrase "unless the system is prepared in an unusual way". So if Rodin and Abergel mean to imply that we had stated that the "founding fathers" were unaware of the limitations of semiclassical relaxation theory, they are misrepresenting our work badly, and that impression should be corrected. The wording used by Rodin and Abergel is not entirely clear, so it is not obvious whether that accusation is their intention. Hopefully not. In any case, any possibility of a misrepresentation of our work should be corrected.
In our work we showed that the techniques of open quantum systems theory, and in particular the use of Lindbladian dissipation superoperators, may be used to formulate a relaxation equation which works outside the weakorder limit, bringing magnetic resonance into alignment with many other forms of spectroscopy, Within their paper, Rodin and Abergel appear to suggest a possibly different way to do this (it is not clear to me whether they propose that the result is different). They are certainly free to propose that, but it is unclear what the advantages would be over the existing and wellunderstood and widely used Lindbladian method.
Since our paper was published, Tom Barbara showed that the difficult exposition by Bloch and Hubbard leads to Lindbladianlike equations, many years before Lindblad. It is very possible that the "founding fathers" of magnetic resonance "prediscovered" this key result of open quantum systems theorists. That is a fascinating analysis of the historical development of magnetic resonance, but does not affect the conclusions we drew in our paper, or the way we represented the work of these giants. Incidentally, if Tom's analysis is correct (which is not in doubt), then Abragam seems to have missed this interesting path in his classic book. On pages 287289, Abragam tries hard to find a formulation of relaxation theory which escapes the weak order approximation, but (in a rare weak passage) his exposition leads nowhere useful. Maybe the Bloch and Hubbard papers were too obscure, even for Abragam.
In summary, the new submission by Rodin and Abergel is a valuable contribution to the debate, but I hope they can pay more attention to the way they refer to our paper, and make sure that any way they represent our work is thoroughly validated and cannot be misread by a casual reader. To give a concrete example, they state in the abstract "..the underlying theory .. has been recently questioned" [by us]. This is not correct. We did not "question the underlying theory", in any way. On the contrary. We proposed a way to +implement+ the underlying theory outside the weakorder limit, making it clear that the proposed method is not our invention, but is standard outside the NMR world. Furthermore, Tom Barbara showed that at least Bloch and Hubbard anticipated a similar method many years before Lindblad.

CC3: 'Reply on CC2', Tom Barbara, 23 Sep 2021
I think that Malcolm makes valid points on aspects of language that can give the wrong impression or appear to misrepresent the "facts of the case", and I did notice these on my first reading as well. There is no doubt that Abragam went astray in his treatment, something I knew of very early on in graduate school. Abragam does appear to be the first to suggest iterating the differential equation for the density matrix and that is an improvement, very clearly recognized for anyone who took the time to slog through the details of the second order approach to the solutions of the equations of motion as was done in the early treatments by Bloch, Hubbard and Redfield. The exposition offered here does obscure the fact that Abragam missed the significance of Bloch and that Redfield only addressed the proper approach in his much later Advances in Magnetic Resonance article, wherein he acknowleges the influence of Bloch.
That Bloch "got it right" from the beginning always intiqued me. It was only while I was working things out for my effort that I learned about a very early use of what is essentially a Lindblad approach by Landau, as revealed in the historical review by Chrusinski and Pascazio on the GKLS equation. Now, one has to appreciate that Bloch was there, if not at the very beginning, but certainly very early on in the applications of the new quantum mechanics and I get the sense that Bloch knew about Landau's paper, and of the form he used, and that Bloch was guided by this result of Landau. As this paper currently reads, it does appear to confuse the issue of the significance of the contribution by Bloch.

CC5: 'Reply on CC3', Malcolm Levitt, 24 Sep 2021
That's yet another fascinating historical insight, Tom. It makes it even more surprising that Abragam completely missed this line. I hope that Rodin and Abergel can modify the wording in their paper so that they don't appear to accuse Christian and myself of claiming to discover something "new under the sun". I had rather hoped that our paper was abundantly clear that we were not calling into question the "fundamental theory" as developed by Bloch, Hubbard, Redfield etc., but merely finding a convenient implementation outside the weakorder limit, that does not fall into the trap of the using the inhomogeneous master equation well outside its domain of validity, a domain which Bloch, Hubbard, Redfield, Abragam were all fully aware of. We were also clear that our work was merely an implementation of wellknown techniques which are taken for granted by mathematical physicists, so definitely not something "new under the sun". I remember even having trouble convincing Christian that these results, which Christian largely extracted from the established literature, were worth publishing at all. Basically NMR lagged decades behind other fields in these matters, because our overwhelmingly common regime of weak spin order let us get away with it  until hyperpolarization became common. I think the only thing we can be accused of is of being somewhat late to press, by many decades it seems.

CC6: 'Reply on CC3', Tom Barbara, 24 Sep 2021
When I was a graduate student (19761981) it was not easy to get Redfields IBM article, but the Advances publication was readily available, as most chemistry department libraries carried the entire series. Today one can find the IBM article on many internet sites and though I have looked at this in the recent past, I did not have a clear picture of its contents in my mind, so I had another look. These two efforts are nearly identical, even down the equation numbering! This was a surprise for me, since I knew the lore of the land, that "Refield Theory" was semiclassical and needed the ad hoc correction, and that was the content of the IBM paper. Redfield acknowleges the influence of Bloch in the IBM paper just as with the Advances publication and also produces Eq.3.15 therein. There is some hint at "entanglement", but on the face of the published literature, Redfield produced his own equation with the correct equilbrium steady state independently. It is even more striking then, that Abragam did not cite Redfield or obtain a definitive result in his attempt to get to the same place as Bloch.
I have always enjoyed reading the history of science, even though at times I felt guilty and I guess, incompetent, that I was not spending that time making scientific history. Now in my mature statue, I can freely enjoy it. Especially with this topic.

CC11: 'Reply on CC6', Tom Barbara, 27 Sep 2021
Once I recovered from my surprise that the IBM and Advances publications by Redfield were nearly identical, the thought occured to me to go back to Wagness and Bloch Phys Rev 89, 728 (1952). This gives a clear prioity for the Bloch school, as the first to write down the proper equations.

CC11: 'Reply on CC6', Tom Barbara, 27 Sep 2021

CC5: 'Reply on CC3', Malcolm Levitt, 24 Sep 2021

AC2: 'Reply on CC2', Daniel Abergel, 24 Sep 2021
Dear Malcolm,
Many thanks for these comments that made me realize that we may have used unfortunate wordings in our manuscript, and I would like to make it clear that your insightful « Bengs & Levitt » paper pinpoints the exact limitations of the semiclassical theory of relaxation (« high spin order », to be short). Although the assumptions that allow one to deduce this semiclassical formulation from the full quantum theory are applicable in many practical situations, the breakdown of this approximate theory, and the way to overcome such limitations by using the Linblad approach was clearly and nicely illustrated by several examples in your paper.
In our manuscript, we mention that your paper shows that the « formulation currently used by NMR spectroscopists », leads to erroneous predictions. This, of course, implicitly refers to the semiclassical theory. I hope this will dissipate any misunderstanding, but we can clarify this in a revised version.
The purpose of our work was essentially to clarify a number of aspects of relaxation theory that may sometimes seem quite obscure, which led us to realize that the approach used by Bloch, Redfield and Abragam essentially leads to the Lindblad formulation. The former seems a more physical approach to the treatment of open quantum systems (a small system coupled to a large system in thermal equilibrium), whereas the latter, by defining structural properties that must be obeyed by a master equation, is more of a mathematical nature. When conditions are imposed to ensure that the system is Markovian, both are equivalent.
We also tried to clarify the point that the detailed balance is not quantummechanical by nature, but a statistical one. However, some properties of the correlation functions, specifically implied by the noncommutativity of the bath operators, make both properties entangled. Note in passing that the Linblad formulation requires the additional Kubo assumption regarding the correlation functions to ensure detailed balance, therefore the correct stationary state, whereas this latter property naturally arises in a conventional derivation from the condition that the lattice is in a Boltzmann thermal equilibrium.

CC7: 'Reply on AC2', Malcolm Levitt, 24 Sep 2021
Thankyou, Daniel. We may have a technical disagreement, though. I do not agree that the RedfieldAbragam approach, meaning that which leads to a relaxation superoperator constructed from double commutation superoperators, can ever lead to the correct thermalization for the case of strong spin order. As far as I know, there is no way to construct a "correct" relaxation superoperator purely from double commutators, whether one uses a semiclassical or a quantum theory. The only way to get it right is to abandon double commutation superoperators, as you and Bogdan in your paper do, where you propose "thermalized double commutators", which are technically not double commutators in the ordinary sense. So I believe that there is still a remaining disagreement between us on this point. However I am not completely sure of my ground, since the topic has formidable technical difficulties. Christian is a far more precise and careful thinker on these matters than myself. Nevertheless I think the right approach is for the contrasting approaches to appear in the literature so that the community can judge.

CC9: 'Reply on CC7', Malcolm Levitt, 25 Sep 2021
Just an additional comment, providing a little personal historical context. Speaking for myself, I had thought (arrogantly I guess) that this topic was essentially solved with Jeener and then myself (with Lorenzo Di Bari) introducing thermalized relaxation superoperators and avoiding the inhomogeneous master equation with its ugly and highly suspicious rho_eq term. It was only when Christian Bengs applied this to the ortho/para conversion of endohedral water in H2O@C60 (doi.org/10.1103/PhysRevLett.120.266001) that we realised there was something fundamentally wrong. I took quite some convincing since I suspected that Christian had just made some sort of mistake (I know better, now). Christian took this opportunity to dig into the open quantum systems literature (which was already an interest of his) and managed, with some trouble, to convince me that the doublecommutator form could never give a relaxation superoperator appropriate for this regime, even with a thermalized correction applied afterwards, and formulated a Lindbladianbased treatment that does address this regime correctly as well as being thoroughly in line with the theoretical developments outside NMR. (Indeed, as you say, Daniel, there is nothing "new under the sun"). I want to put on record here how much credit Christian is due for his insight and persistence, in the face of opposition from myself. We now know, through Tom's work, that at least some of the Lindbladian approach was anticipated much earlier by Bloch and Hubbard, although those papers are really tough, and the significance escaped even Abragam, who apparently overlooked this work, making two failed attempts in his book to address the problem. Apart from its historical interest, I personally see no reason to go over this ground again, since the Lindbladian formulation by Christian appears to have all possible advantages and to be solidly grounded. Clearly some in the community disagree (for example, a hostile referee blocked publication of our paper in J Chem Phys, with its broad readership) so it's very appropriate that the competing interpretations are made available in the open scientific literature, for all to judge.

CC9: 'Reply on CC7', Malcolm Levitt, 25 Sep 2021

CC7: 'Reply on AC2', Malcolm Levitt, 24 Sep 2021

CC3: 'Reply on CC2', Tom Barbara, 23 Sep 2021

CC4: 'Comment on mr202162', Tom Barbara, 23 Sep 2021
Looking over section 3.1 and following, I sense a similarity with the efforts of M. Goldman JMR 149 160187 (2001) in his review article. Apparently the authors are not aware of this publication. It would be great if the authors can make some comments in that regard and also of course cite Maurice.

CC8: 'Comment on mr202162', Tom Barbara, 24 Sep 2021
I cannot resist adding this, which is from Michael Berry's web page.
Three laws of discovery
 Arnold’s law (implied by statements in his many letters disputing priority, usually in response to what he sees as neglect of Russian mathematicians)
Discoveries are rarely attributed to the correct person
(Of course Arnold’s law is selfreferential.) Berry’s law (prompted by the observation that the sequence of antecedents under law 1 seems endless)
Nothing is ever discovered for the first time
 Whitehead’s law (Address to the British Association 2016, quoted by Max Dresden at the beginning of his biography of Kramers)
…to come near to a true theory, and to grasp its precise application, are two very different things, as the history of science teaches us. Everything of importance has been said before by someone who did not discover it.

RC1: 'Comment on mr202162', Christian Bengs, 26 Sep 2021

CC10: 'Reply on RC1', Tom Barbara, 27 Sep 2021
Having more time to look over this effort, I also noted some confusing switching between t1, t2 and then ttau, which seems to come and go during the exposition. The switch of course, can only be made after the extension of the interval of integration to infinity. I think it is important to get this straight and with clarity. As I mentioned in my first comment, one real reason for going over this, is to offer the modern approach rather than the old style one used by the early work with the complicated second order solution method for the magnetic resonance community. Otherwise, the basics are already available in Manzano AIP advances 10,025106 (2020) as I pointed out in the conclusion section of my effort.
 CC12: 'Reply on CC10', Tom Barbara, 27 Sep 2021

AC3: 'Reply on RC1', Daniel Abergel, 08 Nov 2021
The comment was uploaded in the form of a supplement: https://mr.copernicus.org/preprints/mr202162/mr202162AC3supplement.pdf

CC10: 'Reply on RC1', Tom Barbara, 27 Sep 2021

CC14: 'Comment on mr202162', Geoffrey Bodenhausen, 03 Oct 2021
Clearly, the papers about the limitations of Redfield's theory have stirred up much interest and have stimulated a fascinating debate. What are the implications? Should we amend our papers on relaxation of hyperpolarized systems that are far from equilibrium? By way of example, should we overhaul our paper on quadrupolar relaxation in dissolution DNP of deuterated methyl groups by Thomas Kress et al.? More generally speaking, can we continue to trust papers that have been published in this field?

CC15: 'Reply on CC14', Malcolm Levitt, 04 Oct 2021
That is a difficult question to answer. It depends of course on the precise nature of the effects and the analysis that was originally used. My guess is that in the great majority of cases there is nothing to worry about  it is likely that the trajectories of observables never differ by more than their thermal equilibrium values between the (accurate) Lindblad and (suspect) traditional approaches. For strong hyperpolarization effects these differences may often be ignored. Nevertheless there will always be residual doubt given that we now know that the standard dynamical equations do not rest on solid foundations. The safest course of action is to do redo some test calculations using the Lindblad method, or a presumably equivalent method such as the one described in this paper.

CC15: 'Reply on CC14', Malcolm Levitt, 04 Oct 2021

CC16: 'Comment on mr202162', Tom Barbara, 11 Oct 2021
I note that much of the discussion in the derivation leading up to equation 22 is misleading in that the statistical properties required are not actually properly flushed out. A clearer discussion can be found in Goldman in the arguments supplied by him after his equation [17]. These are tricky steps to sort out and some greater care should be exercised in the text. An alternative is to just give the result and cite Goldman for the details.

RC2: 'Comment on mr202162', Anonymous Referee #2, 11 Oct 2021
This paper certainly has generated already considerable discussion and I do not find much to add that has not already been said elegantly by other reviewers and commentators. It is often the case in science that alternative derivations of the same result provide new insights into the underlying physical principles, and I think that this paper, following on recent papers by Bengs and Levitt and Barbara, is a good example of the phenomenon. I found the discussion around the left and right spectral density functions and the interrelationship to Boltzmann factors to be particularly interesting and I suspect will be interesting to the many students of the subject who are, like this reviewer, more familiar with classical spectral density functions.

AC4: 'Reply on RC2', Daniel Abergel, 08 Nov 2021
The comment was uploaded in the form of a supplement: https://mr.copernicus.org/preprints/mr202162/mr202162AC4supplement.pdf

AC4: 'Reply on RC2', Daniel Abergel, 08 Nov 2021

RC3: 'Comment on mr202162', Alberto Rosso, 12 Oct 2021
This paper, together with a recent paper by Bengs and Levitt, discusses and revisits the problem of the relaxational dynamics of a collection of interacting spins coupled to the vibrations of the lattice.It is common to assume that the spins are weakly coupled to the lattice and to use a perturbative approach (BornMarkov approximation) that allows to write a master equation of the Redfield kind for the density matrix reduced to the spin degrees of freedom [equation 21 of the main text]. Equation 21 does not have a Lindblad form. A Lindblad form is required to respect the physical properties of a density matrix (positive, trace conserving…).In practice, in the literature further approximations are proposed for the evolution equation of the spin density matrix. The first two (i.e. the ones discussed in this paper) are:
 To introduce a secular approximation that reduces Eq. 21 to Eq 40 or the identical Eq. 43. The equation has a Lindblad form and, by construction, relaxes the interacting spins to their Boltzmann equilibrium. Namely to \sigma_{Boltzmann) = exp(\beta H_S)
 In NMR (however see the historical paper by Tom Barbara) one is used instead to a semiclassical approximation of the Redfield equation which leads to Equation 65. Unfortunately Eq.65 does not have a Lindblad form. In general, for more than 1 spin this equation leads to misleading results that are discussed in this paper as well as in the previous paper by Bengs and Levitt. In my opinion this equation is very clumsy: on one side it does not reproduce a physical evolution and one the other side it is phenomenological. It works when the physics of the problem is well described by a single spin, but it will fail to capture many body effects, among others. Still within the Markovian assumption, instead of designing further approximations to turn Redfield's equation into a physically sound form (Lindblad), one can go one step back and examine the many possible ways of making the Markovian approximation. This leads to another proposal I would like to mention:
 Recently a more general Lindblad evolution has been proposed (PERLind approaches, see e.g. Nathan & Rudner PRB 2020). It holds at the same level of approximation as Redfield's (Eq.21), i.e. secondorder perturbation with respect to the coupling to the lattice, but it is of Lindblad form. In the limit of very weak coupling with the lattice one recovers the secular approximation of Eq. 40. For moderate coupling it captures the competition between the interactions among the spins and the coupling with the lattice. As a result the stationary state of this equation is not exactly exp(\beta H_S) even if, in a strong magnetic field, the total magnetisation will be indistinguishable from the one predicted by Boltzmann. Indeed Boltzmann equilibrium is not expected to hold outside weak systembath coupling. Recently we used this arguably more general equation to show that the spin temperature (generated by dipolar interactions) can be suppressed at high temperature due to the effect of the lattice vibrations (Maimbourg, Basko, Holtzmann, Rosso, PRL 2021).
In conclusion I think it is important (within the Markovian assumption) to stick with welldefined Lindblad forms and I think that this discussion is important. I also wish to advertise that a lot of physics can be found beyond the secular approximation.

CC17: 'Reply on RC3', Tom Barbara, 20 Oct 2021
Thanks for pointing out these articles on generalizing the case for non secular time dependent situations. Taking the square root of a spectral density is a strange thing for the world of NMR applications. Formalisim aside, the important point is when the time dependent terms can be safely neglected. Bloch and Hubbard treat the case of time dependent terms very generally and it is interesting to compare these formalisms with, for example, equations 75a79b of Hubbards review. Applications of amplitude and phase modulalted RF excitations to relaxation, so called rotating frame relaxation, have a long history that you may find interesting. Examples that come to mind are the series of papers by van der Maarel in journal of chemical physics 91(3) 1989, 94(7) 1991 and 99(8) 1993 for I=3/2 quadrupolar systems. More recently the work of Podkorytov and Shynnkov, Journal of Magnetic Resonance 169 (2004) is an interesting application.

AC5: 'Reply on RC3', Daniel Abergel, 08 Nov 2021
The comment was uploaded in the form of a supplement: https://mr.copernicus.org/preprints/mr202162/mr202162AC5supplement.pdf
Status: closed

CC1: 'Comment on mr202162', Tom Barbara, 23 Sep 2021
This effort gives a modern and more accessable account of many details in the theories of Bloch and Hubbard that were not covered in the exposition I wrote up in MR26892021 on the comparison between these and the Lindbladian master equation. A derivation that does not wrangle through the same path taken by Bloch and Hubbard (and Redfield) can be very helpful to readers interested in learning the fundamentals, especially the starting point of iteration of the density matrix evolution as I mentioned in my summary section of MR26892021. A good emphasis is also made in the discussion of "left" and "right" spectral densities and their thermal symmetrization, which allows one to approach the classical limit, as was done by Hubbard with his SECH scale factor.

AC1: 'Reply on CC1', Daniel Abergel, 24 Sep 2021
Thanks for the careful reading of our manuscript. This work has been the opportunity to realize that the seminal work by Bloch not only presents a comprehensive and fairly general approach to relaxation that extends beyond spin relaxation.

CC13: 'Reply on AC1', Tom Barbara, 28 Sep 2021
Yes it is true that one cannot underestimate the significance of Bloch. For that reason I was surprised that modern terminology ascribes equations like your #21 as "Redfield" in kind. Redfield never wrote down this general form in his IBM article. The closest he came was with his Eq.2.30 in that paper and only then in the extreem narrowing limit. I noticed this "new termiinology" first when I read Manzano's paper. From what I read in the literature, equations like Eq.2.30 were first written down by Abragam in his monograph. From that equation, which is very general, the different treatments follow depending on how one interprets the Hamiltonians and the density operators, which is the difference between Bloch and the semiclassical Redfield treatments. So I believe it is misleading to title your section 3 as "The Redfield equation is equivalent to the Lindblad form..." Even the semiclassical Redfield equation is equivalent to Lindblad, but in that case the identity operator is the steady state and this has to be corrected ad hoc. I believe it is not easy to capture these details in pithy eponymous wordings and that some care is required.

CC13: 'Reply on AC1', Tom Barbara, 28 Sep 2021

AC1: 'Reply on CC1', Daniel Abergel, 24 Sep 2021

CC2: 'Comment on mr202162', Malcolm Levitt, 23 Sep 2021
In this submission Rodin and Abergel comment extensively on the 2020 paper by Christian Bengs and myself entitled "A master equation for spin systems far from equilibrium" (doi.org/10.1016/j.jmr.2019.106645). From the wording used by Rodin and Abergel, many readers will assume that (1) we stated in our paper that the "founding fathers" (Bloch, Redfield, Abragam) got the relaxation theory wrong, and (2) we introduced a new theory based on the Lindbladian equations that give the correct result. If Rodin and Abergel deliberately intended that impression to be conveyed, they would be guilty of a bad misrepresentation of our work. In fact, in our paper we state explicitly, many times, that the "founding fathers" were well aware, and mentioned explicitly many times over, the limitations of the semiclassical relaxation theory to a welldefined regime; and (2) the "founding fathers" were well aware that the crude "fix" that leads to the inhomogeneous master equation is invalid outside the assumptions of weak spin order and high spin temperature. In our paper we even cite extensively Redfield's phrase "unless the system is prepared in an unusual way". So if Rodin and Abergel mean to imply that we had stated that the "founding fathers" were unaware of the limitations of semiclassical relaxation theory, they are misrepresenting our work badly, and that impression should be corrected. The wording used by Rodin and Abergel is not entirely clear, so it is not obvious whether that accusation is their intention. Hopefully not. In any case, any possibility of a misrepresentation of our work should be corrected.
In our work we showed that the techniques of open quantum systems theory, and in particular the use of Lindbladian dissipation superoperators, may be used to formulate a relaxation equation which works outside the weakorder limit, bringing magnetic resonance into alignment with many other forms of spectroscopy, Within their paper, Rodin and Abergel appear to suggest a possibly different way to do this (it is not clear to me whether they propose that the result is different). They are certainly free to propose that, but it is unclear what the advantages would be over the existing and wellunderstood and widely used Lindbladian method.
Since our paper was published, Tom Barbara showed that the difficult exposition by Bloch and Hubbard leads to Lindbladianlike equations, many years before Lindblad. It is very possible that the "founding fathers" of magnetic resonance "prediscovered" this key result of open quantum systems theorists. That is a fascinating analysis of the historical development of magnetic resonance, but does not affect the conclusions we drew in our paper, or the way we represented the work of these giants. Incidentally, if Tom's analysis is correct (which is not in doubt), then Abragam seems to have missed this interesting path in his classic book. On pages 287289, Abragam tries hard to find a formulation of relaxation theory which escapes the weak order approximation, but (in a rare weak passage) his exposition leads nowhere useful. Maybe the Bloch and Hubbard papers were too obscure, even for Abragam.
In summary, the new submission by Rodin and Abergel is a valuable contribution to the debate, but I hope they can pay more attention to the way they refer to our paper, and make sure that any way they represent our work is thoroughly validated and cannot be misread by a casual reader. To give a concrete example, they state in the abstract "..the underlying theory .. has been recently questioned" [by us]. This is not correct. We did not "question the underlying theory", in any way. On the contrary. We proposed a way to +implement+ the underlying theory outside the weakorder limit, making it clear that the proposed method is not our invention, but is standard outside the NMR world. Furthermore, Tom Barbara showed that at least Bloch and Hubbard anticipated a similar method many years before Lindblad.

CC3: 'Reply on CC2', Tom Barbara, 23 Sep 2021
I think that Malcolm makes valid points on aspects of language that can give the wrong impression or appear to misrepresent the "facts of the case", and I did notice these on my first reading as well. There is no doubt that Abragam went astray in his treatment, something I knew of very early on in graduate school. Abragam does appear to be the first to suggest iterating the differential equation for the density matrix and that is an improvement, very clearly recognized for anyone who took the time to slog through the details of the second order approach to the solutions of the equations of motion as was done in the early treatments by Bloch, Hubbard and Redfield. The exposition offered here does obscure the fact that Abragam missed the significance of Bloch and that Redfield only addressed the proper approach in his much later Advances in Magnetic Resonance article, wherein he acknowleges the influence of Bloch.
That Bloch "got it right" from the beginning always intiqued me. It was only while I was working things out for my effort that I learned about a very early use of what is essentially a Lindblad approach by Landau, as revealed in the historical review by Chrusinski and Pascazio on the GKLS equation. Now, one has to appreciate that Bloch was there, if not at the very beginning, but certainly very early on in the applications of the new quantum mechanics and I get the sense that Bloch knew about Landau's paper, and of the form he used, and that Bloch was guided by this result of Landau. As this paper currently reads, it does appear to confuse the issue of the significance of the contribution by Bloch.

CC5: 'Reply on CC3', Malcolm Levitt, 24 Sep 2021
That's yet another fascinating historical insight, Tom. It makes it even more surprising that Abragam completely missed this line. I hope that Rodin and Abergel can modify the wording in their paper so that they don't appear to accuse Christian and myself of claiming to discover something "new under the sun". I had rather hoped that our paper was abundantly clear that we were not calling into question the "fundamental theory" as developed by Bloch, Hubbard, Redfield etc., but merely finding a convenient implementation outside the weakorder limit, that does not fall into the trap of the using the inhomogeneous master equation well outside its domain of validity, a domain which Bloch, Hubbard, Redfield, Abragam were all fully aware of. We were also clear that our work was merely an implementation of wellknown techniques which are taken for granted by mathematical physicists, so definitely not something "new under the sun". I remember even having trouble convincing Christian that these results, which Christian largely extracted from the established literature, were worth publishing at all. Basically NMR lagged decades behind other fields in these matters, because our overwhelmingly common regime of weak spin order let us get away with it  until hyperpolarization became common. I think the only thing we can be accused of is of being somewhat late to press, by many decades it seems.

CC6: 'Reply on CC3', Tom Barbara, 24 Sep 2021
When I was a graduate student (19761981) it was not easy to get Redfields IBM article, but the Advances publication was readily available, as most chemistry department libraries carried the entire series. Today one can find the IBM article on many internet sites and though I have looked at this in the recent past, I did not have a clear picture of its contents in my mind, so I had another look. These two efforts are nearly identical, even down the equation numbering! This was a surprise for me, since I knew the lore of the land, that "Refield Theory" was semiclassical and needed the ad hoc correction, and that was the content of the IBM paper. Redfield acknowleges the influence of Bloch in the IBM paper just as with the Advances publication and also produces Eq.3.15 therein. There is some hint at "entanglement", but on the face of the published literature, Redfield produced his own equation with the correct equilbrium steady state independently. It is even more striking then, that Abragam did not cite Redfield or obtain a definitive result in his attempt to get to the same place as Bloch.
I have always enjoyed reading the history of science, even though at times I felt guilty and I guess, incompetent, that I was not spending that time making scientific history. Now in my mature statue, I can freely enjoy it. Especially with this topic.

CC11: 'Reply on CC6', Tom Barbara, 27 Sep 2021
Once I recovered from my surprise that the IBM and Advances publications by Redfield were nearly identical, the thought occured to me to go back to Wagness and Bloch Phys Rev 89, 728 (1952). This gives a clear prioity for the Bloch school, as the first to write down the proper equations.

CC11: 'Reply on CC6', Tom Barbara, 27 Sep 2021

CC5: 'Reply on CC3', Malcolm Levitt, 24 Sep 2021

AC2: 'Reply on CC2', Daniel Abergel, 24 Sep 2021
Dear Malcolm,
Many thanks for these comments that made me realize that we may have used unfortunate wordings in our manuscript, and I would like to make it clear that your insightful « Bengs & Levitt » paper pinpoints the exact limitations of the semiclassical theory of relaxation (« high spin order », to be short). Although the assumptions that allow one to deduce this semiclassical formulation from the full quantum theory are applicable in many practical situations, the breakdown of this approximate theory, and the way to overcome such limitations by using the Linblad approach was clearly and nicely illustrated by several examples in your paper.
In our manuscript, we mention that your paper shows that the « formulation currently used by NMR spectroscopists », leads to erroneous predictions. This, of course, implicitly refers to the semiclassical theory. I hope this will dissipate any misunderstanding, but we can clarify this in a revised version.
The purpose of our work was essentially to clarify a number of aspects of relaxation theory that may sometimes seem quite obscure, which led us to realize that the approach used by Bloch, Redfield and Abragam essentially leads to the Lindblad formulation. The former seems a more physical approach to the treatment of open quantum systems (a small system coupled to a large system in thermal equilibrium), whereas the latter, by defining structural properties that must be obeyed by a master equation, is more of a mathematical nature. When conditions are imposed to ensure that the system is Markovian, both are equivalent.
We also tried to clarify the point that the detailed balance is not quantummechanical by nature, but a statistical one. However, some properties of the correlation functions, specifically implied by the noncommutativity of the bath operators, make both properties entangled. Note in passing that the Linblad formulation requires the additional Kubo assumption regarding the correlation functions to ensure detailed balance, therefore the correct stationary state, whereas this latter property naturally arises in a conventional derivation from the condition that the lattice is in a Boltzmann thermal equilibrium.

CC7: 'Reply on AC2', Malcolm Levitt, 24 Sep 2021
Thankyou, Daniel. We may have a technical disagreement, though. I do not agree that the RedfieldAbragam approach, meaning that which leads to a relaxation superoperator constructed from double commutation superoperators, can ever lead to the correct thermalization for the case of strong spin order. As far as I know, there is no way to construct a "correct" relaxation superoperator purely from double commutators, whether one uses a semiclassical or a quantum theory. The only way to get it right is to abandon double commutation superoperators, as you and Bogdan in your paper do, where you propose "thermalized double commutators", which are technically not double commutators in the ordinary sense. So I believe that there is still a remaining disagreement between us on this point. However I am not completely sure of my ground, since the topic has formidable technical difficulties. Christian is a far more precise and careful thinker on these matters than myself. Nevertheless I think the right approach is for the contrasting approaches to appear in the literature so that the community can judge.

CC9: 'Reply on CC7', Malcolm Levitt, 25 Sep 2021
Just an additional comment, providing a little personal historical context. Speaking for myself, I had thought (arrogantly I guess) that this topic was essentially solved with Jeener and then myself (with Lorenzo Di Bari) introducing thermalized relaxation superoperators and avoiding the inhomogeneous master equation with its ugly and highly suspicious rho_eq term. It was only when Christian Bengs applied this to the ortho/para conversion of endohedral water in H2O@C60 (doi.org/10.1103/PhysRevLett.120.266001) that we realised there was something fundamentally wrong. I took quite some convincing since I suspected that Christian had just made some sort of mistake (I know better, now). Christian took this opportunity to dig into the open quantum systems literature (which was already an interest of his) and managed, with some trouble, to convince me that the doublecommutator form could never give a relaxation superoperator appropriate for this regime, even with a thermalized correction applied afterwards, and formulated a Lindbladianbased treatment that does address this regime correctly as well as being thoroughly in line with the theoretical developments outside NMR. (Indeed, as you say, Daniel, there is nothing "new under the sun"). I want to put on record here how much credit Christian is due for his insight and persistence, in the face of opposition from myself. We now know, through Tom's work, that at least some of the Lindbladian approach was anticipated much earlier by Bloch and Hubbard, although those papers are really tough, and the significance escaped even Abragam, who apparently overlooked this work, making two failed attempts in his book to address the problem. Apart from its historical interest, I personally see no reason to go over this ground again, since the Lindbladian formulation by Christian appears to have all possible advantages and to be solidly grounded. Clearly some in the community disagree (for example, a hostile referee blocked publication of our paper in J Chem Phys, with its broad readership) so it's very appropriate that the competing interpretations are made available in the open scientific literature, for all to judge.

CC9: 'Reply on CC7', Malcolm Levitt, 25 Sep 2021

CC7: 'Reply on AC2', Malcolm Levitt, 24 Sep 2021

CC3: 'Reply on CC2', Tom Barbara, 23 Sep 2021

CC4: 'Comment on mr202162', Tom Barbara, 23 Sep 2021
Looking over section 3.1 and following, I sense a similarity with the efforts of M. Goldman JMR 149 160187 (2001) in his review article. Apparently the authors are not aware of this publication. It would be great if the authors can make some comments in that regard and also of course cite Maurice.

CC8: 'Comment on mr202162', Tom Barbara, 24 Sep 2021
I cannot resist adding this, which is from Michael Berry's web page.
Three laws of discovery
 Arnold’s law (implied by statements in his many letters disputing priority, usually in response to what he sees as neglect of Russian mathematicians)
Discoveries are rarely attributed to the correct person
(Of course Arnold’s law is selfreferential.) Berry’s law (prompted by the observation that the sequence of antecedents under law 1 seems endless)
Nothing is ever discovered for the first time
 Whitehead’s law (Address to the British Association 2016, quoted by Max Dresden at the beginning of his biography of Kramers)
…to come near to a true theory, and to grasp its precise application, are two very different things, as the history of science teaches us. Everything of importance has been said before by someone who did not discover it.

RC1: 'Comment on mr202162', Christian Bengs, 26 Sep 2021

CC10: 'Reply on RC1', Tom Barbara, 27 Sep 2021
Having more time to look over this effort, I also noted some confusing switching between t1, t2 and then ttau, which seems to come and go during the exposition. The switch of course, can only be made after the extension of the interval of integration to infinity. I think it is important to get this straight and with clarity. As I mentioned in my first comment, one real reason for going over this, is to offer the modern approach rather than the old style one used by the early work with the complicated second order solution method for the magnetic resonance community. Otherwise, the basics are already available in Manzano AIP advances 10,025106 (2020) as I pointed out in the conclusion section of my effort.
 CC12: 'Reply on CC10', Tom Barbara, 27 Sep 2021

AC3: 'Reply on RC1', Daniel Abergel, 08 Nov 2021
The comment was uploaded in the form of a supplement: https://mr.copernicus.org/preprints/mr202162/mr202162AC3supplement.pdf

CC10: 'Reply on RC1', Tom Barbara, 27 Sep 2021

CC14: 'Comment on mr202162', Geoffrey Bodenhausen, 03 Oct 2021
Clearly, the papers about the limitations of Redfield's theory have stirred up much interest and have stimulated a fascinating debate. What are the implications? Should we amend our papers on relaxation of hyperpolarized systems that are far from equilibrium? By way of example, should we overhaul our paper on quadrupolar relaxation in dissolution DNP of deuterated methyl groups by Thomas Kress et al.? More generally speaking, can we continue to trust papers that have been published in this field?

CC15: 'Reply on CC14', Malcolm Levitt, 04 Oct 2021
That is a difficult question to answer. It depends of course on the precise nature of the effects and the analysis that was originally used. My guess is that in the great majority of cases there is nothing to worry about  it is likely that the trajectories of observables never differ by more than their thermal equilibrium values between the (accurate) Lindblad and (suspect) traditional approaches. For strong hyperpolarization effects these differences may often be ignored. Nevertheless there will always be residual doubt given that we now know that the standard dynamical equations do not rest on solid foundations. The safest course of action is to do redo some test calculations using the Lindblad method, or a presumably equivalent method such as the one described in this paper.

CC15: 'Reply on CC14', Malcolm Levitt, 04 Oct 2021

CC16: 'Comment on mr202162', Tom Barbara, 11 Oct 2021
I note that much of the discussion in the derivation leading up to equation 22 is misleading in that the statistical properties required are not actually properly flushed out. A clearer discussion can be found in Goldman in the arguments supplied by him after his equation [17]. These are tricky steps to sort out and some greater care should be exercised in the text. An alternative is to just give the result and cite Goldman for the details.

RC2: 'Comment on mr202162', Anonymous Referee #2, 11 Oct 2021
This paper certainly has generated already considerable discussion and I do not find much to add that has not already been said elegantly by other reviewers and commentators. It is often the case in science that alternative derivations of the same result provide new insights into the underlying physical principles, and I think that this paper, following on recent papers by Bengs and Levitt and Barbara, is a good example of the phenomenon. I found the discussion around the left and right spectral density functions and the interrelationship to Boltzmann factors to be particularly interesting and I suspect will be interesting to the many students of the subject who are, like this reviewer, more familiar with classical spectral density functions.

AC4: 'Reply on RC2', Daniel Abergel, 08 Nov 2021
The comment was uploaded in the form of a supplement: https://mr.copernicus.org/preprints/mr202162/mr202162AC4supplement.pdf

AC4: 'Reply on RC2', Daniel Abergel, 08 Nov 2021

RC3: 'Comment on mr202162', Alberto Rosso, 12 Oct 2021
This paper, together with a recent paper by Bengs and Levitt, discusses and revisits the problem of the relaxational dynamics of a collection of interacting spins coupled to the vibrations of the lattice.It is common to assume that the spins are weakly coupled to the lattice and to use a perturbative approach (BornMarkov approximation) that allows to write a master equation of the Redfield kind for the density matrix reduced to the spin degrees of freedom [equation 21 of the main text]. Equation 21 does not have a Lindblad form. A Lindblad form is required to respect the physical properties of a density matrix (positive, trace conserving…).In practice, in the literature further approximations are proposed for the evolution equation of the spin density matrix. The first two (i.e. the ones discussed in this paper) are:
 To introduce a secular approximation that reduces Eq. 21 to Eq 40 or the identical Eq. 43. The equation has a Lindblad form and, by construction, relaxes the interacting spins to their Boltzmann equilibrium. Namely to \sigma_{Boltzmann) = exp(\beta H_S)
 In NMR (however see the historical paper by Tom Barbara) one is used instead to a semiclassical approximation of the Redfield equation which leads to Equation 65. Unfortunately Eq.65 does not have a Lindblad form. In general, for more than 1 spin this equation leads to misleading results that are discussed in this paper as well as in the previous paper by Bengs and Levitt. In my opinion this equation is very clumsy: on one side it does not reproduce a physical evolution and one the other side it is phenomenological. It works when the physics of the problem is well described by a single spin, but it will fail to capture many body effects, among others. Still within the Markovian assumption, instead of designing further approximations to turn Redfield's equation into a physically sound form (Lindblad), one can go one step back and examine the many possible ways of making the Markovian approximation. This leads to another proposal I would like to mention:
 Recently a more general Lindblad evolution has been proposed (PERLind approaches, see e.g. Nathan & Rudner PRB 2020). It holds at the same level of approximation as Redfield's (Eq.21), i.e. secondorder perturbation with respect to the coupling to the lattice, but it is of Lindblad form. In the limit of very weak coupling with the lattice one recovers the secular approximation of Eq. 40. For moderate coupling it captures the competition between the interactions among the spins and the coupling with the lattice. As a result the stationary state of this equation is not exactly exp(\beta H_S) even if, in a strong magnetic field, the total magnetisation will be indistinguishable from the one predicted by Boltzmann. Indeed Boltzmann equilibrium is not expected to hold outside weak systembath coupling. Recently we used this arguably more general equation to show that the spin temperature (generated by dipolar interactions) can be suppressed at high temperature due to the effect of the lattice vibrations (Maimbourg, Basko, Holtzmann, Rosso, PRL 2021).
In conclusion I think it is important (within the Markovian assumption) to stick with welldefined Lindblad forms and I think that this discussion is important. I also wish to advertise that a lot of physics can be found beyond the secular approximation.

CC17: 'Reply on RC3', Tom Barbara, 20 Oct 2021
Thanks for pointing out these articles on generalizing the case for non secular time dependent situations. Taking the square root of a spectral density is a strange thing for the world of NMR applications. Formalisim aside, the important point is when the time dependent terms can be safely neglected. Bloch and Hubbard treat the case of time dependent terms very generally and it is interesting to compare these formalisms with, for example, equations 75a79b of Hubbards review. Applications of amplitude and phase modulalted RF excitations to relaxation, so called rotating frame relaxation, have a long history that you may find interesting. Examples that come to mind are the series of papers by van der Maarel in journal of chemical physics 91(3) 1989, 94(7) 1991 and 99(8) 1993 for I=3/2 quadrupolar systems. More recently the work of Podkorytov and Shynnkov, Journal of Magnetic Resonance 169 (2004) is an interesting application.

AC5: 'Reply on RC3', Daniel Abergel, 08 Nov 2021
The comment was uploaded in the form of a supplement: https://mr.copernicus.org/preprints/mr202162/mr202162AC5supplement.pdf
Bogdan A. Rodin and Daniel Abergel
Bogdan A. Rodin and Daniel Abergel
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