Magn. Reson., 2, 689–698, 2021 Open Access https://doi.org/10.5194/mr-2-689-2021 © Author(s) 2021. This work is distributed under the Creative Commons Attribution 4.0 License. The Lindbladian form and the reincarnation of Felix Bloch’s generalized theory of relaxation Thomas M. Barbara Advanced Imaging Research Center, Oregon Health and Sciences University, Portland, OR 97239, USA Correspondence: Thomas M. Barbara ([email protected]) Received: 21 May 2021 – Discussion started: 27 May 2021 Revised: 6 July 2021 – Accepted: 7 July 2021 – Published: 26 August 2021 Abstract. The relationship between the classic magnetic resonance density matrix relaxation theories of Bloch and Hubbard and the modern Lindbladian master equation methods are explored. These classic theories are in full agreement with the latest results obtained by the modern methods. A careful scrutiny shows that this also holds true for Redfield’s later treatment, offered in 1965. The early contributions of Bloch and Hubbard to rotating-frame relaxation theory are also highlighted. Taken together, these seminal efforts of Bloch and Hubbard can enjoy a new birth of contemporary relevance in magnetic resonance. 1 Introduction Levitt have resolved a long-standing and long misunderstood issue. In a recent and important publication, Bengs and Levitt This confusion over the results of Bloch and Hubbard is (2019) formalize NMR relaxation theory for systems that de- likely due, in part, to the use of difficult notation by Bloch. viate significantly from the equilibrium state, which are con- Hubbard’s treatment is a significant improvement but also ditions that invalidate the high-temperature weak-ordering possesses a few obscure aspects. In that situation, it is not dif- approximation, a corner stone in the commonly used Red- ficult to understand that nearly all NMR researchers rely on field theory (Redfield, 1957). Bengs and Levitt employed the simpler Redfield formalism, especially given the fact that very modern methods that arose in the late 1970s and are now the conditions under which the approximations are applica- fundamental in the topic of open quantum systems. These ble are those encountered in the vast majority of cases. Bengs methods are often referred to eponymously as the “Lindbla- and Levitt make a detailed comparison between other pro- dian Form”. In addition to the work of Bengs and Levitt, a posals for a more proper relaxation theory that naturally con- useful tutorial on this topic can be found in the work of Man- tains the correct equilibrium steady state, and all are found to zano (2020). Even though many, if not all, of the classic pa- be defective in one way or another. One argument for the im- pers are cited in their offering of their own Lindbladian anal- portance of Bengs and Levitt’s effort is their unequivocal and ysis, Bengs and Levitt were not aware that a result identical independent confirmation of the success of Bloch and Hub- to their own was already offered by Bloch (1957) and mas- bard over the other formulations. The claim, that a significant terly expounded upon by Hubbard (1961). This conclusion aspect of relaxation theory, with an overcast coastline, now can be ascertained by a glance at Table 1 of (Bengs, 2019), enjoys blue sky, is a fair one. where references to neither Bloch nor Hubbard appear in the A brief exposition of Bloch’s main results, by way of Hub- rightmost column. From a study of past efforts, as reviewed bard, using Hubbard’s own notation, appears therefore to be a by Bengs and Levitt along with their own results, this holds worthwhile endeavor, with historical importance and reason- for Abragam, for Jeener, and for Ernst. The significance of able expectations that such an effort will be of some interest Bloch and Hubbard has gone unappreciated by the NMR to NMR researchers in general. How this task has fallen into community for decades. By approaching the problem from the hands of the author may also be of some help in orienting a new perspective and, importantly, with new experimental the reader. measurements on specially prepared spin systems, Bengs and Published by Copernicus Publications on behalf of the Groupement AMPERE. 690 T. M. Barbara: The Lindbladian form The origins reside in the author’s thesis work, a part of mensions (Gorini et al., 1976) as well as for a general which involved the problem of anisotropic spin–lattice relax- Hilbert space (Lindblad, 1976). Because of these com- ation measurements for deuterated molecules dissolved in a bined efforts, the Lindbladian form is also often referred liquid crystalline matrix. This induced the author to study all to as the GKSL equation (Gorini–Kossakowski–Sudarshan– the early papers on relaxation theory very carefully. Discov- Lindblad). For finite dimensional problems, as is pertinent ering the significance of Bloch, and the clarity of Hubbard’s to NMR applications, it is possible to offer a straightfor- exposition, left an indelible impression, even though the au- ward method of construction that is reasonably motivated and thor happily used the simpler Redfield formalism for the task which uses elementary operator algebra. What follows be- at hand. The author abandoned the field in 1988 to pursue low is very much ex post facto, and it was developed by the other interests but always enjoyed reading the current NMR author after reading Bengs and Levitt (2019). It is not with- relaxation literature. Twenty-five years later, a coworker gen- out precedent however. In an interesting historical overview tly persuaded the author to work out aspects of relaxation of the Lindbladian form (Chruscinski and Pascazio, 2017), and exchange during adiabatic sweeps (Barbara, 2016). Af- one can find examples of nearly identical approaches and ter completing that task, the author reacquainted himself with even a very early use of the Lindbladian form by Landau Bloch and Hubbard in order to understand some lingering de- in 1927. For a very readable tutorial and overview, Man- tails. This effort fully prepared the author to recognize how zano and Gyamfi are recommended. For strictly mathemat- the work of Bloch and Hubbard tied into, and formed a prece- ical proofs on the semi-group, Gorini (1976) and Lindblad dent for, the recent efforts of Bengs and Levitt (2019). (1976) should be consulted. The reader should not interpret The main goal is then to go beyond a citation and demon- what follows as a replacement of the rigorous mathemati- strate the equivalence of the major result in Bengs and Levitt cal proofs of the semi-group. The key to the approach is in to that of Bloch and Hubbard. The significant and original the importance of matrix factoring. Indeed, matrix factoring result offered here is in the mathematical analysis required to in terms of Kronecker products is an essential ingredient in reveal the equivalent success of Bloch and Hubbard towards the mathematical proofs, but its role is not often explicitly the problem confronted and solved by Bengs and Levitt by highlighted in the manner given here. Matrix products also their use of Lindbladian methods. The effort is not trivial, arise in a straightforward manner when relaxation theory is and it requires a careful study of Hubbard’s notation and approached by way of weak coupling perturbation theory for the development of symmetry properties that Hubbard does the spin system and the bath, so in this case the spin matri- not provide. Furthermore, Hubbard offers expressions that in ces are already factored. Along the way, the simplest Lind- some ways are superior to the “fully reduced” Lindbladian bladian, i.e., the one used for exchange in NMR and which form, as will be discussed in Sect. 3. In the conclusion sec- involves the factoring of the identity operator, is presented. tion, the relation of these aspects of relaxation theory to later As an ansatz, one can start from a generalized “state vec- work by Redfield is commented on. tor” dynamics, governed by a general complex operator M A discussion of the Lindbladian form is presented before- X hand, from a simple and very direct approach which pos- cPi D Mij cj : (1) sesses useful didactics, thereby allowing non-experts to ap- j preciate the essence of the formalism without going into a In Eq. (1), M generalizes what is usually the Hamiltonian, so full review of the mathematical details involved in a rigor- the equation, in a purely analogous manner, represents a type ous proof. Since the Lindbladian approach is rather new to of “time-dependent Schrodinger equation” for a finite num- the NMR literature, it can be helpful to have a more facile ber of states. From the vector components c , we construct a presentation. Those readers interested in more details can i Hermitian rank-one “density operator” with elements find valuable sources in Bengs and Levitt (2019), Manzano (2020), and Gyamfi (2020). ∗ %ij D cic : (2) In addition to the Lindbladian form of Bloch’s generalized j theory of relaxation, we take advantage of the opportunity The square of all rank-one operators are proportional to to highlight Bloch’s and Hubbard’s early, and also largely 2 D D ∗ themselves, % α%, so in this case α 6cici . However, unrecognized, contributions to the dynamics of spin locking keep in mind that the converse does not hold true. The termi- and rotating-frame relaxation. Together these aspects form a nology of “rank” has a number of variants, and in this section basis for a renewed interest or, at the very least, a new and we adopt the usage common in the theory of linear vector greater appreciation of these classic publications. spaces (Halmos, 1958). The terms “state vector” and “density operator” have been placed in quotation marks to emphasize their heuristic labels 2 A guide to Lindblad at this early stage of the construction.