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.
Highlights
In a recent and important publication, Bengs and Levitt (2019) formalize NMR relaxation theory for systems that deviate significantly from the equilibrium state, which are conditions that invalidate the high-temperature weak-ordering approximation, a corner stone in the commonly used Redfield theory (Redfield, 1957)
Bengs and Levitt employed very modern methods that arose in the late 1970s and are fundamental in the topic of open quantum systems
The Lindbladian form is often written as above, but it is important to recognize that it can be expressed in terms of commutators, as was used in the original work by Gorini et al (1976): N, N † + N, N † = − N †N, + 2N N †
Summary
In a recent and important publication, Bengs and Levitt (2019) formalize NMR relaxation theory for systems that deviate significantly from the equilibrium state, which are conditions that invalidate the high-temperature weak-ordering approximation, a corner stone in the commonly used Redfield theory (Redfield, 1957). A brief exposition of Bloch’s main results, by way of Hubbard, using Hubbard’s own notation, appears to be a worthwhile endeavor, with historical importance and reasonable expectations that such an effort will be of some interest to NMR researchers in general How this task has fallen into the hands of the author may be of some help in orienting the reader. In addition to the Lindbladian form of Bloch’s generalized theory of relaxation, we take advantage of the opportunity to highlight Bloch’s and Hubbard’s early, and largely unrecognized, contributions to the dynamics of spin locking and rotating-frame relaxation Together these aspects form a basis for a renewed interest or, at the very least, a new and greater appreciation of these classic publications
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