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Magnetic Resonance

Spin relaxation: is there anything new under the Sun?

Bogdan A. Rodin, Daniel Abergel

Published Feb 9, 2022• 12 citations
Publisher

Abstract

Spin relaxation has been at the core of many studies since the early days of nuclear magnetic resonance (NMR) and the underlying theory worked out by its founding fathers. This Bloch-Redfield-Abraham relaxation theory has been recently reinvestigated () in the perspective of Lindblad theory of quantum Markovian master equations in order to account for situations where the widely used semi-classical relaxation theory breaks down. 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 Lindblad formulation. We also comment on the debate on semi-classical versus quantum versions of spectral density functions involved in relaxation.

DOI
10.5194/mr-3-27-2022

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Spin relaxation has been at the core of many studies since the early days of nuclear magnetic resonance (NMR) and the underlying theory worked out by its founding fathers. This Bloch-Redfield-Abraham relaxation theory has been recently reinvestigated () in the perspective of Lindblad theory of quantum Markovian master equations in order to account for situations where the widely used semi-classical relaxation theory breaks down. 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 Lindblad formulation. We also comment on the debate on semi-classical versus quantum versions of spectral density functions involved in relaxation.
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 Redfields 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.
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.
Nuclear Magnetic Resonance (NMR) experiments involve coherent and incoherent dynamics. Incoherent dynamics give rise to nuclear spin relaxation typically accounted for by a differential equation known as a quantum master equation. Out of the existing master equations the Lindblad equation is particularly important. A central theorem of open quantum theory states that a relaxation process represents a quantum Markovian process if and only if it can be described by a Lindblad master equation. This in turn raises serious questions regarding the compatibility of conventional NMR relaxation theory and one of the most fundamental results of open quantum theory. The NMR treatment of important Markovian relaxation mechanisms such as Brownian motion or chemical exchange events do in general not result in a master equation in Lindblad form and would therefore be at odds with the mathematically rigorous theory of open quantum systems. We explore this apparent conflict for two important chemical exchange phenomena and show that their conventional treatment may be recast into a suitable Lindblad equation. Similarities between spatial diffusion and multi-site exchange events allow us to extend our results to nuclear spin relaxation due to translational diffusion. As a result we are able to show that the NMR treatment of intramolecular exchange and translational diffusion represent valid quantum Markovian processes compatible with the rigorous definitions of open quantum theory.
This paper is devoted to the development of a rigorous theory of angular-momentum relaxation. It is shown that by incorporating quantum-dynamical semigroups in the quantum Markovian master equation, the theory becomes mathematically sound. The expansion of the initial relaxation process in terms of primary relaxation processes of rank l offers a clearer view of the relaxation process structure. A close look at experimental procedure (the system Hilbert space structure, symmetry of the environment, and multipolarities of the system-environment interaction) gives all the information needed to simplify the structure of the relaxation equation and renders it a powerful tool for considering specific physical effects. The quantum Markovian master equation in the state multipole representation is obtained and the physical meaning of the parameters involved is found. The general approach is used for the study of a Zeeman multiplet. The relaxation rates as functions of the spectral densities of correlations (describing the environment properties) are given in an explicit form. Cross relaxation and cross correlation are taken properly into account there. The important case of spin 1 is discussed in more detail. Some useful expressions and relations among the relaxation rates are given in the Appendixes.
Abstract We derive the semi-classical Lindblad master equation in phase space for both canonical and non-canonical Poisson brackets using the Wigner–Moyal formalism and the Moyal star-product. The semi-classical limit for canonical dynamical variables, i.e. canonical Poisson brackets, is the Fokker–Planck equation, as derived before. We generalize this limit and show that it holds also for non-canonical Poisson brackets. Examples are gyro-Poisson brackets, which occur in spin ensembles, systems of recent interest in atomic physics and quantum optics. We show that the equations of motion for the collective spin variables are given by the Bloch equations of nuclear magnetization with relaxation. The Bloch and relaxation vectors are expressed in terms of the microscopic operators: the Hamiltonian and the Lindblad functions in the Wigner–Moyal formalism.