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 (

Relaxation is the process through which a system loses energy to its environment to eventually reach a state of thermal equilibrium.
Spin–lattice relaxation has been described as the way spins transfer energy to orientation degrees of freedom. Since the early days of nuclear magnetic resonance (NMR), it has been the subject of numerous studies, both theoretical and encompassing a wide range of domains of applications.
NMR relaxation theory really was contemporary in the early days of NMR, and it was formalized by several of its founding fathers

It has been shown recently by

Such an unexpected behavior was ascribed to the fact that some of the assumptions of the theory may not be fulfilled in NMR systems, and the authors solved the problem by making use of the Lindblad operator, which is commonly used in the theory of open quantum systems to account for dissipative Markovian phenomena, i.e., relaxation processes. Among other properties, the structure of the Lindblad operator ensures that the fundamental properties of the density operator

The traditional description of NMR relaxation relies on the description of both the spin system and the lattice as quantum systems, an approach that leads to the celebrated Redfield equation (

It is the purpose of this paper to re-investigate these old questions in order to trace the roles and the consequences of the various assumptions of the traditional approach to relaxation developed in the early days of NMR.

The derivation follows the lines of

The dynamics restricted to the spin system are obtained by eliminating the bath variables. This is achieved by performing a partial trace over the bath degrees of freedom as follows:

Reverting to the Schrödinger representation

In spin relaxation theory, it is customary to express the relaxation equation in the operator form, which often provides a clearer representation of the spin–bath coupling dynamics. Here, the coupling Hamiltonian is assumed to have the form of a sum of terms, each of which factorizes into a product of lattice

It is now straightforward to show that the conventional Bloch–Redfield–Abraham perturbative approach of relaxation is equivalent to the Lindblad formulation of dissipative systems. Indeed, by changing indices in the first term, using the property

The fact that this derivation leads to the Lindblad equation is not obvious.
In principle, one should not expect the perturbative approach to yield an irreversible dissipative operator equivalent to a Lindblad operator. In fact, this equivalence requires the Markovian and the short correlation time assumptions that make the evolution equation depend on the density operator at the present time

Another point is worth mentioning. The properties of the correlation functions that emerge through this procedure reflect the properties of the bath operators of the spin–bath coupling Hamiltonian and, therefore, convey additional properties that are not implied by the structure of the Lindblad in Eqs. (

It is possible to obtain an alternative and completely equivalent form
of the Redfield equation. Expanding Eq. (

A semi-classical version of the master equation can be extremely useful, allowing one to make use of models derived from the framework of classical mechanics to calculate the spectral density functions. In order to obtain such a theory associated to Eqs. (

A semi-classical relaxation theory should provide spectral density functions obeying the general classical mechanics requirements detailed above.
It is clear, however, that in the quantum case, where

A semi-classical version of the Redfield equation is thus obtained by using the spectral density function

The final relaxation super-operator, which defines the relaxation of the density matrix as

Here,

Equation (

The above form of the relaxation master equation (Eqs.

When the largest eigenvalue of the operators

The assumption that the density operator is always close to the fully disordered state,

The differences in the contributions between the first (double commutator) and the second (thermal) series of terms in Eq. (

Expected values of the magnetization spin operator

Same as Fig.

Contributions from both relaxation mechanisms to the expected values in Eq. (

The situation is strikingly different when the spins are initially prepared in a singlet order. Here, the thermal correction (blue) is negligible with respect to the double commutator (red) contribution to the rate of change

The situation depicted in Fig.

Alternatively, when the spins are prepared in the

Interestingly, Fig.

The recent achievement of the Lindblad approach was the description of the magnetization relaxation of a two-spin system prepared in a singlet state (

In the following, we derive the evolution of the magnetization of a two-spin system, using the singlet–triplet population basis, and compare the results obtained by both approaches. As above (and in

In the extreme narrowing regime, where

In Eq. (76),

In the high temperature limit, both terms become approximately equal,

The foregoing discussion has shown that, in the high temperature approximation, the exact thermalization procedure of the spectral density function is irrelevant, as all models are equivalent in these conditions.
Indeed, in a field of 23.5 T (

In the semi-classical viewpoint (as in

Thus, the usual semi-classical inhomogeneous master equation provides erroneous predictions in this case. The latter is obtained when the thermal corrections to the double commutator part of the relaxation operator are retained to the first order in the largest eigenvalue

As shown above, the non-commutation of the bath operators has critical consequences, leading to the lattice-temperature-dependent terms in the master equation, and it is only when the bath operators

The conventional semi-classical approach, where spin–bath interactions are represented by random spin Hamiltonians, has the following two simultaneous consequences: the structure of the relaxation operator is affected in such a way that the master equation takes the form of a double commutator, and since

The correlation functions involved in Eq. (

In the case of a homonuclear spin pair, the main Hamiltonian is defined as follows:

Eigenoperators of Hamiltonian for a homonuclear coupled spin

The coefficients from the Eq. (

The codes used in the simulations can be found at

No data sets were used in this article.

DA designed the research. DA and BAR performed the theoretical derivations and simulations and wrote the paper.

The contact author has declared that neither they nor their co-author have any competing interests.

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This article is part of the special issue “Geoffrey Bodenhausen Festschrift”. It is not associated with a conference.

This paper is dedicated to Geoffrey Bodenhausen, on the occasion of his 70th birthday. Bogdan A. Rodin acknowledges a Vernadski scholarship from the French embassy in the Russian Federation.

This research has been supported by the Russian Foundation for Basic Research (grant no. 20-53-15004) and by a PRC (Projet de Recherche Collaborative) funding from the Centre National de la Recherche Scientifique.

This paper was edited by Jean-Nicolas Dumez and reviewed by Christian Bengs, Alberto Rosso, Malcolm Levitt, Ranajeet Ghose, and one anonymous referee.