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Physical Review

Approach to Equilibrium in Quantal Systems: Magnetic Resonance

A. H. Sher, Henry Primakoff

Published Jul 1, 1960• 49 citations
Publisher

Abstract

The present paper presents a derivation of the "master" or Boltzmann "gain-loss" equation from the Schr\"odinger equation, i.e., a derivation of the equation for the evolution in time of the probabilities of finding a physical system in its various states from the equation for the corresponding probability amplitudes. The "master" equation is derived for an, in effect completely self-enclosed, "supersystem," [$A+B$], consisting of a "system of interest," [$A$], and a "surroundings," [$B$], in relatively weak mutual interaction. A discussion is given of the range of validity of the "master" equation for [$A+B$] and it is shown that the random phase assumption is required for the state vector of [$A+B$] at the initial time only. The normally microcanonical character of the equilibrium statistical configuration of [$A+B$] is demonstrated and a treatment is given of exceptional, "extremely quantal-coherent," initial statistical distributions of [$A+B$] which may evolve away from equilibrium. Derivations are also presented of the "master" equation for [$A$] and of the "master" equation for an individual particle or quasi-particle [$q$], within [$A$]; a discussion of the range of validity of these "master" equations is given and the normally canonical character of the equilibrium statistical configuration of [$A$] is deduced. General solutions of the "master" equations for [$A+B$], [$A$], and [$q$] are worked out and the relation between the principles of "microscopic reversibility" and "detailed balance" and the nonoscillatory character of the approach to equilibrium are exhibited. A theorem is presented regarding the time variation of the entropy of [$A$].As illustrations of the general methods developed two important processes in magnetic resonance---the time variation of the longitudinal magnetization, ${〈\ensuremath{\mu}〉}_{t}$, and the time variation of the transverse magnetization, ${〈{\ensuremath{\mu}}^{\ensuremath{'}}〉}_{t}$---are discussed in some detail. It is shown that the variation of ${〈\ensuremath{\mu}〉}_{t}$ with $t$ and of ${〈{\ensuremath{\mu}}^{\ensuremath{'}}〉}_{t}$ with $t$ for a "nonrigid" lattice can be described by means of the "master" equation for an individual spin [$q$] and several special cases are discussed on the basis of the evaluation of the appropriate transition probabilities; a comparison with the "spin-temperature" procedure is also appended. On the other hand, it is demonstrated that for a "rigid" lattice no description of the variation of ${〈{\ensuremath{\mu}}^{\ensuremath{'}}〉}_{t}$ with $t$ can be given on the basis of a "master" equation; in this case, quantal coherence effects neglected in the derivation of the "master" equation from the Schr\"odinger equation are vital and ${〈{\ensuremath{\mu}}^{\ensuremath{'}}〉}_{t}$ must be evaluated by a rigorous calculation of Trace {[appropriate time dependent density matrix] ${\ensuremath{\mu}}^{\ensuremath{'}}$}.

DOI
10.1103/physrev.119.178

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The present paper presents a derivation of the "master" or Boltzmann "gain-loss" equation from the Schr"odinger equation, i.e., a derivation of the equation for the evolution in time of the probabilities of finding a physical system in its various states from the equation for the corresponding probability amplitudes. The "master" equation is derived for an, in effect completely self-enclosed, "supersystem," [A+BA+BA+B], consisting of a "system of interest," [AAA], and a "surroundings," [BBB], in relatively weak mutual interaction. A discussion is given of the range of validity of the "master" equation for [A+BA+BA+B] and it is shown that the random phase assumption is required for the state vector of [A+BA+BA+B] at the initial time only. The normally microcanonical character of the equilibrium statistical configuration of [A+BA+BA+B] is demonstrated and a treatment is given of exceptional, "extremely quantal-coherent," initial statistical distributions of [A+BA+BA+B] which may evolve away from equilibrium. Derivations are also presented of the "master" equation for [AAA] and of the "master" equation for an individual particle or quasi-particle [qqq], within [AAA]; a discussion of the range of validity of these "master" equations is given and the normally canonical character of the equilibrium statistical configuration of [AAA] is deduced. General solutions of the "master" equations for [A+BA+BA+B], [AAA], and [qqq] are worked out and the relation between the principles of "microscopic reversibility" and "detailed balance" and the nonoscillatory character of the approach to equilibrium are exhibited. A theorem is presented regarding the time variation of the entropy of [AAA].As illustrations of the general methods developed two important processes in magnetic resonance---the time variation of the longitudinal magnetization, 〈\ensuremathμ〉t{〈\ensuremath{\mu}〉}_{t}〈\ensuremathμ〉t​, and the time variation of the transverse magnetization, 〈\ensuremathμ\ensuremath′〉t{〈{\ensuremath{\mu}}^{\ensuremath{'}}〉}_{t}〈\ensuremathμ\ensuremath′〉t​---are discussed in some detail. It is shown that the variation of 〈\ensuremathμ〉t{〈\ensuremath{\mu}〉}_{t}〈\ensuremathμ〉t​ with ttt and of 〈\ensuremathμ\ensuremath′〉t{〈{\ensuremath{\mu}}^{\ensuremath{'}}〉}_{t}〈\ensuremathμ\ensuremath′〉t​ with ttt for a "nonrigid" lattice can be described by means of the "master" equation for an individual spin [qqq] and several special cases are discussed on the basis of the evaluation of the appropriate transition probabilities; a comparison with the "spin-temperature" procedure is also appended. On the other hand, it is demonstrated that for a "rigid" lattice no description of the variation of 〈\ensuremathμ\ensuremath′〉t{〈{\ensuremath{\mu}}^{\ensuremath{'}}〉}_{t}〈\ensuremathμ\ensuremath′〉t​ with ttt can be given on the basis of a "master" equation; in this case, quantal coherence effects neglected in the derivation of the "master" equation from the Schr"odinger equation are vital and 〈\ensuremathμ\ensuremath′〉t{〈{\ensuremath{\mu}}^{\ensuremath{'}}〉}_{t}〈\ensuremathμ\ensuremath′〉t​ must be evaluated by a rigorous calculation of Trace {[appropriate time dependent density matrix] \ensuremathμ\ensuremath′{\ensuremath{\mu}}^{\ensuremath{'}}\ensuremathμ\ensuremath′}.
Starting from the quantum statistical master equation derived in [1] we show how the connection to the semi-classical Boltzmann equation (SCBE) can be established and how irreversibility is related to the problem of separability of quantum mechanics. Our principle goal is to find a sound theoretical basis for the description of the evolution of an electron gas in the intermediate regime between pure classical behavior and pure quantum behavior. We investigate the evolution of one-particle properties in a weakly interacting N-electron system confined to a finite spatial region in a near-equilibrium situation that is weakly coupled to a statistical environment. The equations for the reduced n-particle density matrices, with n < N are hierarchically coupled through two-particle interactions. In order to elucidate the role of this type of coupling and of the inter-particle correlations generated by the interaction, we examine first the particular situation where energy transfer between the N-electron system and the statistical environment is negligible, but where the system has a finite memory. We then formulate the general master equation that describes the evolution of the coarse grained one-particle density matrix of an interacting confined electron gas including energy transfer with one or more bath subsystems, which is called the quantum Boltzmann equation (QBE). The connection with phase space is established by expressing the one-particle states in terms of the overcomplete basis of coherent states, which are localized in phase space. In this way we obtain the QBE in phase space. After performing an additional coarse-graining procedure in phase space, and assuming that the interaction of the electron gas and the bath subsystems is local in real space, we obtain the semi-classical Boltzmann equation. The validity range of the classical description, which introduces local dynamics in phase space is discussed.
Abstract This chapter introduces the fundamentals of the description of the quantum dynamics of open systems in terms of quantum master equations, together with its most important applications. Special emphasis is laid on the theory of completely positive quantum dynamical semigroups, which leads to the concept of a quantum Markov process. It discusses the relaxation to equilibrium and the multi-time structure of quantum Markov processes, as well as their irreversible nature which is characterized by an appropriate entropy functional. Microscopic derivations for various quantum master equations are presented, such as the quantum optical master equation and the master equation for quantum Brownian motion. As a further application, the master equation describing continuous measurements is derived and used to study the quantum Zeno effect. The chapter also contains a treatment of non-linear, mean field quantum master equations together with applications to laser theory and super-radiance.
Abstract Processes at the nanoscale happen far away from the thermodynamic limit, far from equilibrium and are dominated by fluctuations and, perhaps, even quantum effects. This book establishes a consistent thermodynamic framework for such processes by combining tools from non-equilibrium statistical mechanics and the theory of open quantum systems. The book is accessible for graduate students and of interest to all researchers striving for a deeper understanding of the laws of thermodynamics beyond their traditional realm of applicability. It puts most emphasis on the microscopic derivation and understanding of key principles and concepts as well as their interrelation. The topics covered in this book include (quantum) stochastic processes, (quantum) master equations, local detailed balance, classical stochastic thermodynamics, (quantum) fluctuation theorems, strong coupling and non non-Markovian effects, thermodynamic uncertainty relations, operational approaches, Maxwell's demon and time-reversal symmetry, among other topics. Furthermore, the book treats a few applications in detail to illustrate the general theory and its potential for practical applications. These are single-molecule pulling experiments, quantum transport and thermoelectric effects in quantum dots, the micromaser and related set-ups in quantum optics.
In conclusion, by assuming that the initial state is prepared as a microcanonical ensemble, the actual state at time t=T can be replaced by another microcanonical ensemble in the evaluation of macroscopic quantities. The derivation is based on the high dimensionality of the Hilbert space, and restrictions for the system size dependence of observables as well as the perturbative treatment of the unitary transformation of the energy eigenstates. The validity of the uniformly random sampling assumption for [numerical formula] is numerically verified for a nonintegrable spin chain. It is also remarked that the initial state can be out of equilibrium, i.e. the initial density matrix is [numerical formula] with a unitary transformation U_0, since this can be regarded as the state evolved from an actual initial state [numerical formula].
The quantum dynamics of spin systems is often treated by a differential equation known as the master equation, which describes the trajectories of spin observables such as magnetization components, spin state populations, and coherences between spin states. The master equation describes how a perturbed spin system returns to a state of thermal equilibrium with a finite-temperature environment. The conventional master equation, which has the form of an inhomogeneous differential equation, applies to cases where the spin system remains close to thermal equilibrium, which is well satisfied for a wide variety of magnetic resonance experiments conducted on thermally polarized spin systems at ordinary temperatures. However, the conventional inhomogeneous master equation may fail in the case of hyperpolarized spin systems, when the spin state populations deviate strongly from thermal equilibrium, and in general where there is a high degree of nuclear spin order. We highlight a simple case in which the inhomogeneous master equation clearly fails, and propose an alternative master equation based on Lindblad superoperators which avoids most of the deficiencies of previous proposals. We discuss the strengths and limitations of the various formulations of the master equation, in the context of spin systems which are far from thermal equilibrium. The method is applied to several problems in nuclear magnetic resonance and to spin-isomer conversion.