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Physical Review A

Quantum-dynamical semigroup generators for proton-spin relaxation in water

P. Beck, K. Lendi

Published Jan 1, 1993• 10 citations
Publisher

Abstract

Various aspects of a rather general treatment of proton-spin relaxation in water are discussed within the framework of quantum-dynamical semigroup theory for a four-level system coupled to a reservoir in equilibrium. In particular, the specifications of the infinitesimal generator of time evolution, either in Kossakowski or in Davies form, are worked out in detail. With the help of the Lie algebra of SU(4), the results are used to derive, under suitable simplifications, generalized Bloch equations for the static and alternating-field case. The relevant correlation functions are calculated using conventional approaches but supplemented by taking into account explicitly results from a stochastic model for formation and breaking of hydrogen bridges. A further approximate reduction of the coupled general equations to simpler ordinary Bloch equations leads to an identification of the relevant relaxation times. This approach provides a somewhat different interpretation of rotational correlation times whose numerical values are estimated over a wide temperature range.

DOI
10.1103/physreva.47.346

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Various aspects of a rather general treatment of proton-spin relaxation in water are discussed within the framework of quantum-dynamical semigroup theory for a four-level system coupled to a reservoir in equilibrium. In particular, the specifications of the infinitesimal generator of time evolution, either in Kossakowski or in Davies form, are worked out in detail. With the help of the Lie algebra of SU(4), the results are used to derive, under suitable simplifications, generalized Bloch equations for the static and alternating-field case. The relevant correlation functions are calculated using conventional approaches but supplemented by taking into account explicitly results from a stochastic model for formation and breaking of hydrogen bridges. A further approximate reduction of the coupled general equations to simpler ordinary Bloch equations leads to an identification of the relevant relaxation times. This approach provides a somewhat different interpretation of rotational correlation times whose numerical values are estimated over a wide temperature range.
The concept of complete positivity fixes in considerable detail the mathematical structure of quantum Markovian master equations which describe the combined reversible-irreversible dynamics of open systems for the case of time-independent Hamilton operators. For many applications such as, for instance, problems of time-resolved spectroscopy, similar results would be of great value but involve time-varying external fields with a corresponding time-dependent Hamiltonian. For this reason, we propose a suitable extension of the Kossakowski generator to the time-dependent case by maintaining complete positivity and using arguments suggested by experimental facts. As an example, generalized Bloch-like equations can be obtained with three different relaxation times and a nonlinear dependence upon field strength.
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.
The Mori-Zwanzig projection operator formalism is a powerful method for the derivation of mesoscopic and macroscopic theories based on known microscopic equations of motion. It has applications in a large number of areas including fluid mechanics, solid-state theory, spin relaxation theory, and particle physics. In its present form, however, the formalism cannot be directly applied to systems with time-dependent Hamiltonians. Such systems are relevant in many scenarios such as driven soft matter or nuclear magnetic resonance. In this article we derive a generalization of the present Mori-Zwanzig formalism that is able to treat also time-dependent Hamiltonians. The extended formalism can be applied to classical and quantum systems, close to and far from thermodynamic equilibrium, and even in the case of explicitly-time-dependent observables. Moreover, we develop a variety of approximation techniques that enhance the practical applicability of our formalism. Generalizations and approximations are developed for both equations of motion and correlation functions. Our formalism is demonstrated for the important case of spin relaxation in a time-dependent external magnetic field. The Bloch equations are derived together with microscopic expressions for the relaxation times.
A finite-dimensional quantum system is coupled to a bath of oscillators in thermal equilibrium at temperature T>0 . We show that for fixed, small values of the coupling constant λ , the true reduced dynamics of the system is approximated by the completely positive, trace preserving Markovian semigroup generated by the Davies-Lindblad generator. The difference between the true and the Markovian dynamics is O(|λ|1/4) for all times, meaning that the solution of the Gorini-Kossakowski-Sudarshan-Lindblad master equation is approximating the true dynamics to accuracy O(|λ|1/4) for all times. Our method is based on a recently obtained expansion of the full system-bath propagator. It applies to reservoirs with correlation functions decaying in time as 1/t4 or faster, which is a significant improvement relative to the previously required exponential decay.
This work concerns itself with the exact study of the dynamical properties of two model systems. After a brief summary of theory and concepts previous work is discussed, and this provides the motivation for the formulation of the first model. This quantum mechanical lattice model and some of its equilibrium properties are described in Chapter II. The dynamical problem to be studied is formulated in Chapter III; this is essentially the study of the time evolution (generated by the Hamiltonian) of a finite system at temperature T1 coupled to an infinite copy of itself at a temperature T2 and acting as a heat bath for the system. The problem is solved for the special case, when the coupling as scaled by the parameter γ, takes the value γ = 1. The general case for arbitrary γ values is treated in Chapter IV. It is shown that the system approaches the equilibrium state of the heat bath in a non-exponential manner, provided the spectrum of the Hamiltonian is continuous and does not have a discrete part. This result is in complete accord with the findings of other work summarised in Chapter I. The mixing properties of the model and behaviour of the relaxation rate in the weak coupling limit are studied in Chapter V. The model is shown to fail to behave as a calorimeter and in view of this result the relevance of the concept of mixing to irreversible behaviour is discussed. The main conclusions and results for the model are summarised at the end of Chapter V. The second model discussed, was first introduced by R.J. Glauber to study the dynamics of the Ising chain. The main feature here is that the time evolution is defined through a Master equation, and the associated stochastic operator. It is shown in Chapter VI that exploiting fully the free fermion character of the stochastic operator for the Glauber model, it is possible to provide a simple method to the study of the dynamics of the Ising Chain.
The approach to equilibrium of a quantum mechanical system in interaction with a bath is studied from a practical as well as a conceptual point of view. Explicit memory functions are derived for given models of bath couplings. If the system is a harmonic oscillator representing a molecule in interaction with a reservoir, the generalized master equation derived becomes an extension into the coherent domain of the well-known Montroll–Shuler equation for vibrational relaxation and unimolecular dissociation. A generalization of the Bethe–Teller result regarding energy relaxation is found for short times. The theory has obvious applications to relaxation dynamics at ultra-short times as in observations on the femtosecond time scale and to the investigation of quantum coherence at those short times. While vibrational relaxation in chemical physics is a primary target of the study, another system of interest in condensed matter physics, an electron or hole in a lattice subjected to a strong DC electric field that gives rise to well-known Wannier–Stark ladders, is naturally addressed with the theory. Specific system–bath interactions are explored to obtain explicit details of the dynamics. General phenomenological descriptions of the reservoir are considered rather than specific microscopic realizations.