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American Journal of Physics

Simple Derivation of the Bloch Equation

D. ter Haar

Published Dec 1, 1966• 2 citations
Publisher

Abstract

It is shown how one can derive Bloch's induction equation from the equation of motion of the density matrix by coarse-graining in time. The relaxation of the magnetization is assumed to be due to a fluctuating magnetic field acting upon the spins. The derivation does not provide any new information but is sufficiently simple that it may be used as a lecture application of the density matrix.

DOI
10.1119/1.1972541

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It is shown how one can derive Bloch's induction equation from the equation of motion of the density matrix by coarse-graining in time. The relaxation of the magnetization is assumed to be due to a fluctuating magnetic field acting upon the spins. The derivation does not provide any new information but is sufficiently simple that it may be used as a lecture application of the density matrix.
The multiplicative stochastic process treatment of the time development of the density matrix for a subsystem in contact with a heat reservoir is applied to the specific problem of the relaxation of a nuclear magnetic moment which is interacting with a fluctuating magnetic environment. A model for the fluctuating interaction Hamiltonian, appropriate for the magnetic moment case, is presented, and the Bloch equations for nuclear magnetic relaxation are constructed as a consequence. Agreement with empirical observations is noted.
We present a derivation of the Markovian master equation using the time coarse graining method. Starting from a naive perturbative solution of the von Neumann equation, the reduced density matrix with the coarse grained time steps is obtained using the assumption of short correlation time of the bath field. Then applying the renormalization group method, it is shown that dependence of the specific initial time in the solution can be removed and the Markovian semi-group master equation in Lindblad form is attained in the weak coupling limit.
Abstract One of the most powerful tools for calculating quantum device performance is based on the density matrix operator. The operator is unique because it is time dependent in the Schrödinger picture. The approach is quite general, but in the systems of interest here, the Hilbert space of the operator includes both the quantum system such as a nano-vibrator or two-level system and the quantized vacuum radiation field. The equation of motion follows from the time dependent Schrödinger equation. It is possible, as we show, to include the generation of spontaneous emission in this system and then, because observables of interest do not depend on the vacuum field, trace over the vacuum field to create a new density matrix called the reduced density matrix. The resulting equations of motion are the Bloch equations. We use these equations to analyze several problems involving two- and three-level systems.
The arguments of Chapter 2, as well as of those of subsequent chapters, have as their foundation the formalism based on the state vector of a system of which the evolution is described by the Schrödinger equation. In fact, such an approach is badly suited to the case in which the coupling between an atom and its environment (for example through collisions with other atoms or spontaneous emission into formerly empty modes of the electromagnetic field) cannot be neglected. If the correlations induced by these interactions between the atom and its environment do not concern us and we are only interested in the evolution of the atom, the formalism of the density matrix must be employed. This provides a description at all times of the state of the atom, although a state vector for the atom alone cannot be defined. In this formalism the effect of the environment on the atom is accounted for by the introduction of suitable relaxation terms (Section 2C.1) in the equation of evolution of the density matrix. An important application of the density matrix is to the case of a two-level atomic system for which the relaxation terms lead to its deexcitation to a level of lower energy. We shall show that in this case the density matrix can be represented by a vector, known as the Bloch vector , which will allow us to give simple geometrical pictures of the evolution of the system.
The density matrix is a useful concept which will allow one to deal with a large collection of spinors with a statistical distribution at time t = 0 and follow their time evolution by tracking the time evolution of a new operator (the density matrix). This will require the extension of the Bloch sphere to the Bloch ball concept, in which the vector representing the collection of spinors will be able to reach points in the interior of the Bloch sphere as a function of time. This requires the inclusion of damping and scattering mechanisms representing interactions among spinors, and also their interactions with the environment. The phenomenological equations allowing a description of these relaxation mechanisms are known as Bloch’s equations. They are introduced here through a connection to the Bloch sphere and Bloch ball concepts. The longitudinal and transverse relaxation times, T1 and T2, respectively, are also introduced and some numerical examples (results of Monte-Carlo simulations) are presented to illustrate their physical meaning.
The density matrix is a useful concept which will allow one to deal with a large collection of spinors with a statistical distribution at time t = 0 and follow their time evolution by tracking the time evolution of a new operator (the density matrix). This will require the extension of the Bloch sphere to the Bloch ball concept, in which the vector representing the collection of spinors will be able to reach points in the interior of the Bloch sphere as a function of time. This requires the inclusion of damping and scattering mechanisms representing interactions among spinors, and also their interactions with the environment. The phenomenological equations allowing a description of these relaxation mechanisms are known as Bloch’s equations. They are introduced here through a connection to the Bloch sphere and Bloch ball concepts. The longitudinal and transverse relaxation times, T1 and T2, respectively, are also introduced and some numerical examples (results of Monte-Carlo simulations) are presented to illustrate their physical meaning.