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Physical Review

Density-Operator Theory of Harmonic Oscillator Relaxation

W. H. Louisell, L. R. Walker

Published Jan 11, 1965• 86 citations
Publisher

Abstract

The reduced density operator for a driven electromagnetic field mode relaxing to a bath of oscillators is formally found in closed form. In the limit of weak coupling, the density operator which maximizes the entropy subject to the constraints of a given average energy and average electric and magnetic field remains invariant under interaction with the bath. The quantum characteristic function is found in the weak-coupling case and agrees with previous results. The method may be generalized to many modes relaxing to a bath of oscillators.

DOI
10.1103/physrev.137.b204

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The reduced density operator for a driven electromagnetic field mode relaxing to a bath of oscillators is formally found in closed form. In the limit of weak coupling, the density operator which maximizes the entropy subject to the constraints of a given average energy and average electric and magnetic field remains invariant under interaction with the bath. The quantum characteristic function is found in the weak-coupling case and agrees with previous results. The method may be generalized to many modes relaxing to a bath of oscillators.
The quantum theory of a driven damped harmonic oscillator is presented. By using iteration methods the time development of the density operator for the oscillator is obtained and its normally ordered generating functional is written. The field of the oscillator appears to be the superposition of three fields. The first field, which shows the damping of the oscillator, is statistically similar to the one before the interaction; the second one is a coherent field that depends on the driving field; and the third describes a thermal field. In order to evaluate two-time averages by differentiation, a two-time generating functional of the oscillator is obtained. This functional allows two-time averages to be computed from one-time averages before the interaction.
The harmonic oscillator with dissipation is studied within the framework of the Lindblad theory for open quantum systems. By using the Wang-Uhlenbeck method, the Fokker-Planck equation, obtained from the master equation for the density operator, is solved for the Wigner distribution function, subject to either the Gaussian type or the δδδ-function type of initial conditions. The obtained Wigner functions are two-dimensional Gaussians with different widths. Then a closed expression for the density operator is extracted. The entropy of the system is subsequently calculated and its temporal behaviour shows that this quantity relaxes to its equilibrium value.
A closed expression for the density operator of the damped harmonic oscillator is extracted from the master equation based on the Lindblad theory for open quantum systems. The entropy and effective temperature of the system are subsequently calculated and their temporal behaviour is surveyed by showing how these quantities relax to their equilibrium values. The entropy for a state characterized by a Wigner distribution function which is Gaussian in form is found to depend only on the variance of the distribution function.
Abstract This chapter introduces the density operator and the Liouville equation and outlines how reduced dynamical description of open quantum systems naturally leads to the appearance of relaxation processes. (1) The density operator and the quantum Liouville equation (the density matrix for a pure system, statistical mixtures, representations, coherences, thermodynamic equilibrium). (2) An example: The time evolution of a two-level system in the density matrix formalism. (3) Reduced descriptions. (4) Time evolution equations for reduced density operators: The quantum master equation (projection operators, The Nakajima–Zwanzig equation, Derivation of the quantum master equation using the thermal projector, The quantum master equation in the interaction representation, The Markovian limit—the Redfield equation, The secular approximation and the Lindblad equation). (5) The Lindblad equation in the ‘Heisenberg representation’. (6) Relaxation of a two-level system and a harmonic oscillator revisited. (7) The optically driven two-level system in a thermal environment—the Bloch equations. (8) The oscillating dipole model of a two-level molecule. (9) Analogy of a coupled two-level system to a spin ½ system in a magnetic field.
This preprint contains the English translation of the paper "Relaxation dynamics of a quantum chain of harmonic oscillators", which was published by the author in 1980 in the Ukrainian Physical Journal. Comments describing its motivations, background ideas and results are presented as well. This paper addressed to the problem of approach to the thermal equilibrium in an isolated macroscopic quantum system, which was studied on the example of the quantum chain of weakly interacting harmonic oscillators. In the initial state, macroscopic energy was supplied to one oscillator (atom) in the chain. Subsequent evolution of the quantum state of the whole chain was determined due to the model integrability. The main subject of interest was the time evolution of the reduced density operators characterizing the quantum state of a particular atom. On the short time-scale, the energy perturbation expands along the chain with the velocity of the fastest phonon mode. On the long-time scale, the single-atom density operators display strong fluctuations around the canonical Gibbs distribution. These fluctuations are caused by the degeneracy of the energy level differences (presence of resonances) in the model of coupled harmonic oscillators. After lifting this degeneracy, fluctuations are suppressed providing, that the reduced density operator of each atom in the chain becomes very close to the Gibbs distribution at almost any time moment.
A translation-invariant state (a quantum Markov chain) is associated with a nearest-neighbor interaction on a one-dimensional lattice by a new technique which provides closed forms for all the correlation functions. When applied to an Ising-type perturbation of a chain of harmonic oscillators, the dynamics can be computed explicitly. The resulting translation-invariant distribution is substantially different from the Planck distribution when the temperature and the coupling constant are large. For the evolution of the field operators on a given mode, we obtain a natural nonlinear generalization of the theorem which states that the free evolution of the field operators is obtained by second quantization of the classical free evolution.