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Physical Review

Dissipation in Quantum Mechanics. The Harmonic Oscillator. II

I. R. Senitzky

Published Nov 1, 1961• 140 citations
Publisher

Abstract

The development of a quantum-mechanical formalism for systems with dissipation that was presented in an earlier article, and intended mainly for application to the electromagnetic field in a cavity, is extended. The problem of the harmonic oscillator with dissipation is shown to be the same as that of a harmonic oscillator coupled to a thermal reservoir, and the need of the formalism to contain the appropriate statistical mechanics is discussed. The derivation of relationships which permit the calculation of all moments of the oscillator coordinate and momentum provides the necessary extension of the theory. The formal resemblance of the completed theory to that of classical Brownian motion, some differences due to quantum mechanics, and the fact that certain fundamental relationships which are assumed in the latter are derived in the present analysis, are pointed out. The application of the theory is illustrated by the consideration of three problems: the proof of Ott's formula, and the derivation of both the probability density and energy distribution of the oscillator in equilibrium with a thermal reservoir.

DOI
10.1103/physrev.124.642

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Physical Review A•Aug 1, 1971•208 citations
The development of a quantum-mechanical formalism for systems with dissipation that was presented in an earlier article, and intended mainly for application to the electromagnetic field in a cavity, is extended. The problem of the harmonic oscillator with dissipation is shown to be the same as that of a harmonic oscillator coupled to a thermal reservoir, and the need of the formalism to contain the appropriate statistical mechanics is discussed. The derivation of relationships which permit the calculation of all moments of the oscillator coordinate and momentum provides the necessary extension of the theory. The formal resemblance of the completed theory to that of classical Brownian motion, some differences due to quantum mechanics, and the fact that certain fundamental relationships which are assumed in the latter are derived in the present analysis, are pointed out. The application of the theory is illustrated by the consideration of three problems: the proof of Ott's formula, and the derivation of both the probability density and energy distribution of the oscillator in equilibrium with a thermal reservoir.
The need for a quantum-mechanical formalism for systems with dissipation which is applicable to the radiation field of a cavity is discussed. Two methods that have been used in this connection are described. The first, which starts with the classical Newtonian equation of motion for a damped oscillator and applies the conventional formal quantization techniques, leads to an exact solution; but subsequent discussion shows that this method is invalid, the results being unacceptable from a quantum-mechanical viewpoint. The second method, which considers the interaction of two systems, the lossless oscillator and the loss mechanism, is adopted in the present article. No special model is used for the loss mechanism, but this mechanism is assumed to have a large number of densely-spaced energy states.The approximations with respect to the loss mechanism that underlie the concept of dissipation are discussed. These approximations are then applied to the analysis, and a differential equation for a coordinate operator of the harmonic oscillator is obtained which has the formal appearance of the Newtonian equation of motion for a driven damped harmonic oscillator, the driving term being an operator referring to the loss mechanism. The presence of the driving term is responsible for the difference between the present theory and that of the first method mentioned above. A solution of the differential equation for the coordinate operator is given explicitly. An examination of the physical significance of the solution shows that the driving term is responsible not only for the thermal fluctuations which are due to the loss mechanism, but also for the proper commutation relationship of the conjugate coordinates of the oscillator and for its zero-point fluctuations.A generalization of the solution to provide for a classical driving force and coupled atomic systems is given. The results are then restated in a form that refers to the loss mechanism only through the two parameters by which it is usually described---the dissipation constant and the temperature.
The quantum theory of a damped harmonic oscillator is developed by considering an exactly solvable model, previously discussed by Unruh and Zurek [Phys. Rev. D 40, 1071 (1989)]. A one-dimensional harmonic oscillator is coupled to a scalar field. The field is assumed to be initially in thermal equilibrium. The Heisenberg equations of motion are solved without approximation, and the first and second moments, 〈q(t)〉, 〈p(t)〉, 〈q2{\mathit{q}}^{2}q2(t)〉, 〈p2{\mathit{p}}^{2}p2(t)〉, and 〈q(t)p(t)+p(t)q(t)〉 of the coordinate and momentum of the oscillator are calculated. The high-temperature, zero-temperature, free-particle, and weak-damping limits are discussed. A Wigner function of Gaussian form is constructed. The density-matrix equation derived by Louisell $[---Quantum Statistical Properties of Radiation (Wiley, New York, 1973)] is examined and shown to have a solution in agreement with our weak-damping limit.
This chapter provides a summary of all results concerning the fluctuation-dissipation theorem needed to understand many aspects of noisy systems. The proof is a succinct one in density matrix language, with a review of the latter provided for those who wish to follow the proof. Callen and Welton (1951) were the first to emphasise the universal nature of the relation between fluctuations and dissipation. They provided a quantum-mechanical derivation that involved detailed sums over the eigenstates. In this chapter, density operators are introduced to deal with an ensemble of systems rather than a single system. The response function, equilibrium theorems, hermiticity and time reversal, and application of the fluctuation-dissipation theorem to a harmonic oscillator are discussed. To make some of the abstract concepts of dissipative motion become meaningful, they are illustrated by using a reservoir consisting of a continuum of harmonic oscillators.
We extend the quantum theory of dissipation in the context of a system-reservoir model, where the reservoir in question is kept in a non-equilibrium condition. Based on a systematic separation of time scales involved in the dynamics, appropriate generalizations of the fluctuation-dissipation and Einstein's relations have been pointed out. We show that the Wigner-Weisskopf decay of the system mode results in a rate constant which, depending on the relaxation of a non-equilibrium bath, is dynamically modified. We also calculate the time-dependent spectra of a cavity mode with a suitable gain when the cavity is kept in contact with a non-equilibrium bath.
Abstract Almost all traditional physical formalisms are developed by using conservative forces, and the microscopic implementation of dissipation involves a sort of unusual process, mainly in quantum systems. In this work, we study the quantum harmonic model endowed with a non-Hermitian term responsible for dissipation. In addition, we also include an oscillating field that drives the model to a coherent state, which is dominated by fluctuation in a specific frequency, while regular thermal states are lowly occupied. The usual coherent state formalism at zero temperature is extended to treat dissipative models at finite temperature. We define a generating function that is used in the evaluation of the most relevant statistical averages, such as the particle distribution. Then, we successfully employ the developed formalism to discuss two well-known applications; the damped quantum harmonic oscillator, and the precession magnetization in a ferromagnetic sample.
The theory of Brownian motion of a quantum oscillator is developed. The Brownian motion is described by a model Hamiltonian which is taken to be the one describing the interaction between this oscillator and a reservoir. Use is made of the master equation recently derived by the author, to obtain the equation of motion for the various reduced phase-space distribution functions that are obtained by mapping the density operator onto ccc-number functions. The equations of motion for the reduced phase-space distribution functions are found to be of the Fokker-Planck type. On transforming the Fokker-Planck equation to real variables, it is found to have the same form as the Fokker-Planck equation obtained by Wang and Uhlenbeck to describe the Brownian motion of a classical oscillator. The Fokker-Planck equation is solved for the conditional probability (Green's function) which is found to be in the form of a two-dimensional Gaussian distribution. This solution is then used to obtain various time-dependent quantum statistical properties of the oscillator. Next, the entropy for a quantum oscillator undergoing Brownian motion is calculated and we show that this system approaches equilibrium as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. Finally we show that in the weak-coupling limit the Fokker-Planck equation reduces to the one obtained by making the usual rotating-wave approximation.