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Journal of Physics A Mathematical Nuclear and General

Collective radiation by harmonic oscillators

W. Żakowicz, Kazimierz Rza̧żewski

Published May 1, 1974• 21 citations
Publisher

Abstract

The radiation of a small system of harmonic oscillators is analysed. The exact solution of the problem in the dipole and rotating wave approximation is discussed. It is shown that only vibrations of the centre of charge are radiatively damped and that their damping constant is N times larger than for a single oscillator. Due to the collective emission, broadening and shift of the line, dependent on the number of oscillators, also occur.

DOI
10.1088/0305-4470/7/7/012

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The radiation of a small system of harmonic oscillators is analysed. The exact solution of the problem in the dipole and rotating wave approximation is discussed. It is shown that only vibrations of the centre of charge are radiatively damped and that their damping constant is N times larger than for a single oscillator. Due to the collective emission, broadening and shift of the line, dependent on the number of oscillators, also occur.
Spontaneous emission from two systems, namely, NNN identical harmonic oscillators and NNN identical two-level atoms, is studied without the use of the rotating-wave approximation. Certain new features of spontaneous emission, for instance, the dependence of the radiation rate on the initial dipole moment phase, are discussed.
The present review is devoted to the study of certain aspects of anharmonic, time-dependent and damped oscillator(s) system using different theoretical techniques. A theoretical understanding of these systems is important for application in many problems in physics, mechanics and other fields. We discuss in detail the difficulties in the theoretical analysis of the problem. In particular we discuss here the regular, well-behaved perturbative solution, the large quantum number behaviour of anharmonic oscillator(s) using the technique of coherent states, exact solution of quantum anharmonic oscillators, the electromagnetic radiation emitted by a charged particle executing damped anharmonic oscillator motion using Krylov-Bogoliubov approximation method, use of invariants to obtain solution and coherent states of time-dependent oscillator(s), the derivation of perturbative frequencies of a damped coupled anharmonic oscillators system using suitable canonical transformation in the framework of Hamilton-Jacobi formalism and the quantisation and construction of coherent states of a damped oscillator using time-dependent operators.
Coherent emission from two or more classical damped oscillators leads to an apparent violation of conservation of energy if the oscillators are assumed to emit independently. To resolve this apparent paradox, it is necessary to include effects related to cooperative decay, wherein each oscillator modifies the damping rate of the other oscillators. We show how these effects are included for two radiating oscillators and for a line of N oscillators. The relation between cooperative decay, optical coherent transients, and superradiance is discussed.
The evolution of a small spherical system of many harmonic oscillators, initially in a coherent superradiant state, is considered. The dipole oscillators interact via their mutual fields, both longitudinal and transverse, including both radiation- and near-zone contributions. In principle, this interaction can dephase the vibration of oscillators and thereby decrease or eliminate the coherent superradiant contribution to the emitted radiation. It is shown, however, that if the system is sufficiently small, this dephasing makes only small corrections to the decay of the initial state. Thus the emission of a superradiant pulse and cooperative radiative decay are major processes in the system.
The wave equations for the forced, damped, and forced and damped oscillators are solved in closed form for an arbitrary forcing function; the solutions produced being in one-to-one correspondence with the stationary states of the unforced, undamped Hamiltonian H 0 . The quantal motion is closely connected with the classical: for the forced oscillator the probability density is that of H 0 but moves as a whole with the classical motion; for the forced and damped oscillator this motion is accompanied by a contraction progressing eventually into a delta function at the classical position. Transition probabilities between states of H 0 are computed in the case of forced motion and depend solely on the classically acquired energy of the oscillator at any time. The transition probability vanishes strictly only when this energy has a value falling at the roots of a Laguerre polynomial associated with the transition. The classical dipole radiation emitted by a disturbed oscillator is, when the damping force is identified with the force of radiation damping, that of the classical oscillator: a shifted and broadened line.
The classical problem of a damped harmonic motion is formulated wave mechanically, with particular reference to the charge in a tuned circuit, and the electromagnetic field in a resonant cavity. Results in close agreement with the classical problem are obtained, without using perturbation theory. The classical frequency appears in the time dependence of the eigenfunctions and the classical damping appears as a decay in the eigenvalues. Zero point energy and thermal noise are discussed. The formulation is expected to break down, at very high frequencies, for tuned circuits and it is suggested that the same may be true for cavity oscillations and travelling electromagnetic waves.