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Quantum Electronics

DYNAMIC BEHAVIOR OF QUANTUM MECHANICAL OSCILLATORS

George Makhov, O. Risgin

Published Dec 1, 1963• 0 citations
OpenAlex

Abstract

Abstract : The time variation of amplitude of a maser oscillator is derived from a study of the rate equations connecting the population inversion and the electromagnetic energy stored in the maser resonator. The methods of nonlinear mechanics are used to derive a qualitative description of the amplitude variation in the absence of a complete solution of the rate equations. Modifications of the simple rate equations necessary to account for the existence of undamped relaxation oscillations are introduced for both microwave and optical masers, and their physical significance is discussed. The production of 'giant pulses' of radiation is also treated. (Author)

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Reports on Progress in Physics•Jan 1, 1964•11 citations
Abstract : The time variation of amplitude of a maser oscillator is derived from a study of the rate equations connecting the population inversion and the electromagnetic energy stored in the maser resonator. The methods of nonlinear mechanics are used to derive a qualitative description of the amplitude variation in the absence of a complete solution of the rate equations. Modifications of the simple rate equations necessary to account for the existence of undamped relaxation oscillations are introduced for both microwave and optical masers, and their physical significance is discussed. The production of 'giant pulses' of radiation is also treated. (Author)
The equations are generalized governing coherent emission from quantum mechanical amplifiers (microwave, infrared, and optical masers) using either electric or magnetic dipole transitions. Earlier analyses by Gordon, Zeiger and Townes, Feynman, Vernon and Hellworth, and Wang and Singer, are incorporated in this generalization. The principal new result of this analysis is that an amplitude modulation of the output is to be expected from all maser oscillators excepting those in which excited atoms are supplied at a notably greater rate than the depopulation rate due to coherent induced emission. This analysis also suggests that an energy population inversion of electric dipole moments can be performed using coherent light sources in the same manner as with magnetic dipole moments. It appears therefore that modulation of an optical or infrared maser (e.g., for communication purposes) may be easily accomplished by varying the population of excited states through control of the pumping energy. This modulation may be both by absorption and amplification. (Author)
A theoretical model for the behavior of an optical maser is presented in which the electromagnetic field is treated classically, and the active medium is made up of thermally moving atoms which acquire nonlinear electric dipole moments under the action of the field according to the laws of quantum mechanics. The corresponding macroscopic electric polarization of the medium acts as a source for an electromagnetic field. The self-consistency requirement that a quasistationary field should be sustained by the induced polarization leads to equations which determine the amplitudes and frequencies of multimode oscillation as functions of the various parameters characterizing the maser. Among the results obtained are: threshold conditions, single-mode output as a function of cavity tuning, frequency pulling and pushing, mode competition phenomena including frequency locking, production of combination tones, and population pulsations. A more approximate discussion of maser action using rate equations is also given in which the concept of "hole burning" plays a role.
We discuss the quantum theory of a simple model of a maser oscillator, consisting of one radiation-field mode interacting with a large number of stationary three-level atoms. The field and the atoms also interact with separate heat reservoirs which represent dissipation mechanisms and an incoherent pumping mechanism. The model is sufficiently simple that some analytical progress can be made with the nonlinear quantal equations before further approximation is necessary. We start from the quantal equation of motion of the field-atom density operator. We make immediate use of a diagonal coherent-state expansion for the field part of the density operator and a somewhat similar expansion for the atom part. This yields an exact equation of the Fokker-Planck form for a ccc-number weight distribution, which retains all the significance of the original operator equation, and which has the semiclassical equation for the same model as a first, fluctuation-free approximation. We make use of our basic Fokker-Planck equation in a variety of ways. We discuss the reduction of the equation under conditions that the atomic decay constants are large (large atomic linewidth), arriving finally at an equation of motion for a field-only weight function which serves to demonstrate the basic coherence properties of a maser. We derive and discuss the equation of motion of the generalized Wigner density for the maser model. The generalized Wigner density is a smoothed version (a convolution) of our basic weight distribution, and from it we derive an equivalent classical model including noise sources. Finally, we discuss other useful weight distributions and the number representation for the field. The equations we derive in these discussions make contact with the rate equations of Shimoda, Takahasi, and Townes, as well as with the more recent work of Lax and Louisell, Lax, and of Scully and Lamb.
A simple maser model consisting of a single-mode field coupled to N identical two-level atoms is presented. The assumption of negligible statistical correlation between the atoms and the field, permits the Heisenberg equations of motion to be replaced by a self-consistent set of ordinary nonlinear differential equations. Relaxation terms and an energy source are introduced phenomenologically. The resulting equations exhibit a threshold, stable and unstable steady states, relaxation oscillations and, sometimes, spiking. The dynamic equations for the hydrogen maser and the rate equations of laser theory are derived as special cases. Also discussed are the maser amplifier, locking of a maser to an external signal, and the effects of cavity thermal noise.
The subject of masers is introduced by giving an historical account of events which led to the amplification of microwave radiation by an inverted population in a system of weakly interacting ammonia molecules. The introduction also outlines the development of masers up to the present. Sections 2 and 3 are concerned with maser fundamentals. Methods for obtaining inverted populations are discussed in § 2. Section 3 is devoted to the theory of maser gain and noise. This is followed by a discussion of microwave solid state masers. Microwave maser materials and the development of unidirectional travelling wave maser amplifiers are emphasized. Optical masers are treated in § 5. In this section the theory of optical amplifiers and oscillators is outlined. Also included in the theoretical discussion is a rather lengthy consideration of pump power requirements in optically pumped masers. This is followed by a survey of solid state, semiconductor and gaseous optical masers. The review ends with some comments on the extension of masers to new wavelength regions.