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Physical Review

Validity of the Semiclassical Approximation in Maser Theory

J.H. Shirley

Published May 10, 1969• 19 citations
Publisher

Abstract

A simple model of a maser or laser consisting of a single-mode field coupled to $N$ identical two-level atoms is considered. The semiclassical approximation is equivalent to neglecting the statistical correlations between the atoms and the field. The accuracy of this approximation is investigated by writing equations of motion for the correlations. To obtain a complete, self-consistent set of equations, it is necessary to include correlations of the field with itself (related to the coherence of the field) and correlations between different atoms. Relaxations and an energy source are introduced phenomenologically into the equations of motion. They are then solved for the case of steady-state oscillation. It is found that the correlations are smaller than the terms kept in the semiclassical theory by the order of 1 over the number of photons present in the field, larger if thermal photons abound, but smaller if the field relaxation is dominant. Expressions are also found for the amplitude and phase fluctuations of the field. The latter yield a linewidth for the maser oscillator in agreement with earlier calculations, but obtained by a different method.

DOI
10.1103/physrev.181.600

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A simple model of a maser or laser consisting of a single-mode field coupled to NNN identical two-level atoms is considered. The semiclassical approximation is equivalent to neglecting the statistical correlations between the atoms and the field. The accuracy of this approximation is investigated by writing equations of motion for the correlations. To obtain a complete, self-consistent set of equations, it is necessary to include correlations of the field with itself (related to the coherence of the field) and correlations between different atoms. Relaxations and an energy source are introduced phenomenologically into the equations of motion. They are then solved for the case of steady-state oscillation. It is found that the correlations are smaller than the terms kept in the semiclassical theory by the order of 1 over the number of photons present in the field, larger if thermal photons abound, but smaller if the field relaxation is dominant. Expressions are also found for the amplitude and phase fluctuations of the field. The latter yield a linewidth for the maser oscillator in agreement with earlier calculations, but obtained by a different method.
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.
A quantum statistical analysis of an optical maser is presented in generalization of the recent semiclassical theory of Lamb. Equations of motion for the density matrix of the quantized electromagnetic field are derived. These equations describe the irreversible dynamics of the laser radiation in all regions of operation (above, below, and at threshold). Nonlinearities play an essential role in this problem. The diagonal equations of motion for the radiation are found to have an apparent physical interpretation. At steady state, these equations may be solved via detailed-balance considerations to yield the photon statistical distribution ${\ensuremath{\rho}}_{n,n}$. The resulting distribution has a variance which is significantly larger than that for coherent light. The off-diagonal elements of the radiation density matrix describe the effects of phase diffusion in general and provide the spectral profile ${| E(\ensuremath{\omega}) |}^{2}$ as a special case. A detailed discussion of the physics involved in this paper is given in the concluding sections. The theory of the laser adds another example to the short list of solved problems in irreversible quantum statistical mechanics.
This paper extends the calculations of the semiclassical Lamb theory of a Doppler-broadened gas laser (optical maser) to arbitrary intensities for the case of single-mode operation. The coupled equations of the classical electromagnetic field and an ensemble of two-level atoms are set up, and a solution is obtained in the form of a continued fraction. This is used to compute intensity-detuning curves and atomic population inversion densities as functions of both the velocity and the position in the laser. When the laser is tuned to resonance, the velocity dependence of the inversion density shows a previously unknown fine structure consisting of a "bump" in the bottom of the hole. The calculations are compared to results obtained from a perturbation expansion in powers of the field and exact results known for atoms with zero velocity. The response of the laser output to a slow modulation and the buildup of oscillations are also discussed.
Previous theories of laser radiation have described the electromagnetic field in terms of a discrete set of quasimodes (Fox and Li "modes") of the laser cavity. Each quasimode was assumed to have a finite quality factor Q. In steady state (when gain due to lasing atoms cancels the losses) and in the absence of noise sources, the semiclassical theory predicts a $\ensuremath{\delta}$-function line shape. In the present semiclassical analysis (fluctuations are ignored), this theory is generalized for a maser with a cavity having a semitransparent wall as one of the mirrors so that there are now many modes of the universe corresponding to each Fox-Li-type quasimode. It is demonstrated that the normal modes of the universe associated with a single Fox-Li "mode" may, under proper conditions, lock together and the $\ensuremath{\delta}$-function laser line shape may be regained. Specifically, we investigate the conditions under which the coupled multimode equations associated with laser oscillation can be reduced to an equation of the form found in the usual single-quasimode theory.
This paper presents a semiclassical theory of a single-mode gas laser valid for essentially arbitrary values of field intensity. The theory is formulated in terms of an ensemble-averaged form of the density-matrix equations of motion involving a single set of coordinates. The polarization and the population-inversion density are expanded in spatial Fourier components. One obtains a set of coupled difference equations in the Fourier coefficients yn{y}_{n}yn​, which may be solved subject to appropriate boundary conditions on yn{y}_{n}yn​ for large nnn. The solutions may be expressed in terms of continued fractions in the general case, and in closed form in important special cases. Quantities of interest are evaluated by computer. The steady-state laser intensity and frequency are obtained from the coupled electromagnetic field equations as functions of the cavity parameters and the linear gain of the amplifying medium. At moderate values of field intensity, the velocity distribution of the space-averaged population inversion density exhibits a fine structure for low-velocity atoms superimposed on the usual broader depletion. This structure is a manifestation of the coherent ringing of the amplifying medium. The results are compared with those obtained in the rate-equation approximation, and with the recent work of Stenholm and Lamb. Discrepancies with the latter paper are explained.