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Physical Review A

Modifications to the Scully-Lamb laser master equation

H. J. Carmichael, Daniel F. Walls

Published Jun 1, 1974• 17 citations
Publisher

Abstract

A quantum theory of the laser is presented in which the coupling between atoms and field is retained in the treatment of the irreversible behavior of the atoms. A master equation for the optical field is obtained and compared with that resulting from the laser theory of Scully and Lamb, where atom-field correlations are neglected in describing the relaxation of the atoms. It is confirmed that for laser powers currently available the approximation employed by Scully and Lamb is valid.

DOI
10.1103/physreva.9.2686

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A quantum theory of the laser is presented in which the coupling between atoms and field is retained in the treatment of the irreversible behavior of the atoms. A master equation for the optical field is obtained and compared with that resulting from the laser theory of Scully and Lamb, where atom-field correlations are neglected in describing the relaxation of the atoms. It is confirmed that for laser powers currently available the approximation employed by Scully and Lamb is valid.
A master equation of a Laser system is presented from a dynamical point of .It is assumed that the Laser system is coupled with other independent subsystems.They p!ay a role as distinct reservoirs in our Laser system.As an application of this approach, the master equation is obtained for a system that corresponds to a Laser model considered by Scully-Lamb.The equation obtained heye includes the dynamical effect of an atomic system, i. e., atomic coherence. gl. IntroductionRecently various statistical mechanical theories have been developed by many authors in order to explain the properties of Laser systems.Especially, Haken et al.i) presented a model in which the Laser system is in contact with several independent heat reservoirs of different temperatures, undergoing irreversible processes among them.In particular, Weidlich and Haake2) showed the existence of the lasing state which might be considered as a revelation of the phase change in an open system by the method of a master equation similar to the Wangnness-Bloch equation of nuclear spin induction.In this theory the relaxation processes are lntroduced phenomenologica!ly in the master equation, without taking into account its detailed derivation from a microscopic point of view.On the other hand, Scully-Lamb3) independently developed a quantal theory of Laser, which is an extention of the famous semiclassic theory by Lamb, and obtained the statistical properties of radiation field of Laser systems.In g2, one of the purposes of this paper is presented.Here, from a dynamical point of view, a generalized master equation was setforth for a system coupled with several independent subsystems.Thus the discussion in g2 is analogous to Argyres' theory") for the relaxation of spin system, in wh!ch Wangness-Bloch equation was derived from a dynamical point of view.The second purpose of this paper is to present a generalized master equation corresponding to the Laser system considered by Scully-Lamb as described in g3.gZ. Derivation of the master equation Now let us consider the following system.The system with which we are mainly concerned shali be called A and is described by dynamical variables {ai}.
The quantum theory of a laser developed by Scully and Lamb (1967) is reformulated to include the effects of atomic motion and detuning. The equations are solved to fourth order in the atom-field coupling constant. An approximate treatment of the mode structure is shown to reproduce the results of Lamb's semi-classical theory in the Doppler limit. In addition the photon distribution may be studied as a function of the detuning. The width of the photon distribution is found to increase monotonically with increasing detuning.
The results of the semi-classical theory for the laser are compared with the results of the quantum theory given by Scully and Lamb (1967). It is found that the spatial structure of the cavity mode leads to important corrections. For moving atoms the cavity mode is smeared, and a generalized form of the Scully-Lamb theory (for resonance between the atomic transitions and the cavity mode) leads to agreement with the semi-classical theory. The photon distribution for the case with moving atoms is obtained and compared with the results of the Scully-Lamb theory.
The quantum-statistical properties of the optical field of a two-mode laser with two coupled transitions have been studied by means of a generalization of the Scully and Lamb theory. The photon number distribution and the mode intensity distribution are obtained in the steady state for a system of homogeneously broadened atoms in resonance with the laser field. It is shown that the mode-coupling constant $\ensuremath{\xi}$ is unity and that near threshold the results of earlier treatments are recovered. Furthermore, it is shown that certain limit measures for the relative intensity fluctuations predicted by the semiclassical Fokker-Planck treatments based on the third-order theory are valid even in the limit of arbitrarily high gain levels.
A fully quantum-mechanical theory of bistable inversionless lasers is developed by generalizing the Scully-Lamb laser theory to this kind of laser systems. By converting the quantum master equation of the system into a Fokker-Planck equation, we obtain the steady-state properties and the fluctuation properties of the laser field for both coherently and incoherently injected atomic systems. It is shown that both the intensity and phase fluctuations of the laser system with coherent atomic injection are reduced corresponding to their counterpart of the incoherent atomic injection. The phase noise of bistable inversionless lasers can be quenched down to the vacuum-noise level for both atomic injections.