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Memoirs of the Faculty of Engineering, Hokkaido University = 北海道大学工学部紀要

Master Equation for a Laser Model

Tomomasa Nagashima

Published Mar 1, 1975• 0 citations
OpenAlex

Abstract

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}.

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Physical Review A•Dec 1, 1998•2 citations
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 two-level atom single-mode laser system of Sargent, Scully and Lamb is extended to two-mode laser in a three-level atomic system with common lower level, and master equation is obtained for the system with all the three levels being pumped. There appear four terms in the master equation that are absent in the single-mode case and those interpreted as 2-photon processes play an especial role in the present case. The master equation is represented by a probability flow diagram in two-dimensions with the photon numbers n1 and n2 of the two-mode as variables. The probability flow diagram can be extended to infinity, and each arrow of it represents a term of the right hand side of master equation. By taking sum over one of the variables n1 or n2, the master equation is reduced to two equations, each of them can be represented by an one-dimensional probability flow diagram. The equation of motion under steady state is obtained by considering the correspondence between the macroscopic balance and the microscopic balance.A parameter H is introduced to reduce the equation of motion and to facilitate the mathematical procedure and a formal solution is thus obtained. Using the equation of motion and its formal solution various operation characteristics are discussed. Among them are: the threshold condition for each mode; the condition of only one mode operation; the variation of photon statistics between single-mode operation and two-mode operation and the variation of photon statistics with lower level being pumped and not pumped.
The single-mode laser model as formulated by Haken and co-workers is considered. Within the approximation which takes into account only the interaction (at all orders in the coupling constant) between the mode and the single atoms, this model leads to a linear time-evolution equation for the Glauber quasiprobability distribution $P(\ensuremath{\beta},{\ensuremath{\beta}}^{*},t)$ of the field mode, containing derivatives of all orders in $\ensuremath{\beta}$ and ${\ensuremath{\beta}}^{*}$. This equation is solved exactly in the stationary situation. Thus the quantum effects due to the terms with derivatives of order higher than the second one are fully taken into account. It is shown that these terms have a shrinking effect on the stationary distribution above threshold, counteracting the broadening effect due to the second-derivative term. The photon and photoelectron distributions are given. Exact relations are deduced, which allow the calculation of all the moments of the steady-state distribution from the first one. This distribution is compared with the stationary solutions of Risken's equation, of P. Mandel's equation, and with the stationary distribution deduced by Weidlich, Risken, and Haken. The connection with the single-mode laser theory of Scully and Lamb is analyzed in the high-intensity region. The complete Scully-Lamb master equation is translated into the Glauber representation, yielding a linear equation for $P(\ensuremath{\beta},{\ensuremath{\beta}}^{*},t)$ containing derivatives of all orders in $\ensuremath{\beta}$ and ${\ensuremath{\beta}}^{*}$. Suitably connecting the parameters AAA, BBB, and CCC of the Scully-Lamb equation with the parameters of the model of Haken and co-workers, it is shown that there is perfect agreement between the strong-signal Scully-Lamb steady-state distribution and that of Weidlich, Risken, and Haken for all values of the pump parameter.
A scheme for deriving a field kinetic equation for the problems of macroscopic quantum optics has been formulated. The general master equation obtained for the s-ordered quasi-probability, and Glauber-Sudarshan P-function in particular, has been concretized for the case of three-photon parametric processes in a multi-level medium. Thus the three-photon parametric phenomena have been examined without exploiting the traditional effective Hamiltonian. Discussed here is the problem of formation of field statistics depending on the type of excitation medium with two-level lasing levels that reduce to the Lamb-Scully or Haken models. The form of the kinetic equation obtained is shown to depend on the types of atomic transitions.
The density operator theory and method are used to extend the fully quantum-mechanical theory of the two-level single-mode laser advanced by Sargent, Scully and Lamb (1974) to the homogeneously broadened four-level atomic system so that the master equation of the three-mode laser operation for this system is obtained. The steady-state operation characteristics of the laser in different cases are discussed and compared. Especially, pictorial comparison and discussion of the quantum-statistical properties of the laser are carried out by means of numerical computation. Meanwhile some experimental results obtained by Otsuka (1978) are qualitatively explained.
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.