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Journal of the Physical Society of Japan

A Microscopic Laser Theory. I. Basic Formulation

Toshihico Arimitsu, Fumiaki Shibata

Published Mar 1, 1983• 13 citations
Publisher

Abstract

A basic formulation of the laser system is given in the form most convenient to study the time-dependent cooperative ( symmetry breaking ) behavior of the system in the far-from-equilibrium states from the microscopic and fully quantum-statistical-mechanical point of view. The quantum mechanical master equation, derived from a microscopic model Hamiltonian, is mapped onto a c-number function space with the use of the Boson and spin coherent states representations. The distribution function (mapped from the density operator) is expanded in terms of the Hermite polynomials and the spherical harmonics to obtain differential equations for the expansion coefficients with which time evolution of the laser system can be determined completely.

DOI
10.1143/jpsj.52.772

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A basic formulation of the laser system is given in the form most convenient to study the time-dependent cooperative ( symmetry breaking ) behavior of the system in the far-from-equilibrium states from the microscopic and fully quantum-statistical-mechanical point of view. The quantum mechanical master equation, derived from a microscopic model Hamiltonian, is mapped onto a c-number function space with the use of the Boson and spin coherent states representations. The distribution function (mapped from the density operator) is expanded in terms of the Hermite polynomials and the spherical harmonics to obtain differential equations for the expansion coefficients with which time evolution of the laser system can be determined completely.
A laser master equation is derived microscopically by using the time-convolution-less formulation of the damping theory up to the fourth-order terms with respect to the coupling constant between the laser field and the atoms, in order to investigate the quantum effect and the system size (number of the active atoms) dependence of the kinetic operator . The master equation is compared with the equation given by the time-convolution formalism which is essentially equal to the master equation derived by Haake.
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 fully quantum-mechanical model of a single-mode laser is examined. A previously introduced method of solution where both the pump and loss reservoirs are treated simultaneously as a weak perturbation of the atom-light system is employed. This method is similar to perturbation theory for degenerate systems in that the reservoirs remove the nonuniqueness (degeneracy) of the steady state of the unperturbed atom-light system. In the limit of weak reservoir coupling, the exact steady-state solution of the master equation is derived. This solution still contains the atomic variables explicitly and therefore remains valid even in situations where the conventional adiabatic elimination of the atomic variables is no longer possible. A compact discrete representation of the master equation in terms of atomic occupation numbers and photon numbers is developed which allows one to perform all calculations in an elementary manner, requiring only appropriate summations over photon and occupation numbers. General exact relations are obtained for the steady-state probability distribution function p(n,m)p(n,m)p(n,m) to find nnn photons in the light field and mmm atoms in the excited state. Our solution constitutes the first steady-state laser distribution function that is valid for an arbitrary number of laser atoms and in the complete region from far below to far above threshold. It provides a quantitative criterion for the range of validity of approximate methods in laser theory.
For systems which obey quantum mechanics, the formulation of the problem of treating large numbers of particles is, of course, somewhat different than it is for classical systems. The microscopic description of the system is provided by a wave function which (in the absence of spin) is a function of the classical coordinates { q i }. The mathematical model is provided by a Hamiltonian operator H which is often obtained from the corresponding classical Hamiltonian by the replacement p i → ( ħ /i)(∂/∂ q i ). In other cases the form of the Hamiltonian operator is simply postulated. The microscopic dynamics are provided by the Schrödinger equation i ħ (∂Ψ/∂ t ) = H Ψ which requires as initial condition the knowledge of the wave function Ψ({ q i }, t ) at some initial time t 0 . (Boundary conditions on Ψ({ q i }, t ) must be specified as part of the description of the model as well.) The results of experiments in quantum mechanics are characterized by operators, usually obtained, like the Hamiltonian, from their classical forms and termed observables. Operators associated with observables must be Hermitian. In general, the various operators corresponding to observables do not commute with one another. It is possible to find sets of commuting operators whose mutual eigenstates span the Hilbert space in which the wave function is confined by the Schrödinger equation and the boundary conditions. A set of such (time independent) eigenstates, termed ψ ν ( q ), is a basis for the Hilbert space.