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Quantum Optics Journal of the European Optical Society Part B

Master equations for quantum optics problems

В. Н. Горбачев, A. I. Zhiliba

Published Aug 1, 1993• 15 citations
Publisher

Abstract

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.

DOI
10.1088/0954-8998/5/4/001

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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.
We discuss a generalized version of the Scully-Lamb laser master equation for a collection of driven multilevel atoms in a lambda-type configuration. We are especially interested in the effect of the external driving field on the laser process with or without a population inversion between the lasing levels. We provide an exact solution of the photon probability distribution in steady state and discuss regimes of operation that can lead to sub-Poissonian statistics, as measured by the ratio between the variance and the average photon number. We solve the field master equation in the transient regime and analyze the effects of the external driving field on the linewidth of the emitted radiation. The largest reduction of the photon variance relative to the mean number corresponds to the injection into the cavity of atoms prepared in the nonlasing level. Injection of atoms in an incoherent mixture of the three energy states normally leads to super-Poissonian statistics.
The quantum statistics of light in a transparent medium with cubic nonlinearity is considered. Two types of transparent medium are treated, namely, a cold transparent medium with a ground working level and an inversion-free medium with upper and lower levels of the lasing transition having the same population. The spectra of light fluctuations are derived on the basis of both the Scully-Lamb and the Haken theories. The conditions for the use of an effective Hamiltonian of interaction are determined. Based on the exact solution of the Fokker-Planck equation for the Glauber-Sudarshan P -function, the inversion-free medium is shown to be a source of spontaneous radiation with non-Gaussian statistics.
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.
A consistent quantum formalism based on the localized basis of the Wannier functions in the Heisenberg and Schrödinger pictures to describe propagation of an electromagnetic field in a three-dimensional media including diffraction is presented. In the Schrödinger picture the Fokker-Planck equation for the Glauber-Sudarshan quasi-probability and the corresponding Langevin equations are given. As a result the spacetime description is obtained by a simple change of variables in the temporal master equation of the field. Using this formalism it is shown that the existence of integrals of motion in the propagation of light in a medium under the condition of non-degenerated parametric and two-photon interactions results in amplification of modes when non-classical properties of the light are conserved. Quantum propagation of light in a linear medium taking into account the diffraction is considered and its solution is found.
A unified approach to the quantum statistics of single-mode linear processes described by the general Fokker-Planck equation and the generalized initial superposition of coherent and chaotic fields is developed to reveal the common traits in the mathematical models of several quantum optical systems. Processes such as the degenerate parametric amplification and the harmonic oscillator eventually coupled to the thermal or rigged reservoir are involved as special cases. A sufficient condition for the validity of the master equation is formulated as its being formally independent of possible reservoirs. The dependence of non-classical phenomena such as the sub-Poisson photon statistics and squeezing on the drift and diffusion coefficients is discussed.