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Journal of Physics A Mathematical Nuclear and General

A master equation approach to the Raman effect

Daniel F. Walls

Published Apr 1, 1973• 36 citations
Publisher

Abstract

A quantum-mechanical model of Raman scattering from a phonon bath is formulated. A master equation is derived for the density operator of the light fields alone. This model correctly describes the stochastic coupling of Stokes and anti-Stokes radiation through the phonon bath. Since quantum fluctuations are included, this treatment contains the spontaneous and stimulated Raman effects in one self-consistent formalism. The photon statistics of the light fields are obtained by the solution of a Fokker-Planck equation.

DOI
10.1088/0305-4470/6/4/014

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A quantum-mechanical model of Raman scattering from a phonon bath is formulated. A master equation is derived for the density operator of the light fields alone. This model correctly describes the stochastic coupling of Stokes and anti-Stokes radiation through the phonon bath. Since quantum fluctuations are included, this treatment contains the spontaneous and stimulated Raman effects in one self-consistent formalism. The photon statistics of the light fields are obtained by the solution of a Fokker-Planck equation.
Raman scattering from a great number of phonon modes is described from a quantum-statistical point of view within the standing-wave model. The master equation for the completely quantum case, including laser pump depletion and stochastic coupling of Stokes and anti-Stokes modes, is derived and converted to classical equations: either into a generalized Fokker-Planck equation and an equation of motion for the characteristic function or into the master equation in Fock representation. These two approaches are developed both in linear and nonlinear regime. A detailed analysis of scattering into Stokes and anti-Stokes modes in linear regime, i.e., under parametric approximation, is presented. The existence of s-parametrized quasiprobability distributions, in particular the Glauber-Sudarshan P-function, is investigated. An analysis of Raman scattering into separate Stokes and anti-Stokes modes in nonlinear regime, thus including pump depletion, is given. The master equation in Fock representation is solved exactly for the complete density matrix using the Laplace transform method. Short-time solutions, steady-state solutions and approximate compact form solutions are obtained. Relations between the quasidistribution approach based on the Fokker-Planck equation and the density matrix approach based on the master equation in Fock representation are presented. The photocount distribution and its factorial moments as well as variances and extremal variances of quadratures are calculated in both approaches giving the basis for the analysis of the quantum properties of radiation such as sub-Poissonian photon-counting statistics and squeezing. Various statistical moments obtained from numerical calculations utilizing our exact solutions, from the approximate relations for short times, and those obtained under parametric approximation are compared.
The basis of this dissertation is a review of the quantum mechanical formulation of damping and stimulation, particularly in non-linear optical processes. The basic problems of the quantum theory of damping are discussed, and the formalism for the quantum theory is introduced. The study of a particular example, that of a damped simple harmonic oscillator, provides an introduction to the handling of the basic tool, the master equation. The physical aspects of the non-linear processes are contained in the systems' photon statistics, and the master equation provides several approaches for obtaining these. This theory is then used to formulate a quantum mechanical model of Raman scattering by phonons, and to thus obtain the photon statistics of the scattered radiation.
We use the dressed-state representation to generate the full quantum-mechanical treatment of stimulated Raman scattering (SRS). Equations of motion and propagation are obtained from the Heisenberg equations of motion in the plane-wave limit and under conditions such that anti-Stokes and higher-order Stokes processes can be neglected. Equations for a stochastic model, based upon antinormal ordering and a quantum-classical correspondence are developed from them, suitable for numerical ensemble calculations for the study of amplified quantum fluctuations and spontaneously generated phase waves and solitons. Several alternative approaches, which have more general validity but present greater difficulty in numerical implementation, are also developed by introduction of quasiprobability functions and their associated Fokker-Planck equations from which corresponding stochastic equations of motion and propagation are derived. The Q representation, whose moments correspond to expectation values of products of antinormally ordered fields, and normally ordered atomic operators yield stochastic equations identical to the equations for the stochastic model developed directly from the Heisenberg equations for the same initial and boundary conditions, giving strong credence to the stochastic model. The Wigner representation, whose moments correspond to expectation values of symmetrically ordered operators, leads to a set of equations that are valid where the stochastic model fails, namely, where conditions are such that quantum fluctuations and nonlinearity are simultaneously important. The stochastic model, which is shown to be, by far, the simplest to implement in numerical calculations, is shown also to be valid for the conditions of SRS.
We present a propagational formulation of resonant Raman scattering in terms of a master equation. An equivalent Fokker-Planck type equation for a classical quasi-probability distribution function is derived through the use of the positive P representation. Under the diffusion approximations in terms of a system size expansion over a coarse grained volume element, a set of Ito classical stochastic differential equations are obtained in a doubled space, which can be numerically integrated. Dynamic transition behavior for the Stokes field from spontaneous emission to stimulated emission through propagation can be studied when the intermediate states are not adiabatically eliminated.
A photon theory of the stimulated Raman effect is developed. A system of coupled equations for the amplitudes of scattered photons and molecular vibrations are derived. The molecular vibration coordinates are then eliminated using the random phase approximation. The equations which result assume the form of coupled equations for the first Stokes and anti-Stokes lights. The secular equation which characterizes the solution of these coupled equations is shown to be equivalent to that of Shen and Bloembergen. Using this secular equation, the condition for temporal growth of the scattered lights is obtained and is compared with that for the spatial growth, which was obtained previously by Shen and Bloembergen. It is shown that both types of growth (temporal and spatial) proceed with comparable rates if the damping of the molecular vibration is large. If its damping is small, however, the temporal growth prevails over the spatial growth. Typical line-shape of the anti-Stokes line is obtained in the case of small damping. The result shows a strong directionality, as is the case of large damping.
The nonlinear interaction of light with matter is described from a quantum-statistical point of view. The phenomena of two-photon emission and two-photon absorption including both the single- and two-mode cases and the Raman effect are discussed in detail. A master equation for the density operator of the light fields alone is derived. This operator equation is converted to a c number equation and analytic solutions are obtained for the diagonal matrix elements of the density operator in the Fock representation. No linearizing approximation is introduced. These solutions allow one to compute the moments of the photon distribution for the above nonlinear processes.