  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.
