Photon distribution function for stimulated Raman scattering

The process of stimulated Raman scattering is described by the model of two-dimensional oscillator. The photons of Stokes mode are described by one mode of the oscillator and phonons of the medium are described by the other mode of oscillator. The interaction of photons and phonons is taken to be quadratic in the creation and annihilation operators of the photons and phonons. New time-dependent integrals of motion are found for stimulated Raman scattering within the framework of this model. The photon-phonon probability-distribution function is found in terms of Hermite polynomials of four variables. It is shown that the squeezing phenomenon can be transfered form one mode to the other due to interaction. The statistical dependence appears due to interaction as well. An explicit formula for the photon-number probability distribution as a function of the laser-field intensity and the medium parameters is obtained both in terms of Hermite polynomials of two variables and in terms of Legendre polynomials after the averaging of the photon-phonon probability-distribution function over the phonon mode. The mean photon number and its dispersion in the Stokes mode are expressed as functions of the medium parameters (initial dispersions of phonons) and a parameter of the interaction (coupling constant). It is shown that if the medium is initially prepared (the phonons of the medium are squeezed), then after the interaction with laser field the photons are created in squeezed state. The classical propagator for stimulated Raman scattering is found within the framework of the symplectic-tomography scheme.