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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Photon distribution function for stimulated Raman scattering

Olga V. Man’ko

Published Feb 18, 2000• 5 citations
Publisher

Abstract

The process of stimulated Raman scattering is described by the model of 2D 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 transferred from one mode to the other due to interaction. The statistical dependence appears due to interaction as well. An explicit formula of 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 and a parameter of the interaction. It is shown that if the medium is initially prepared, 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.

DOI
10.1117/12.378116

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The process of stimulated Raman scattering is described by the model of 2D 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 transferred from one mode to the other due to interaction. The statistical dependence appears due to interaction as well. An explicit formula of 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 and a parameter of the interaction. It is shown that if the medium is initially prepared, 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.
Several aspects of the stimulated Raman scattering process are studied using the model of a two-dimensional oscillator. The photons of the Stokes mode are taken as one mode of the oscillator and the phonons of the medium are taken as the other mode of the oscillator. The interaction Hamiltonian of the photons and phonons is taken to be quadratic in the annihilation and creation operators of photons and phonons. The laser is considered as a classical light source and its depletion is neglected. The photon?phonon joint probability distribution function is found and expressed in terms of Hermite polynomials of four variables. The general formula for the photon number probability distribution as a function of the laser-field intensity and the medium parameters is obtained in terms of Hermite polynomials of two variables. It is shown that the phenomenon of entanglement appears in photon?phonon modes due to the interaction of the light with the medium. The tomograms of the photon?phonon states and (averaged over the phonon mode) photon states are investigated within the framework of a symplectic tomography scheme.
The quantum aspects like entanglement and quantum distances between quantum states in the Stimulated Raman Scattering (SRS) process are studied using the simpliest model of two-dimensional oscillator with quadratic interaction. The two modes of the oscillator represent the photons of the Stokes mode and the phonons of the medium respectively. The SRS process is discussed in the probability representation of quantum mechanics. The exact formulae for tomograms are given for the symplectic and photon number tomography schemes. The phenomenon of entanglement is investigated. It is shown that after the interaction of laser field with the medium the state of photon-phonon mode is in entangled states, in view of Peres-Horodecki-Simon criterion. The tomographic entropy associated with photon-phonon mode state is introduced. The quantum distance between the states of photon-phonon mode appearing in the case of different initial states of the medium is discussed.
The non-classical effect of squeezing of light in spontaneous Raman and stimulated Raman processes are investigated under a short-time approximation based on a fully quantum mechanical approach. A Raman process is looked upon as a two-photon and one-phonon interaction process. Squeezing in amplitude, amplitude squared and amplitude cubed have been investigated. The occurrence of squeezing is observed in the fundamental mode in spontaneous Raman and stimulated Raman scattering. The squeezing in a Stokes field is found to be dependent on squeezing of the initial pump field.
Photon statistics of stimulated Raman scattering of squeezed light are considered including additive noise. The phonon system is assumed to be in chaotic or squeezed chaotic states. The influence of nonlinear dynamics is established on the initial non-classical features of light adopting quasi-distribution related to anti-normal ordering, photocount distribution, its factorial moments and principal squeezing parameter. It is shown that the initial non-classical effects are increased by the nonlinear dynamics provided that the Stokes mode is less amplified than the anti-Stokes mode is damped, whereas in the opposite case the non-classical effects are swiftly smoothed out. The frequency detuning can be optimized to have maximum non-classical effects, which can periodically return. However, above a certain bound the non-classical effects are missing with frequency detuning, and also with increasing phonon number, even if the phonon system is squeezed. Small initial squeezing can be increased by the nonlinear dynamics, whereas strong squeezing can only be diminished.