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Journal of Optics B Quantum and Semiclassical Optics

Photon distribution function, tomograms and entanglement in stimulated Raman scattering

Sergey Vladimirovich Kuznetsov, Olga V Man apos ko, Nicolaii V. Tcherniega

Published Aug 1, 2003• 25 citations
Publisher

Abstract

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.

DOI
10.1088/1464-4266/5/4/357

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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 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.
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.
Using the approximation of small fluctuations around a stationary point, photon statistics and squeezing of vacuum fluctuations are derived in stimulated Raman scattering including the coupling of photon-photon and photon-phonon modes. The depletion of pump light is described and it is shown that a regime of inversion of the process is related to an increase in quantum fluctuations, which are finally reduced again, more in the anti-Stokes mode than in the Stokes mode. If the anti-Stokes coupling is stronger than the Stokes coupling, then an oscillating behaviour of the statistical characteristics occurs and substantial squeezing can be found in photon-phonon modes.