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arXiv:quant-ph/9705004

Photon Distribution Function for Stocks Wave for Stimulatted Raman Scattering

Man'ko, O. V., Nicolaii V. Tcherniega

Published May 6, 1997• 1 citation
arXivHTMLPDF

Abstract

New time-dependent integrals of motion are found for stimulated Raman scattering. Explicit 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.
DOI
10.48550/arxiv.quant-ph/9705004

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Stimulated Raman Scattering

Raman scattering is a two-photon process [1]. As illustrated in Fig. la, an incident photon is annihilated and a new photon of a different frequency is created. This implies that the scattering is inelastic, and the medium is left in a different energy state. When the scattered photon is lower in energy, or frequency, it is said to be Stokes shifted. The shift in frequency of the Stokes photon is related to a characteristic frequency of the medium. This is illustrated in Fig. 2a. Calling this characteristic material excitation frequency ω)v, the Stokes frequency is then ωs = ΩL — ωv, where ωL is the incident laser frequency. Thus, Raman (Stokes) scattering results in an excitation from the ground state of the medium to an excited state mediated by a two-photon inelastic scattering. The internal material excitation can involve the creation of an excited electronic state, an excited vibrational-rotational state, a lattice vibration, a spin flip in semiconductors, or electron waves in plasmas [2]. The phenomenon occurs in gases, liquids, solids, and plasmas.

CRC Press eBooks•Apr 22, 2003•1 citation
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 nonlinear process of the stimulated Raman effect is treated theoretically. Simple expressions are derived for the initial rates of change and limiting steady state values of the first two moments, the degrees of second-order coherence and the correlation function of the pump and Stokes beams. An analytical solution for the time dependence of the joint photon probability distribution of the two light beams is obtained and then employed to calculate some numerical results for various special cases of interest, and the significance of these results is discussed in physical terms. Simpler approximate expressions are also derived for the case where the pump beam is much more intense than the Stokes beam.
Raman scattering is a two-photon process [1]. As illustrated in Fig. la, an incident photon is annihilated and a new photon of a different frequency is created. This implies that the scattering is inelastic, and the medium is left in a different energy state. When the scattered photon is lower in energy, or frequency, it is said to be Stokes shifted. The shift in frequency of the Stokes photon is related to a characteristic frequency of the medium. This is illustrated in Fig. 2a. Calling this characteristic material excitation frequency ω)v, the Stokes frequency is then ωs = ΩL — ωv, where ωL is the incident laser frequency. Thus, Raman (Stokes) scattering results in an excitation from the ground state of the medium to an excited state mediated by a two-photon inelastic scattering. The internal material excitation can involve the creation of an excited electronic state, an excited vibrational-rotational state, a lattice vibration, a spin flip in semiconductors, or electron waves in plasmas [2]. The phenomenon occurs in gases, liquids, solids, and plasmas.