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Journal of Physics A Mathematical and General

Solution to the spherical Raman-Nath equation

C T Lee

Published Dec 21, 1985• 3 citations
Publisher

Abstract

The spherical Raman-Nath equation, which describes the stimulated Compton scattering, is transformed into a Fokker-Planck type equation for 'diffusion' over the spherical surface. This is done by using the Q representation in the phase space of the atomic coherent states. With the quantum electron recoil as the perturbation parameter, a perturbative solution is obtained up to arbitrarily high order, hence approaching an exact solution.

DOI
10.1088/0305-4470/18/18/005

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The spherical Raman-Nath equation, which describes the stimulated Compton scattering, is transformed into a Fokker-Planck type equation for 'diffusion' over the spherical surface. This is done by using the Q representation in the phase space of the atomic coherent states. With the quantum electron recoil as the perturbation parameter, a perturbative solution is obtained up to arbitrarily high order, hence approaching an exact solution.
We present a propagational formulation of resonant Raman scattering in terms of a master equation. An equivalent Fokker-Planck type equation for a classical quasi-probability distribution function is derived through the use of the positive P representation. Under the diffusion approximations in terms of a system size expansion over a coarse grained volume element, a set of Ito classical stochastic differential equations are obtained in a doubled space, which can be numerically integrated. Dynamic transition behavior for the Stokes field from spontaneous emission to stimulated emission through propagation can be studied when the intermediate states are not adiabatically eliminated.
The "superfluorescence equations" deduced in a previous paper are generalized to the case of finite temperature and translated into the atomic coherent-state representation of Arecchi et al. We get three coupled partial differential equations involving the quasiprobability distribution $Q(\ensuremath{\theta},t)$ of the atomic system and two suitable functions $\mathcal{F}(\ensuremath{\theta},t)$ and $\mathfrak{M}(\ensuremath{\theta},t)$ describing the stimulated processes which are relevant in oscillatory superfluorescence. Neglecting $\mathcal{F}(\ensuremath{\theta},t)$ and $\mathfrak{M}(\ensuremath{\theta},t)$, the equation for $Q(\ensuremath{\theta},t)$ reduces to the Fokker-Planck equation of Narducci et al. corrected by the contribution of thermal noise. The stationary solution is given by the unperturbed canonical state, which is shown to be the exact stationary solution of the single-mode model from which the superfluorescence equations have been deduced. These equations prescribe that $\mathfrak{M}(\ensuremath{\theta},t)$ always remain equal and opposite to $\mathcal{F}(\ensuremath{\theta},t)$, so that the superfluorescence equations reduce automatically to two equations. Eliminating $\mathcal{F}(\ensuremath{\theta},t)$ between the two remaining equations, one obtains a non-Markovian equation for $Q(\ensuremath{\theta},t)$ with derivatives of all orders in $\ensuremath{\theta}$. In the case of pure superfluorescence or super-radiance, such an equation reduces to a Markovian Fokker-Planck equation with derivatives of first and second order in $\ensuremath{\theta}$ which exhibits an interesting fourth-order contribution in the diffusion term. In fact, if one excludes a very small region around the north pole of the Bloch sphere, this term is much larger than the second-order contribution in the diffusion term of the equation of Narducci et al. In the case of oscillatory superfluorescence, the presence of memory effects enhances fluctuations, thus forbidding the existence of "classical" time evolutions.
We use the dressed-state representation to generate the full quantum-mechanical treatment of stimulated Raman scattering (SRS). Equations of motion and propagation are obtained from the Heisenberg equations of motion in the plane-wave limit and under conditions such that anti-Stokes and higher-order Stokes processes can be neglected. Equations for a stochastic model, based upon antinormal ordering and a quantum-classical correspondence are developed from them, suitable for numerical ensemble calculations for the study of amplified quantum fluctuations and spontaneously generated phase waves and solitons. Several alternative approaches, which have more general validity but present greater difficulty in numerical implementation, are also developed by introduction of quasiprobability functions and their associated Fokker-Planck equations from which corresponding stochastic equations of motion and propagation are derived. The Q representation, whose moments correspond to expectation values of products of antinormally ordered fields, and normally ordered atomic operators yield stochastic equations identical to the equations for the stochastic model developed directly from the Heisenberg equations for the same initial and boundary conditions, giving strong credence to the stochastic model. The Wigner representation, whose moments correspond to expectation values of symmetrically ordered operators, leads to a set of equations that are valid where the stochastic model fails, namely, where conditions are such that quantum fluctuations and nonlinearity are simultaneously important. The stochastic model, which is shown to be, by far, the simplest to implement in numerical calculations, is shown also to be valid for the conditions of SRS.