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Journal of Physics A Mathematical and General

General solution to the spherical Raman-Nath equation

C T Lee

Published Nov 11, 1987• 3 citations
Publisher

Abstract

The spherical Raman-Nath equation, a difference-differential equation describing stimulated Compton scattering, is transformed into a partial differential equation for the probability amplitude in spherical phase space by using the Q representation of the atomic coherent states. With the quantum electron recoil as the perturbation parameter, a perturbative solution up to indefinite high order is obtained. This is a general solution because the initial condition can be arbitrary.

DOI
10.1088/0305-4470/20/16/021

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The spherical Raman-Nath equation, a difference-differential equation describing stimulated Compton scattering, is transformed into a partial differential equation for the probability amplitude in spherical phase space by using the Q representation of the atomic coherent states. With the quantum electron recoil as the perturbation parameter, a perturbative solution up to indefinite high order is obtained. This is a general solution because the initial condition can be arbitrary.
Abstract We present a generalized quantum theory of stimulated Raman scattering, which is based on coupled Heisenberg–Langevin equations. Analytical solutions in previous models neglect the effects of the frequency dispersion, assume far-off resonance and only consider slowly varying terms in the atomic operator. In obtaining the more general theory, we have derived partial integro-differential equation containing polarization kernels that include all the temporal nonlocal effects using systematic integral method over time. Analytical solutions of the Stokes field with greater generality for arbitrary spatial-temporal laser fields have been obtained using Laplace transform in the retarded space frame with minimal approximations made. The solutions are very general and potentially useful for studying systems in which highly accurate modeling is required such as quantum memories of a wide range of Raman systems.
This paper obtains the exact analytical solution of atomic Raman–Nath diffraction in the coordinate representation and discusses the influence of different initial conditions and detunings on the atomic spatial population distribution. The phase difference between the dipole matrix element and initial atomic population may influence the atomic spatial population distribution after diffraction, which has never been discussed before as far as we know. It offers a method to measure the phase by the spatial population distribution, which is interesting in the study of quantum optics.
The diffraction of a coherent, stationary, beam of atoms from a standing wave of light is considered. Particular attention is paid to long interaction times for which the motion of the atoms inside the light field must be taken into account. This is the regime of dynamical diffraction, and is treated within the framework of a set of differential difference equations due to Raman and Nath. The popular Raman-Nath approximation, in which the kinetic energy imparted to the atoms by the light wave is ignored, is no longer valid and the full equations must be solved. When the interaction between the atoms and the light is very weak then few beams are produced. To describe spontaneous emission from the atom one can invoke a complex potential as a model for dissipation. If the atom beam is obliquely incident one finds anomalously high transmission close to the Bragg angles which is the physical manifestation of degeneracies in non-Hermitian matrices. The classical limit is approached when the atom-light interaction is strong, producing many beams. The classical dynamics (geometrical rays) reveal that the farfield intensity pattern is dominated by caustics. These proliferate with increasing interaction distance. Two methods are presented for analytically treating the semiclassical limit of the wave mechanics. The first begins with W.K.B. solutions to the continuised Raman-Nath equations. Mapping onto parabolic cylinder functions removes the spurious divergences of the W.K.B. solutions, giving eigenvectors which are uniformly valid for all scattering angles and may be summed to give the wavefunction. The second method transforms the original W.K.B. eigensum, using the Poisson summation formula, giving a new sum, each term of which has classical significance. This method shows that wave theory decorates the classical caustics with an Airy function intensity profile, as predicted by catastrophe theory.