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Journal of the Optical Society of America B

Asymptotic perturbation theory applied to diffractive atomic scattering

J. Mizerski, W. Miklaszewski, Stig Stenholm

Published Mar 1, 1991• 7 citations
Publisher

Abstract

We use the multiple-time-scale perturbation technique to treat the long interaction time limit of atomic scattering from a standing light wave, i.e., the atomic Kapitza–Dirac effect. The use of the method is restricted to an exactly resonant interaction and an oblique angle of incidence. We develop the theory in some detail in view of its general applicability to a large class of physical phenomena. The accuracy of the ensuing analytical expressions is tested against exact numerical computations of the effect, and the allowed ranges of the parameters are given with reference to the actual atomic experiments. For moderately large values of the parameters that characterize the field strength and the photon recoil, the method is found to give excellent approximations for the long-time behavior, whereas straightforward perturbation expansions diverge because of unbounded secular terms.

DOI
10.1364/josab.8.000509

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We use the multiple-time-scale perturbation technique to treat the long interaction time limit of atomic scattering from a standing light wave, i.e., the atomic Kapitza–Dirac effect. The use of the method is restricted to an exactly resonant interaction and an oblique angle of incidence. We develop the theory in some detail in view of its general applicability to a large class of physical phenomena. The accuracy of the ensuing analytical expressions is tested against exact numerical computations of the effect, and the allowed ranges of the parameters are given with reference to the actual atomic experiments. For moderately large values of the parameters that characterize the field strength and the photon recoil, the method is found to give excellent approximations for the long-time behavior, whereas straightforward perturbation expansions diverge because of unbounded secular terms.
Theoretical spin properties of the Kapitza-Dirac effect beyond the plane-wave description are not known in detail. We develop a method for computing electron diffraction of the two-photon Kapitza-Dirac effect in a two-dimensional Gaussian beam standing light wave within a relativistic formulation. The solutions are computed on the basis of time-dependent perturbation theory, where a momentum space formulation with the use of a Fourier transformation of the external potential allows for the solving the perturbative time-integrals. An iteration over each possible quantum state combination leads to a quadratic scaling of our method with respect to spacial grid resolution, where time-stepping does not occur in the numeric implementation. The position- and momentum space grids are adapted to the two-photon interaction geometry at low resolution, for which our study only finds partial convergence of the simulated diffraction pattern. Further, the method has the advantage of having an easy implementable parallelization layout.
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.
The standard theoretical treatment of the Kapitza-Dirac effect---that is, the scattering of an electron passing through a standing-wave laser field---is extended here through the use of the Pauli equation to account for the interaction of the electron spin with the magnetic field of the standing wave. Prescriptions for determining unitarity-preserving approximations for the transition probabilities for scattering both with and without rotation of the electron spin direction are provided. This formalism is used to develop a perturbation theory for the spin-flip probability which, in the strong-field limit of interest here, reduces to a fairly simple relation between SSS-matrix elements for scattering with and without change in spin orientation, each expressed in terms of a Bessel function. A similar perturbative procedure is applied to estimate corrections to the standard theory for scattering in the absence of spin-flip processes, in which interactions that change the net number of photons in the field are ignored.
The multiple-linear-time-scale method is used to construct a perturbation theory for NNN-level quantum systems subjected to time-dependent perturbations. The secular and small-denominator terms which plague conventional time-dependent perturbation theory are avoided; consequently, the theory is useful for treating long-term behavior and resonant interactions. The introduction of a self-energy operator allows the shifts in energy levels to be displayed explicitly. When the perturbation is the dipole interaction with an electromagnetic field, the successive approximations yield the well-known rotating-wave approximation, corrections due to counter rotating terms, and the Bloch-Siegert frequency shift. The formalism is applicable to arbitrary pulse shapes, provided that $\ensuremath{\parallel}{H}_{1}\ensuremath{\parallel}\ensuremath{\ll}\ensuremath{\hbar}\overline{\ensuremath{\omega}}$, where $\ensuremath{\parallel}{H}_{1}\ensuremath{\parallel}$ represents the inter-action strength and $\overline{\ensuremath{\omega}}$ is a characteristic frequency of the field. Rabi oscillations induced by multiphoton resonances are automatically included, and this effect is demonstrated in the case of a square pulse with frequency approximately one half the resonance frequency for a transition between two vibrational levels of a molecule. These calculations are compared to an exact numerical solution in order to find the limits of validity of the approximation. Even at high intensities ($I\ensuremath{\sim}{10}^{14}$ W/cm2{\mathrm{cm}}^{2}cm2) the shifted resonance frequency is quite accurate; however, the approximate and exact solutions are slightly out of phase, so that the approximate solution can only be trusted for a few Rabi cycles. For much lower intensities ($I\ensuremath{\sim}{10}^{10}$ W/cm2{\mathrm{cm}}^{2}cm2), the approximate solution is valid for many thousands of Rabi cycles.
The semiclassical perturbation method developed by Henkel et al. [J. Phys. II 4, 1955 (1994)] to model cold-atom diffraction by optical standing waves, is applied to the diffraction of fast atoms on crystal surfaces at grazing incidence (GIFAD or FAD). We first show that the interaction time and interaction length embedded in the obliquity factor is well suited to explain the transition from three-dimensional to two-dimensional (2D) diffraction. The situation of a slightly misaligned primary beam, corresponding to oblique incidence in the effective 2D system, is addressed pointing out discrepancies such as the absence of net deflection of the atomic beam. Guided by time-reversal considerations, we propose an arbitrarily symmetrized form significantly improving the agreement with experimental data recorded in oblique incidence.