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Physical Review A

Quantum calculation of cold-atom diffraction using periodic magnetic fields

Yin Hung, Bradley Schuller, John T. Giblin, Janine Shertzer

Published Jun 29, 2006• 1 citation
Publisher

Abstract

When cold atoms approach a periodically magnetized surface, they are scattered by the effective repulsive potential $U(x,y)={\ensuremath{\mu}}_{B}{m}_{F}gB(x,y)$. If the period $a$ of the magnetized surface is larger than the de Broglie wavelength $\ensuremath{\lambda}$ of the atoms, the atoms are diffracted. We have developed a method for calculating the location and intensity of the diffraction peaks using formal scattering theory. We solve the exact two-dimensional Schr\"odinger equation with the finite element method and calculate the scattering amplitude $f(\ensuremath{\phi})$ using the integral formula; a plot of $\ensuremath{\mid}f(\ensuremath{\phi}){\ensuremath{\mid}}^{2}$ provides a visualization of the diffraction pattern. By varying the experimental parameters within a realistic range, we predict the optimal conditions for observing atom diffraction.

DOI
10.1103/physreva.73.062722

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When cold atoms approach a periodically magnetized surface, they are scattered by the effective repulsive potential U(x,y)=\ensuremathμBmFgB(x,y)U(x,y)={\ensuremath{\mu}}_{B}{m}_{F}gB(x,y)U(x,y)=\ensuremathμB​mF​gB(x,y). If the period aaa of the magnetized surface is larger than the de Broglie wavelength \ensuremathλ\ensuremath{\lambda}\ensuremathλ of the atoms, the atoms are diffracted. We have developed a method for calculating the location and intensity of the diffraction peaks using formal scattering theory. We solve the exact two-dimensional Schr"odinger equation with the finite element method and calculate the scattering amplitude f(\ensuremathϕ)f(\ensuremath{\phi})f(\ensuremathϕ) using the integral formula; a plot of \ensuremath∣f(\ensuremathϕ)\ensuremath∣2\ensuremath{\mid}f(\ensuremath{\phi}){\ensuremath{\mid}}^{2}\ensuremath∣f(\ensuremathϕ)\ensuremath∣2 provides a visualization of the diffraction pattern. By varying the experimental parameters within a realistic range, we predict the optimal conditions for observing atom diffraction.
The method of quantum trajectories proposed by de Broglie and Bohm is applied to the study of atom diffraction by surfaces. As an example, a realistic model for the scattering of He off corrugated Cu is considered. In this way, the final angular distribution of trajectories is obtained by box counting, which is in excellent agreement with the results calculated by standard SSS matrix methods of scattering theory. More interestingly, the accumulation of quantum trajectories at the different diffraction peaks is explained in terms of the corresponding quantum potential. This nonlocal potential ``guides'' the trajectories causing a transition from a distribution near the surface, which reproduces its shape, to the final diffraction pattern observed in the asymptotic region, far from the diffracting object. These two regimes are homologous to the Fresnel and Fraunhofer regions described in undulatory optics. Finally, the turning points of the quantum trajectories provide a better description of the surface electronic density than the corresponding classical ones, usually employed for this task.
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