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arXiv:1001.5368 · Physical Review A

Quality factor of a matter-wave beam

François Impens

Published Jan 25, 2008• 11 citations
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Abstract

Imperfections in matter-wave beams have been recently characterized experimentally through the measurement of a parameter presented as a beam quality factor. While this parameter is adequate to propagate the transverse width of a cylindrical atomic beam in the paraxial regime, it is not well suited for the description of matter waves in general atom-optical systems. A different quality factor, consistent with the general propagation of matter waves, is proposed.

DOI
10.1103/physreva.77.013619

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Imperfections in matter-wave beams have been recently characterized experimentally through the measurement of a parameter presented as a beam quality factor. While this parameter is adequate to propagate the transverse width of a cylindrical atomic beam in the paraxial regime, it is not well suited for the description of matter waves in general atom-optical systems. A different quality factor, consistent with the general propagation of matter waves, is proposed.
A hidden symmetry of the nonlinear wave equation is exploited to analyze the propagation of paraxial and uniform atom-laser beams in time-independent and quadratic transverse potentials with cylindrical symmetry. The quality factor and the paraxial ABCDABCDABCD formalism are generalized to account exactly for mean-field interaction effects in such beams. Using an approach based on moments, these theoretical tools provide a simple yet exact picture of the interacting beam profile evolution. Guided atom laser experiments are discussed. This treatment addresses simultaneously optical and atomic beams in a unified manner, exploiting the formal analogy between nonlinear optics, nonlinear paraxial atom optics, and the physics of two-dimensional Bose-Einstein condensates.
We present a theoretical model for the propagation of non-self-interacting atom laser beams. We start from a general propagation integral equation and we use the same approximations as in photon optics to derive tools to calculate the atom-laser-beam propagation. We discuss the approximations that allow one to reduce the general equation whether to a Fresnel-Kirchhoff integral calculated by using the stationary phase method, or to the eikonal. Within the paraxial approximation, we also introduce the ABCDABCDABCD matrices formalism and the beam quality factor. As an example, we apply these tools to analyze the recent experiment by Riou et al. [Phys. Rev. Lett. 96, 070404 (2006)].
We study the propagation of a noninteracting atom laser distorted by the strong lensing effect of the Bose-Einstein condensate (BEC) from which it is outcoupled. We observe a transverse structure containing caustics that vary with the density within the residing BEC. Using the WKB approximation, Fresnel-Kirchhoff integral formalism, and ABCD matrices, we are able to describe analytically the atom-laser propagation. This allows us to characterize the quality of the nonideal atom-laser beam by a generalized M2 factor defined in analogy to photon lasers. Finally we measure this quality factor for different lensing effects.
Within the frame of the ABCD formulation for the propagators in a time-dependent quantum system, the ABCD law has been extended to the so-called effective ABCD system in which the effective beam quality factor is conservative. We discussed the system as the propagation of the atom laser beam. The effective beam quality factor and the effective parameter are introduced to obtain the effective ABCD formulation of the propagators by means of Heisenberg picture. The atom-atom interactions in the atom laser beam make the beam quality factor nonconservative, and thus the effective beam quality factor can describe the combined effect of the beam quality factor and the atom-atom interaction.
This paper draws a comparison between atom optics and light optics, based on the observation that one can derive a Schrödinger equation for the slowly varying amplitude of the electromagnetic field in the paraxial approximation, the `optical Schrödinger equation'. A correction scheme between (monochromatic) light waves and matter waves is established and applied to cases of interest in atom or fibre optics. First the propagation in a periodic -potential is investigated, where well known effects, such as Bragg scattering or channelling are re-examined from a `matter wave versus light wave' point of view. In a second application the general theory is extended to axisymmetrical materials such as step-index fibres or graded-index fibres. In particular, a matter wave analogue of the so-called non-diffractive beams of electrodynamics is discussed.
We study experimentally and theoretically a beam splitter setup for guided atomic matter waves. The matter wave is a guided atom laser that can be tuned from quasimonomode to a regime where many transverse modes are populated, and propagates in a horizontal dipole beam until it crosses another horizontal beam at 45°. We show that depending on the parameters of this X configuration, the atoms can all end up in one of the two beams (the system behaves as a perfect guide switch), or be split between the four available channels (the system behaves as a beam splitter). The splitting regime results from a chaotic scattering dynamics. The existence of these different regimes turns out to be robust against small variations of the parameters of the system. From numerical studies, we also propose a scheme that provides a robust and controlled beam splitter in two channels only.