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Acta Physica Sinica

Effective ABCD formulation of the propagation of the atom laser

Jianling Yin, Chengyi Liu, Yang You-yuan, Liu Jiang, Fan Guang-Han, (1)华南师范大学激光运动医学实验室,广州 510631;华南师范大学信息光电子科技学院，广州 510631; (2)华南师范大学激光运动医学实验室,广州 510631;香港教育学院科学系，香港; (3)华南师范大学信息光电子科技学院，广州 510631; (4)香港教育学院科学系，香港

Published Jan 1, 2004• 4 citations

Abstract

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.

DOI: 10.7498/aps.53.356

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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.

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

ABCD

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)].

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

ABCD

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 treatment of the nonlinear matter-wave propagation inspired by optical methods, which includes interaction effects within the atom-optics equivalent of the aberrationless approximation. The atom-optical

ABCD

-matrix formalism, considered so far for noninteracting clouds, is extended perturbatively beyond the linear regime of propagation. This approach, applied to discuss the stability of a matter-wave resonator involving a free-falling sample, agrees very well with the predictions of the full nonlinear paraxial wave equation. An alternative optical treatment of interaction effects, based on the aberrationless approximation and suitable for cylindrical paraxial beams of uniform linear density, is also adapted for matter waves.