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arXiv:cond-mat/0509281 · Physical Review Letters

Beam Quality of a Nonideal Atom Laser

J.-F. Riou, William Guerin, Yann Le Coq, M. Fauquembergue, Vincent Josse, Philippe Bouyer, Alain Aspect

Published Feb 24, 2006• 76 citations
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Abstract

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.

DOI
10.1103/physrevlett.96.070404

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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.
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)].
The transverse mode of an atom laser beam that is outcoupled from a Bose--Einstein condensate is investigated and is found to be strongly determined by the mean-field interaction of the laser beam with the condensate. For repulsive interactions, the condensate acts as a diverging lens for the outcoupled atoms, and transverse interference fringes in the atomic beam are predicted due to the coherent addition of amplitudes for atoms emitted from different regions in the condensate. This effect can be used to investigate the transverse coherence of an atom laser beam.
We present a three-dimensional quantum-mechanical theory of radio-frequency-outcoupled atom lasers from trapped atomic gases in the presence of the gravitational force. Predictions for the total outcoupling rate as a function of the radio frequency and for the beam wave function are given. We establish a sum rule for the energy-integrated outcoupling, which leads to a separate determination of the coupling strength between the atoms and radiation field. For a noninteracting Bose-Einstein condensate analytic solutions are derived which are subsequently extended to include the effects of atomic interactions. The interactions enhance interference effects in the beam profile and modify the outcoupling rate of the atom laser. We provide a complete quantum-mechanical solution which is in line with experimental findings and allows us to determine the validity of commonly used approximative methods. We also extend the formalism to a fermionic atom laser and analyze the effect of superfluidity on the outcoupling of atoms.
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.
Focusing on the cigar-shaped trap model, we provide longitudinal profiles of a weakly outcoupled atom laser propagation both inside and outside the Bose-Einstein condensate regions. The propagation itself is generally represented by inhomogeneous Schrödinger equation which is derived from a set of Gross-Pitaevskii equations by applying the available conditions. We also show that by imposing boundary condition and using quantum oscillator model, energy of the outcoupled atom laser outside the Bose-Einstein condensate region is quantized while there is no analytical solution for the propagation of the outcoupled atom laser inside the Bose-Einstein condensate region.