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Journal of Physics B Atomic Molecular and Optical Physics

Scattering of atoms on the Bose-Einstein condensate

Zbigniew Idziaszek, Kazimierz Rza̧żewski, Martin Wilkens

Published Jan 1, 1999• 12 citations
Publisher

Abstract

We investigate the scattering of slowly moving atoms on the Bose-Einstein condensate. The condensate excitations are described by Bogolyubov-de Gennes equations. We derive the analytic expressions for the differential cross section for both elastic and inelastic channels. For the elastic channel we obtain analytic results for total cross sections, and discuss their scaling with the number of condensed atoms. For inelastic channels we present numerical results for the total cross section.

DOI
10.1088/0953-4075/32/9/103

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We investigate the scattering of slowly moving atoms on the Bose-Einstein condensate. The condensate excitations are described by Bogolyubov-de Gennes equations. We derive the analytic expressions for the differential cross section for both elastic and inelastic channels. For the elastic channel we obtain analytic results for total cross sections, and discuss their scaling with the number of condensed atoms. For inelastic channels we present numerical results for the total cross section.
The scattering of an incident electron by a pure Bose-Einstein condensate of alkali-metal atoms confined to a three-dimensional harmonic-oscillator potential is investigated to give the corresponding differential cross section. It is shown that for the elastic scattering process the differential cross section is proportional to the square of the number of condensed atoms, and hence can be called the ``coherent scattering.'' For the inelastic scattering process, the stopping power is obtained by considering the state of the center of mass and the internal electronic state of the condensed atom being excited separately and simultaneously, which is found to be proportional to the 75\frac{7}{5}57​ power of the number of condensed atoms when the binary collision interaction between atoms is taken into account.
Bragg diffraction of atoms by light waves can create high momentum components in a Bose-Einstein condensate. Collisions between atoms from two distinct momentum wave packets cause elastic scattering that can remove a significant fraction of atoms from the wave packets and cause the formation of a spherical shell of scattered atoms. We develop a slowly varying envelope technique that includes the effects of this loss on the condensate dynamics described by the Gross-Pitaevski equation. Three-dimensional numerical calculations are presented for two experimental situations: passage of a moving daughter condensate through a nonmoving parent condensate, and four-wave mixing of matter waves.
Bragg diffraction divides a Bose-Einstein condensate into two overlapping components, moving with respect to each other with high momentum. Elastic collisions between atoms from distinct wave packets can significantly deplete the condensate. Recently, Zi\ifmmode \acute{n}\else \'{n}\fi{} et al. [Phys. Rev. Lett. 94, 200401 (2005)] introduced a model of two counterpropagating atomic Gaussian wave packets incorporating the dynamics of the incoherent scattering processes. Here we study the properties of this model in detail, including the nature of the transition from spontaneous to stimulated scattering. Within the first-order approximation, we derive analytical expressions for the density matrix and anomalous density that provide excellent insight into correlation properties of scattered atoms.
During a collision of Bose-Einstein condensates correlated pairs of atoms are emitted. The scattered massive particles, in analogy to photon pairs in quantum optics, might be used in the violation of Bell's inequalities, demonstration of Einstein-Podolsky-Rosen correlations, or sub-shot-noise atomic interferometry. Usually, a theoretical description of the collision relies either on stochastic numerical methods or on analytical treatments involving various approximations. Here, we investigate elastic scattering of atoms from colliding elongated Bose-Einstein condensates within the Bogoliubov method, carefully controlling performed approximations at every stage of the analysis. We derive expressions for the one- and two-particle correlation functions. The obtained formulas, which relate the correlation functions to the condensate wave function, are convenient for numerical calculations. We employ the variational approach for condensate wave functions to obtain analytical expressions for the correlation functions, whose properties we analyze in detail. We also present a useful semiclassical model of the process and compare its results with the quantum one. The results are relevant for recent experiments with excited helium atoms, as well as for planned experiments aimed at investigating the nonclassicality of the system.