We present experimental and theoretical studies of the coherence properties of a Bose-Einstein condensate (BEC) using an interference technique. Two optical standing wave pulses of duration 100 ns and separation Dt are applied to a condensate. Each standing wave phase grating makes small copies of the condensate displaced in momentum space. The quantum mechanical amplitudes of each copy interfere, depending on Dt and on spatial phase variations across the condensate. We find that the behavior of a trapped BEC is consistent with a uniform spatial phase. A released BEC, however, exhibits large phase variation across the condensate. PACS numbers: 03.75.Fi, 05.30.Jp, 32.80.Qk Since the first demonstrations of Bose-Einstein condensation in dilute atomic gases [1], there have been many efforts to explore the nature of such condensates [2,3]. The phase properties of Bose-Einstein condensates are of particular interest because they affect how BECs interfere. The characterization of atoms extracted from a BEC as constituting an "atom laser" beam is related to these phase properties. In this Letter we present direct measurements and theoretical calculations of phase variations across a BEC, a property related to spatial coherence. For a trapped, pure BEC one expects the phase to be spatially uniform because the condensate is in a stationary state of the system with no angular momentum. On the other hand, in an incompletely formed BEC, one might expect differences in the phase between different regions of the condensate [4]. A released BEC, composed of atoms with a positive scattering length, develops phase variations as it explosively expands due to the atom-atom (mean-field) interaction [2]. Understanding these phase variations is essential for characterizing condensates as sources of coherent matter waves.Matter-wave interference between two condensates was reported in 1997 [5] where two condensates initially localized in different regions of a double-well potential were released and allowed to spread and overlap. That experiment, equivalent to a Young's double slit experiment, showed that two independent condensates interfere, as do two separate lasers [6]. Here we describe a novel method of self-interfering a BEC to extract information about its phase.We measure the spatial coherence of the BEC by creating and interfering two spatially displaced, coherently diffracted "copies" of the original BEC in the same momentum state. An optical standing wave pulse diffracts [7-9] a small fraction of the condensate into momentum states 62nh k, where n is an integer and k ͑2p͞l͒ẑ is the optical wave vector. For our conditions, a negligible fraction is diffracted into momentum states with n . 1. (Because the process is symmetric, the discussion that follows refers to only the 12hk copy.) A second diffraction pulse, applied Dt after the first pulse, creates a second overlapping 2hk copy displaced from the first by D z ͑2hkDt͞m͒ẑ 2y r Dtẑ, with m the atomic mass and y r the recoil velocity. The amplitudes of the wave functions rep...

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We examine the long-range part of the rare-gas diatomic potentials that connect to the R͕(nϪ1)p 5 ns͖ ϩR͕(nϪ1)p 5 np͖ atomic states in the separated atom limit ͑nϭ3, 4, 5, and 6 for Ne, Ar, Kr, and Xe, respectively͒. We obtain our potentials by diagonalization of a Hamiltonian matrix containing the atomic energies and the electric dipole-dipole interaction, with experimentally determined parameters ͑atomic energies, lifetimes, transition wavelengths, and branching ratios͒ as input. Our numerical studies focus on Ne and Kr in this paper, but apply in principle to all other rare gases lacking hyperfine structure. These diatomic potentials are essential for applications in which homonuclear rare-gas pairs interact at large internuclear separations, greater than about 20 Bohr radii. Among such applications are the study of cold atomic collisions and photoassociative spectroscopy.

We study the coherence properties of an atom laser, which operates by extracting atoms from a gaseous Bose-Einstein condensate via a two-photon Raman process, by analyzing a recent experiment [(Hagley et al., submitted to Phys. Rev. Lett. (1999)]. We obtain good agreement with the experimental data by solving the time-dependent Gross-Pitaevskii equation in three dimensions both numerically and with a Thomas-Fermi model. The coherence length is strongly affected by the space-dependent phase developed by the condensate when the trapping potential is turned off.