letters to nature 218 NATURE | VOL 398 | 18 MARCH 1999 | www.nature.com optimizing the performance of the interferometers at low frequencies, where both equations (4) and (6) become more signi®cant. It appears natural to perform such studies in the quiet environment of space, perhaps through future re®nements of LISA-type set-ups 27 .The above discussion of gravity-wave interferometers shows that the smallness of the Planck length does not preclude the possibility of direct investigations of space-time fuzziness. This complements the results of studies 28,29 which have shown that indirect evidence of quantum space-time¯uctuations could be obtained by testing the predictions of theories consistent with a given picture of thesē uctuations. Additional encouragement for experiment-driven progress in understanding the interplay between gravity and quantum mechanics comes from recent studies 30,31 in the area of gravitationally induced phases, the signi®cance of which has been emphasized in refs 32 and 33. M

We identify all possible classes of solutions for two-component Bose-Einstein condensates (BECs) within the Thomas-Fermi (TF) approximation, and check these results against numerical simulations of the coupled Gross-Pitaevskii equations (GPEs). We find that they can be divided into two general categories. The first class contains solutions with a region of overlap between the components. The other class consists of non-overlapping wavefunctions, and contains also solutions that do not possess the symmetry of the trap. The chemical potential and average energy can be found for both classes within the TF approximation by solving a set of coupled algebraic equations representing the normalization conditions for each component. A ground state minimizing the energy (within both classes of the states) is found for a given set of parameters characterizing the scattering length and confining potential. In the TF approximation, the ground state always shares the symmetry of the trap. However, a full numerical solution of the coupled GPEs, incorporating the kinetic energy of the BEC atoms, can sometimes select a broken-symmetry state as the ground state of the system. We also investigate effects of finite-range interactions on the structure of the ground state.

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...

It has been proposed that the adiabatic loading of a Bose-Einstein Condensate (BEC) into an optical lattice via the Mott-insulator transition can be used to initialize a quantum computer [D. Jaksch, et al., Phys. Rev. Lett. 81, 3108 (1998)]. The loading of a BEC into the lattice without causing band excitation is readily achievable; however, unless one switches on an optical lattice very slowly, the optical lattice causes a phase to accumulate across the condensate. We show analytically and numerically that a cancellation of this effect is possible by adjusting the harmonic trap force-constant of the magnetic trap appropriately, thereby facilitating quick loading of an optical lattice for quantum computing purposes. A simple analytical theory is developed for a non-stationary BEC in a harmonic trap.