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Physical Review Letters

Measurement of the Coherence of a Bose-Einstein Condensate

E. W. Hagley, L. Deng, Mikio Kozuma, Marek Trippenbach, Y. B. Band, Mark Edwards, M. R. Doery, Paul S. Julienne, Kristian Helmerson, S. L. Rolston, William D. Phillips

Published Oct 18, 1999• 198 citations
Publisher

Abstract

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 $\ensuremath{\Delta}t$ 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 $\ensuremath{\Delta}t$ 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.

DOI
10.1103/physrevlett.83.3112

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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 \ensuremathΔt\ensuremath{\Delta}t\ensuremathΔt 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 \ensuremathΔt\ensuremath{\Delta}t\ensuremathΔt 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.
The coherence properties of phase fluctuating Bose-Einstein condensates are studied both theoretically and experimentally. We derive a general expression for the N-particle correlation function of a condensed Bose gas in a highly elongated trapping potential. The second-order correlation function is analyzed in detail, and an interferometric method to directly measure it is discussed and experimentally implemented. Using a Bragg diffraction interferometer, we measure intensity correlations in the interference pattern generated by two spatially displaced copies of a parent condensate. Our experiment demonstrates how to characterize the second-order correlation function of a highly elongated condensate and to measure its phase coherence length.
We probe the spatial coherence properties of a magnetically trapped Bose gas. Two matter wave beams are extracted from two spatially separated regions of the trap and overlap outside the trapping region. The visibility of the resulting interference pattern measures the phase coherence between the regions of extraction. By varying the spatial separation between the two regions the first-order spatial correlation function of the trapped Bose gas can be measured. The location of the minima of the interference pattern is reproducible, which experimentally confirms that the trapped Bose-Einstein condensate is not fragmented into individual condensates.
A trapped-atom interferometer was demonstrated using gaseous Bose-Einstein condensates coherently split by deforming an optical single-well potential into a double-well potential. The relative phase between the two condensates was determined from the spatial phase of the matter wave interference pattern formed upon releasing the condensates from the separated potential wells. Coherent phase evolution was observed for condensates held separated by 13 microm for up to 5 ms and was controlled by applying ac Stark shift potentials to either of the two separated condensates.
Bose-Einstein condensates of rubidium atoms are stored in a two-dimensional periodic dipole force potential, formed by a pair of standing wave laser fields. The resulting potential consists of a lattice of tightly confining tubes, each filled with a 1D quantum gas. Tunnel coupling between neighboring tubes is controlled by the intensity of the laser fields. By observing the interference pattern of atoms released from more than 3000 individual lattice tubes, the phase coherence of the coupled quantum gases is studied. The lifetime of the condensate in the lattice and the dependence of the interference pattern on the lattice configuration are investigated.
The criteria for validity of adiabaticity for nonlinear wave equations are considered within the context of atomic matter waves tunneling from macroscopically populated optical standing-wave traps loaded from a Bose-Einstein condensate. We show that, even when the optical standing wave is slowly turned on and the condensate behaves adiabatically during this turn-on, once the tunneling time between wells in the optical lattice becomes longer than the nonlinear time scale, adiabaticity breaks down and a significant spatially varying phase develops across the condensate wave function from well to well. This phase drastically affects the contrast of the fringe pattern in Josephson-effect interference experiments, and the condensate coherence properties in general.
We suggest an experiment to investigate interference effects between two spatially separated Bose-Einstein condensates. In the presence of coherence, the dynamic structure factor, measurable through inelastic photon scattering, is predicted to exhibit, at high momentum transfer qqq, interference fringes with frequency period $\ensuremath{\Delta}\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}q/\mathrm{md}$, where ddd is the distance between the condensates. We show that this coherent configuration corresponds to an eigenstate of the physical observable measured in the experiment and that the relative phase of the condensates can be, hence, created through the measurement process.