Measurement of the Coherence of a Bose-Einstein Condensate

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 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...

Spinor Condensates and Light Scattering from Bose-Einstein Condensates

“…Our measurements using Bragg scattering provide a more quantitative measure of this property. Recently, two other groups have made measurements of the coherence length [38,42]. These measurements give evidence of a decay of the coherence length over two length scales.…”

“…the condensate is one "coherent matter wave." An analogous measurement in the time domain has been done in Gaithersburg [38]. The error bars are 1σ errors of the Gaussian fits to the data.…”

“…The strength of the confining field could be easily modified by adjusting the currents through the magnetic trapping coils, allowing us to vary the peak condensate density in the range (0.5 -6) ×10 14 cm −3 and the radial half-width of the condensate (x c ) between 7 and 15 µm. The two laser beams used for Bragg scattering were derived from a common source, and had a detuning to the red of the 3S 1/2 , |F = 1 → 3P 3/2 , |F ′ = 0, 1, 2 optical transitions of about ∆ = 1.7 GHz (similar parameters were used for experiments at NIST, Gaithersburg [14,15,38,39]). A small frequency difference ω between the counter-propagating beams, on the order of 2π × 100 kHz, was introduced by using two independently-controlled acousto-optic modulators (AOM's) to control the frequency of two independent beams 2 .…”

Time-orbiting potential trap for Bose-Einstein condensate interferometry

“…Condensate interferometers using atoms confined by either magnetic [5,9,13] or optical [4,7,10] fields have been demonstrated. Measurement times in these devices have been limited for a variety of reasons, but a common concern is the effect of interatomic interactions, which can introduce phase noise and cause spatial distortions in the cloud [4,10,14].…”

“…Slow velocities and long coherence lengths allow condensates to exhibit greater sensitivity per atom than a thermal cloud with similar number of atoms. Though producing and working with condensates remains challenging, several groups have demonstrated Mach-Zehnder or Michelson interferometers using condensates [3,4,5,6,7,8,9,10]. However, all these experiments have been limited to measurement times of roughly ten milliseconds or less.…”