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We measure the axial momentum distribution of Bose-Einstein condensates with an aspect ratio of 152 using Bragg spectroscopy. We observe the Lorentzian momentum distribution characteristic of one-dimensional phase fluctuations. The temperature dependence of the width of this distribution provides a quantitative test of quasi-condensate theory. In addition, we observe a condensate length consistent with the absence of density fluctuations, even when phase fluctuations are large.PACS numbers: 03.75. Fi,05.30.Jp One of the most striking features of Bose-Einstein condensates is their phase coherence. Extensive experimental work on dilute atomic gases has demonstrated a uniform phase of three-dimensional (3D) trapped condensates [1,2], even at finite temperature [3]. In low dimensional systems, however, phase fluctuations of the order parameter are expected to destroy off-diagonal long range order (see [4,5] and references therein). This phenomenon also occurs in sufficiently anisotropic 3D samples, where phase coherence across the axis (long dimension) is established only below a temperature T φ , that can be much lower than the critical temperature T c [6]. In the range T φ < T < T c , the cloud is a "quasi-condensate", whose incomplete phase coherence is due to thermal excitations of 1D axial modes, with wavelengths larger than its radial size. Quasi-condensates in elongated traps have been observed by Dettmer et al. [7], who measured the conversion, during free expansion, of the phase fluctuations into ripples in the density profile. Although the conversion dynamics is well understood [8], the measured amplitude of density ripples was a factor of two smaller than expected.In this Letter, we report on the measurement of the axial coherence properties of quasi-condensates via momentum Bragg spectroscopy. In previous work using this technique [2,9], the finite size and mean-field energy were the primary contributors to the spectral width. By contrast, the dominant broadening in our conditions results from thermally driven fluctuations of the phase, which reduce the coherence length [4,6]. Indeed, the axial momentum distribution is the Fourier transform of the spatial correlation function, where u z is the axial unit vector. When phase fluctuations dominate (i.e. T ≫ T φ ), the axial momentum width is hence proportional to /L φ , where L φ is the characteristic axial decay length of C(s). Experimentally, for 6 < T /T φ < 36, we find momentum distributions with Lorentzian shapes, whose widths increase with T . Such a shape is charac- † UMR 8501 du CNRS teristic of large phase fluctuations in 1D [11], which result in a nearly exponential decay of C(s). Moreover, the momentum width agrees quantitatively with theoretical predictions to within our 15% experimental uncertainty. This implies, in this temperature range, a coherence length substantially smaller than the quasicondensate length 2L, from about L/18 to L/4. We have also checked an important feature of quasi-condensates: the suppression of density fluctuations ...