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 ...

We report on measurements of the critical temperature of a harmonically trapped, weakly interacting Bose gas as a function of atom number. Our results exclude ideal-gas behavior by more than two standard deviations, and agree quantitatively with mean-field theory. At our level of sensitivity, we find no additional shift due to critical fluctuations. In the course of this measurement, the onset of hydrodynamic expansion in the thermal component has been observed. Our thermometry method takes this feature into account.PACS numbers: 03.75.Hh,03.75.Kk Degenerate atomic Bose gases provide an ideal testing ground for the theory of quantum fluids. First, their diluteness makes possible first-principles theoretical approaches [1]. Second, thanks to the powerful experimental techniques of atomic physics, static and dynamic properties can be studied quantitatively through a wide range of temperature and densities. Furthermore, the inhomogeneity induced by the external trapping potential leads to entirely new behavior, when compared to bulk quantum fluids.Atomic interactions have previously been found to affect deeply the dynamical behavior of trapped Bose gases at finite temperatures [2,3]. By contrast, the influence of interactions on thermodynamics is less pronounced [4], and has been less studied experimentally. Pioneering work on thermodynamics [5] concentrated essentially on the ground state occupation, and the role of interactions was somewhat hidden by finite size effects [1]. Though several such measurements have been reported [3,6], to our knowledge a decisive test of the role of interactions is still lacking. In this Letter, we focus on the critical temperature T c of a harmonically trapped 87 Rb Bose gas to demonstrate the influence of interactions on the thermodynamics. We study the behavior of T c as a function of the number of atoms at the transition, for a fixed trapping geometry. We find a deviation from ideal-gas behavior, towards lower critical temperatures, whose signification will be discussed below. In the course of this study, we have observed that collisions induce an anisotropy in the free expansion of the cloud even far from the hydrodynamic regime [7,8]. We correct for this effect in our temperature measurement.For an ideal Bose gas in a harmonic trap, the critical temperature is [1]

We have investigated experimentally the finite-temperature properties of a Bose-Einstein condensed cloud of 87 Rb atoms in a harmonic trap. Focusing primarily on condensed fraction and expansion energy, we measure unambiguous deviations from ideal-gas thermodynamics and obtain good agreement with a Hartree-Fock description of the mixed cloud. Our results offer clear evidence of the mutual interaction between the condensed and thermal components. To probe the low-temperature region unaccessible to the usual time-of-flight technique, we use coherent Bragg scattering as a filtering technique for the condensate. This allows us to separate spatially the condensed and normal components in time of flight and to measure reliably temperatures as low as 0.2T c 0 and thermal fractions as low as 10%. Finally, we observe evidence for the limitations of the usual image analysis procedure, pointing out to the need for a more elaborate model of the expansion of the mixed cloud.