The occurrence of phase fluctuations due to thermal excitations in Bose-Einstein condensates (BECs) is studied for a variety of temperatures and trap geometries. We observe the statistical nature of the appearence of phase fluctuations and characterize the dependence of their average value on temperature, number of particles and the trapping potential. We find pronounced phase fluctuations for condensates in very elongated traps in a broad temperature range. The results are of great importance for the realization of BEC in quasi 1D geometries, for matter wave interferometry with BECs, as well as for coherence properties of guided atom laser beams. 03.75.Fi, 32.80.Pj, 05.30.Jp Since the first experimental realization of BoseEinstein condensation in dilute atomic gases [1], there has been enormous interest in the coherence properties of BECs. In particular, the phase coherence is essential for applications of BEC in matter wave interferometry; it also sets limits on the coherence of atom lasers, and guided atom laser beams. For a trapped 3D condensate well below the BEC transition temperature T c , recent experiments have confirmed the phase coherence, e.g., it was shown that the coherence length is equal to the condensate size [2,3]. However, phase coherence is not an obvious property of BEC. In particular, it is expected that low-dimensional (1D and 2D) quantum gases differ qualitatively from the 3D case in this respect [4][5][6][7]. Recently, it was shown theoretically [8] that for very elongated condensates phase fluctuations can be pronounced already in the equilibrium state of the usual 3D ensemble, where the density fluctuations are suppressed. The phase coherence length in this case can be smaller than the axial size of the sample. This is referred to as the regime of quasicondensation [9]. The detailed characterization of phase fluctuations in condensates is thus of great importance for applications of BEC, especially for recent attempts to reach BEC in elongated micro-circuit geometries [12]. The temperature dependence of the coherence of an atom laser beam was studied in [13].In this Letter we report on systematic studies of BEC of 87 Rb atoms in the regime of phase-fluctuating condensates. We achieved this regime in highly anisotropic traps leading to a strongly elongated shape of the condensate. We observe the phase fluctuations by measuring the density distribution of the released cloud after ballistic expansion. By varying the temperature and the aspect ratio of the trapping potential, we study the continuous transition from the usual 3D regime, where phase fluctuations of the condensate are low, into the regime of strong phase fluctuations. We show that a phase coherent matter wave is not a direct outcome of BEC in strongly elongated geometries but can only be achieved for very low temperatures, well below the BEC transition temperature T c .Fluctuations of the phase of a Bose condensate are related to thermal excitations and always appear at finite temperature. However, as shown in Ref. [8], th...

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We demonstrate the existence of phase fluctuations in elongated Bose-Einstein Condensates (BECs) and study the dependence of those fluctuations on the system parameters. A strong dependence on temperature, atom number, and trapping geometry is observed. Phase fluctuations directly affect the coherence properties of BECs. In particular, we observe instances where the phase coherence length is significantly smaller than the condensate size. Our method of detecting phase fluctuations is based on their transformation into density modulations after ballistic expansion. An analytic theory describing this transformation is developed.