Fluctuations of Bose-Einstein Condensate

The recent achievement of Bose-Einstein condensation (BEC) in trapped atoms of dilute gases [1–3] has been quickly followed by the quantitative measurements of the condensate’s properties. In particular the dependence of the fraction of atoms in the condensate on the temperature has been found to be in agreement with the theory of the noninteracting Bose gas in a harmonic trap [4]. From the point of view of the textbook thermodynamics the atomic condensate studied by Boulder and MIT groups is an exotic system: (1) It consists of a finite number of particles. This number does not fluctuate after the complicated cooling process is over. (2) It is highly nonuniform. The trapping harmonic potential changes within the condensate. And yet the textbook theory of the Bose-Einstein condensation is based on the thermodynamic limit of the uniform, noninteracting gas described by the grand canonical ensemble. Perhaps the first person to worry about this was Schrödinger [5], long before the present experiments. He considered, however, only the case of a finite number of noninteracting bosons in a cell. Later the grand canonical ensemble predictions for the harmonically trapped finite number of noninteracting bosons was made [6–8]. According to this theory kN0lykNl ­ 1 2 sTyTcd, where kN0l is the mean number of atoms in the condensate, kNl is the mean total number of trapped atoms, T is the temperature, and Tc is the critical temperature. The recent experimental results agree with the above formula. But the fluctuations of the condensate fraction certainly do not agree with the results of the grand canonical ensemble. This last quantity suffers large relative fluctuations which approach 100% as the temperature tends to zero. Hence the theory of the harmonically trapped condensate based on a microcanonical ensemble is needed. In the recent Letter [9] these fluctuations were computed in a one dimensional model. This cannot be directly compared to the experiment, which is obviously three dimensional. Moreover, in one dimension there is no phase transition in the limit of number of particles going to infinity. It is the purpose of this Letter to estimate for the first time the fluctuations of the condensate fraction according to the