Abstract
We analytically calculate the statistics of Bose-Einstein condensate (BEC) fluctuations in an interacting gas trapped in a three-dimensional cubic or rectangular box with the Dirichlet, fused or periodic boundary conditions within the mean-field Bogoliubov and Thomas-Fermi approximations. We study a mesoscopic system of a finite number of trapped particles and its thermodynamic limit. We find that the BEC fluctuations, first, are anomalously large and non-Gaussian and, second, depend on the trap’s form and boundary conditions. Remarkably, these effects persist with increasing interparticle interaction and even in the thermodynamic limit—only the mean BEC occupation, not BEC fluctuations, becomes independent on the trap’s form and boundary conditions.
Highlights
QuasiparticlesIn the thermal equilibrium the mesoscopic system of N interacting particles is described by the statistical operator (density matrix)
The overall picture of the effects of the trap’s form and boundary conditions on the noncondensate occupation statistics for a weakly interacting gas within the basic model (10)–(12) is similar to the one known for an ideal gas. (For example, statistics of Bose-Einstein condensate (BEC) in an ideal gas trapped in the anisotropic box or slabs was discussed in [19,21,22,27,46].) As is illustrated in Figure 2, a typical change in the thermodynamic-limit asymptotics of the noncondensate occupation probability distribution due to a change of the trap’s form or boundary conditions is about 10% or so
This basic model yields a series of interesting conclusions about the probability distribution, moments and cumulants of BEC fluctuations as the functions of the interaction strength, temperature, the number of trapped particles, the dimension of the system, the form of the trap and the imposed boundary conditions
Summary
In the thermal equilibrium the mesoscopic system of N interacting particles is described by the statistical operator (density matrix). The condensate is well developed and the particle number constraint of the canonical ensemble N = N0 + Nex does not play any significant role for the occupation numbers of the quasiparticle states and different blocks of quasiparticles (which are, pairs of quasiparticles in the case of the box trap), which have mutually nonzero overlapping integrals in the excited particle occupation operator (7), produce independent contributions to the noncondensate occupation This means that the characteristic function is the product of the partial characteristic functions, corresponding to the occupation contributions from such blocks of quasiparticles, and the probability distribution ρ(n) of the condensate depletion operator Nex could be effectively obtained via the Fourier transform of this characteristic function. This statistics is non-Gaussian as is clearly seen from the formulas for the higher-order cumulants which establish nonzero values
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