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arXiv:cond-mat/0004208 · Physical Review A

Rotational properties of trapped bosons

F. Brosens, J. T. Devreese, L. F. Lemmens

Published Mar 1, 1997• 12 citations
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Abstract

Based on the Hellman-Feynman theorem, it is shown that the average square radius of a cloud of interacting bosons in a parabolic well can be derived from their free energy. As an application, the temperature dependence of the moment of inertia of noninteracting bosons in a parabolic trap is determined as a function of the number of bosons. Well below the critical condensation temperature, the Bose-Einstein statistics are found to substantially reduce the moment of inertia of this system, as compared to a gas of ``distinguishable'' particles in a parabolic well.

DOI
10.1103/physreva.55.2453

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Journal of Experimental and Theoretical Physics Letters•Oct 1, 1997•0 citations
Based on the Hellman-Feynman theorem, it is shown that the average square radius of a cloud of interacting bosons in a parabolic well can be derived from their free energy. As an application, the temperature dependence of the moment of inertia of noninteracting bosons in a parabolic trap is determined as a function of the number of bosons. Well below the critical condensation temperature, the Bose-Einstein statistics are found to substantially reduce the moment of inertia of this system, as compared to a gas of ``distinguishable'' particles in a parabolic well.
The superfluid fraction of an atomic cloud can be defined using the cloud's response to a rotation of the external potential, i.e. the moment of inertia. In this paper we derive analytical results for the moment of inertia of a small number of non-interacting Bosons using the canonical ensemble. The required symmetrized averages are obtained via a representation of the partition function by permutation cycles. Our results are useful to discriminate purely quantum statistical effects from interaction effects in studies of superfluidity and phase transitions in finite samples.
We compute the full probability distribution of the moment of inertia I \propto \sum_{i=1}^N \vec{r}_i^{\,2} I ∝ ∑ i = 1 N r → i 2 of a gas of N N noninteracting bosons trapped in a harmonic potential V(r) = (1/2)\, m\, \omega^2 r^2 V ( r ) = ( 1 / 2 ) m ω 2 r 2 , in all dimensions and at all temperature. The appropriate thermodynamic limit in a trapped Bose gas consists in taking the limit N\to ∞ N → ∞ and \omega\to 0 ω → 0 with their product \rho = N \omega^d ρ = N ω d fixed, where \rho ρ plays the role analogous to the density in a translationally invariant system. In this thermodynamic limit and in dimensions d;gt 1 d > 1 , the harmonically trapped Bose gas undergoes a Bose-Einstein condensation (BEC) transition as the density \rho ρ crosses a critical value \rho_c(\beta) ρ c ( β ) , where \beta β denotes the inverse temperature. We show that the nature of the condensation is different for 1<d≤ 2 1 < d ≤ 2 and d;gt 2
The momentum distribution function of a parabolically confined gas of bosons with harmonic interparticle interactions is derived. In the Bose-Einstein condensation region, this momentum distribution substantially deviates from a Maxwell-Boltzmann distribution. It is argued that the determination of the temperature of the boson gas from the Bose-Einstein momentum distribution function is more appropriate than the currently used fitting to the high-momentum tail of the Maxwell-Boltzmann distribution.
The topic of Bose–Einstein condensation in a uniform, non-interacting gas of bosons is treated in most textbooks on statistical mechanics. In the present chapter we discuss the properties of a non-interacting Bose gas in a trap. We shall calculate equilibrium properties of systems in a semi-classical approximation, in which the energy spectrum is treated as a continuum. For this approach to be valid the temperature must be large compared with Δ∊/ k , where Δ∊ denotes the separation between neighbouring energy levels. As is well known, at temperatures below the Bose–Einstein condensation temperature, the lowest-energy state is not properly accounted for if one simply replaces sums by integrals, and it must be included explicitly. The statistical distribution function is discussed in Sec. 2.1, as is the single-particle density of states, which is a key ingredient in the calculations of thermodynamic properties. Calculations of the transition temperature and the fraction of particles in the condensate are described in Sec. 2.2. In Sec. 2.3 the semi-classical distribution function is introduced. From this we obtain the density profile and the velocity distribution of particles, and use these to determine the shape of an anisotropic cloud after free expansion. Thermodynamic properties of Bose gases are calculated as functions of the temperature in Sec. 2.4. Finally, Sec. 2.5 treats corrections to the transition temperature due to a finite particle number.