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Chinese Physics Letters

Roles of Collective Excitations in the Anomalous Fluctuations of One-Dimensional Interacting Bose-Condensed Gases

Shujuan Liu, Xiong Hong-Wei, Xu Zhi-Jun, Huang Lin

Published Sep 24, 2003• 3 citations
Publisher

Abstract

The particle-number fluctuations due to collective excitations are calculated for a one-dimensional interacting Bose-condensed gas confined in a harmonic potential. By using the Thomas-Fermi approximation, the behaviour of the anomalous fluctuations is found to be ⟨δ 2 N 0 ⟩~ N 10/3 /(ln N ) 2 with N being the total number of particles.

DOI
10.1088/0256-307x/20/10/305

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The particle-number fluctuations due to collective excitations are calculated for a one-dimensional interacting Bose-condensed gas confined in a harmonic potential. By using the Thomas-Fermi approximation, the behaviour of the anomalous fluctuations is found to be ⟨δ 2 N 0 ⟩~ N 10/3 /(ln N ) 2 with N being the total number of particles.
The particle-number fluctuations originated from collective excitations are investigated for a three-dimensional, repulsively interacting Bose-Einstein condensate (BEC) confined in a harmonic trap. The contribution due to the quantum depletion of the condensate is calculated and the explicit expression of the coefficient in the formulas denoting the particle-number fluctuations is given. The results show that the particle-number fluctuations of the condensate follow the law $〈\ensuremath{\delta}{N}_{0}^{2}〉\ensuremath{\sim}{N}^{22/15}$ and the fluctuations vanish when temperature approaches the BEC critical temperature.
We investigate the particle-number fluctuations due to the collective excitations created in a two-dimensional (2D) and a quasi-2D Bose-Einstein condensates (BECs) at low temperature. We find that the fluctuations display an anomalous behavior, i.e., for the 2D BEC they are proportional to N2,{N}^{2},N2, where N is the total number of particles. For the quasi-2D BEC, the particle-number fluctuations are proportional not only to N2{N}^{2}N2 but also to the square root of the trapping frequency in the strongly confined direction.
Following the idea of the density-functional approach, we develop a generalized Bogoliubov theory of an interacting Bose gas confined in a one-dimensional harmonic trap by using a local chemical potential, calculated with the Lieb-Liniger exact solution, as the exchange energy. At zero temperature, we use the theory to describe collective modes of a finite-particle system in all interaction regimes from the ideal gas limit to the mean-field Thomas-Fermi regime and to the strongly interacting Tonks-Girardeau regime. At finite temperature, we investigate the temperature dependence of collective modes in the weak-coupling regime by means of a Hartree-Fock-Bogoliubov theory with Popov approximation. By emphasizing the effects of finite particle number and nonzero temperature on collective-mode frequencies, we compare our results with a recent experimental measurement [E. Haller et al., Science 325, 1224 (2009)] and some previous theoretical predictions. We show that the experimental data are still not fully explained within current theoretical framework.
We find that the fluctuations of the condensate in a weakly interacting Bose gas confined in a box of volume VVV follow the law $〈\ensuremath{\delta}{N}_{0}^{2}〉\ensuremath{\sim}{V}^{4/3}$. This anomalous behavior arises from the occurrence of infrared divergencies due to phonon excitations and holds also for strongly correlated Bose superfluids. The analysis is extended to an interacting Bose gas confined in a harmonic trap where the fluctuations are found to exhibit a similar anomaly.