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Bulletin of the American Physical Society

The spectrum of low-lying states of an interacting Bose gas

Brett D. Esry, Chris H. Greene

Published May 1, 1996• 0 citations
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Abstract

The authors calculate the energy spectrum for N interacting bosons confined by a harmonic trap. Due to the rapidly growing number of states available, the authors are limited practically to relatively low-lying states. They consider parameters for the atoms and the trap similar to those from the recent Bose-Einstein condensation experiment of Anderson et al. and discuss the relevance of their results to the experiment.

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The authors calculate the energy spectrum for N interacting bosons confined by a harmonic trap. Due to the rapidly growing number of states available, the authors are limited practically to relatively low-lying states. They consider parameters for the atoms and the trap similar to those from the recent Bose-Einstein condensation experiment of Anderson et al. and discuss the relevance of their results to the experiment.
An analytic expression is presented for the elementary excitation spectrum of the Bose-Einstein condensate of a trapped boson system in the weakly interacting, low-density limit. Explicit analytic formulas for the elementary excitation spectrum are obtained for harmonic-oscillator traps. These formulas provide information about the behavior of the elementary excitation levels as a function of the number of atoms and their interaction strength for a given trap geometry. They also provide a low-density benchmark for results of fully numerical calculations.
We investigate the low-lying excitations of a weakly interacting, harmonically trapped Bose-Einstein condensed gas under rotation, in the limit where the angular mometum L of the system is much less than the number of the atoms N in the trap. We show that in the asymptotic limit $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}$ the excitation energy, measured from the energy of the lowest state, is given by ${27N}_{3}{(N}_{3}\ensuremath{-}{1)v}_{0}/68,$ where N3{N}_{3}N3​ is the number of octupole excitations, and v0{v}_{0}v0​ is the unit of the interaction energy.
We study analytically the ground-state stability of a Bose–Einstein condensate (BEC) confined in an harmonic trap with repulsive or attractive zero-range interaction by minimizing the energy functional of the system. In the case of repulsive interaction the BEC mean radius grows by increasing the number of bosons, instead in the case of attractive interaction the BEC mean radius decreases by increasing the number of bosons: to zero if the system is one-dimensional and to a constant minimum radius, with a maximum number of bosons, if the system is three-dimensional.