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Physical Review A

Bose-Einstein condensation in an atomic gas with attractive interactions

Hualin Shi, Wei‐Mou Zheng

Published Apr 1, 1997• 31 citations
Publisher

Abstract

The behavior of a trapped atomic gas undergoing a Bose-Einstein condensation is strongly influenced by the attractive interaction between particles. The role played by such an interaction is analyzed by means of a variation method. The critical number of atoms for a metastable state to exist is determined. The aspect ratio of condensed bosons in an anisotropic trap, the loss rate of atoms, and the excitation spectrum are studied. The vortex state is discussed.

DOI
10.1103/physreva.55.2930

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The behavior of a trapped atomic gas undergoing a Bose-Einstein condensation is strongly influenced by the attractive interaction between particles. The role played by such an interaction is analyzed by means of a variation method. The critical number of atoms for a metastable state to exist is determined. The aspect ratio of condensed bosons in an anisotropic trap, the loss rate of atoms, and the excitation spectrum are studied. The vortex state is discussed.
The properties and stability of a trapped Bose-Einstein condensate are strongly influenced by attractive interactions between the particles. We describe the spatial distribution, stability, and collisional loss rates for a weakly interacting gas in the mean-field limit. We show how the condensate contracts and becomes unstable as the number of condensate atoms increases. We further show how the number of atoms is limited by the collisional loss rates associated with the contraction of the condensate; this loss is in addition to the particle ejection decay indentified by Kagan et al. \textcopyright{} 1996 The American Physical Society.
We discuss Bose-Einstein condensation in a trapped atomic gas and analyze how the sign of the scattering length a and the ratio \ensuremath{\eta} of the interaction between particles to the level spacing in the trap influence the behavior of the condensate wave function ${\ensuremath{\psi}}_{0}$. We find that for a<0a<0a<0 and $\ensuremath{\eta}\ensuremath{\ll}1$ it is possible to form a metastable Bose condensate, with a long characteristic lifetime with respect to contraction and transitions of particles to excited trap states. For $\ensuremath{\eta}\ensuremath{\gg}1$ a negative scattering length prevents the formation of the condensate. If a>0a>0a>0, then an increase of density is accompanied by the evolution of ${\ensuremath{\psi}}_{0}$ to a comparatively wide quasihomogeneous distribution.
We discuss various properties of the ground state of a Bose-condensed dilute gas confined by an external potential. We devote particular attention to the role played by the interaction in determining the kinetic energy of the system and the aspect ratio of the velocity distribution. The structure of the wave function near the classical turning point is discussed and the drawback of the Thomas-Fermi approximation is explicitly pointed out. We consider also states with quantized vorticity and calculate the critical angular velocity for the production of vortices. The presence of vortex states is found to increases the stability of the condensate in the case of attractive interactions.
Among the species that have Bose-condensed, lithium 7 and rubidium 85 form unique condensates in that atoms interact attractively and that the energy barrier that stabilizes the gaseous Bose-Einstein condensates originates from the zero-point motion of atoms. This paper reviews some unique features of these new states of matter that are distinguished from those of other species with repulsive interactions. These include the density instability, macroscopic quantum tunneling, intermittent implosion, shellstructure formation, and partial quantization of circulation. 1. METASTABILITY AND DENSITY INSTABILITY OF THE CONDENSATE Atoms with attractive interactions are believed not to undergo Bose-Einstein condensation in the uniform 3D system. It is therefore appropriate to begin by discussing why attractive Bose-Einstein condensates (BEes) have been realized in trapped systems [1, 2]. A trapped BEe has three characteristic energies, namely, the kinetic, potential and interaction energies. The kinetic energy arises from the uncertainty principle, and is proportional to R-2 where R is the linear dimension of BEe. The trapping potential is usually harmonic, and therefore the potential energy is proportional to R2. The mean-field interaction energy is proportional to NoaR-3, where No and a denote the number of condensate atoms and the s-wave scattering length, respectively. The interaction energy and hence a are negative for attractive systems. The sum of these three energies F = No(AR-2 + BR2 CNolaIR-3), where A, B, Care positive constants, gives the total energy of the system which has a metastable minimum if No is smaller than a certain critical value Nc.lt is at this local minimum where a metastable BEe is believed to be formed. If No is larger than Nc' there is no metastable minimum and the condensate is bound to collapse to the R = 0 state, which is a metallic phase. It is clear from the above argument that the zero-point energy due to confinement serves as a kinetic obstacle against collapse, allowing the metastable BEe to be formed. For the case of attractive interactions, the lowest-energy collective mode is the
We apply controlled perturbation theory to calculate the spectrum of the Gross-Pitaevskii equation for a system composed of attractive bosons confined in an anisotropic harmonic trap. The energy spectrum is calculated as a function of the coupling parameters for traps going from cigar to pancake shapes. The critical number of particles that ensures real values for the energy spectrum is obtained as a function of the potential anisotropic parameter, showing strong dependence of the critical number on the anisotropy of the trap. For a number of particles above the critical value the metastability of the system is characterized through the calculation of the condensate lifetime, using the imaginary part of the energy values. The obtained results are relevant for experiments where highly anisotropic traps are considered.