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Physical Review A

Properties of a Bose-Einstein condensate in an anisotropic harmonic potential

Mark Edwards, R. J. Dodd, C. W. Clark, P. A. Ruprecht, Keith R. Burnett

Published Apr 1, 1996• 133 citations
Publisher

Abstract

We present results on the size, density, chemical potential, and lifetime of a Bose-Einstein condensate as a function of the number of condensate atoms for parameters appropriate to the time-averaged orbiting potential trap in which a Bose condensate of $^{87}\mathrm{Rb}$ atoms was recently formed [M. H. Anderson et al., Science 269, 198 (1995)]. These results were obtained by solving the time-independent nonlinear Schr\"odinger equation with anisotropic harmonic trapping potential. We also find vortex solutions for the trap and the critical rotation frequency required for their formation.

DOI
10.1103/physreva.53.r1950

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We present results on the size, density, chemical potential, and lifetime of a Bose-Einstein condensate as a function of the number of condensate atoms for parameters appropriate to the time-averaged orbiting potential trap in which a Bose condensate of 87Rb^{87}\mathrm{Rb}87Rb atoms was recently formed [M. H. Anderson et al., Science 269, 198 (1995)]. These results were obtained by solving the time-independent nonlinear Schr"odinger equation with anisotropic harmonic trapping potential. We also find vortex solutions for the trap and the critical rotation frequency required for their formation.
We study the vortex lattices in a Bose-Einstein condensate in a rotating anisotropic harmonic trap. We first investigate the single-particle wave functions obtained by the exact solution of the problem and give simple expressions for these wave functions in the small anisotropy limit. Depending on the strength of the interactions, a few or a large number of vortices can be formed. In the limit of many vortices, we calculate the density profile of the cloud and show that the vortex lattice stays triangular. We also find that the vortex lattice planes align themselves with the weak axis of the external potential. For a small number of vortices, we numerically solve the Gross-Pitaevskii equation and find vortex configurations that are very different from the vortex configurations in an axisymmetric rotating trap.
Vortex states of weakly interacting Bose-Einstein condensates confined in three-dimensional rotating harmonic traps are investigated numerically at zero temperature. The ground state in the rotating frame is obtained by propagating the Gross-Pitaevskii equation for the condensate in imaginary time. The total energies between states with and without a vortex are compared, yielding critical rotation frequencies that depend on the anisotropy of the trap and the number of atoms. Vortices displaced from the center of nonrotating traps are found to have long lifetimes for sufficiently large numbers of atoms. The relationship between vortex stability and bound core states is explored.
We found an analytical solution for the vortex structure in a rapidly rotating trapped Bose-Einstein condensate in the lowest Landau level approximation. This solution is exact in the limit of a large number of vortices and is obtained for the case of a condensate in a anisotropic harmonic potential. The solution describes as limiting cases both a triangle vortex lattice in the symmetric potential trap and a quasi-one-dimensional structure of vortex rows in an asymmetric case, when the rotation frequency is very close to the lower trapping potential frequency. The shape of the density profile is found to be close to the Thomas-Fermi inverted paraboloid form, except in the vicinity of edges of a condensate cloud.
We solve the Gross-Pitaevskii equations for a dilute atomic gas in a magnetic trap, modeled by an anisotropic harmonic potential. We evaluate the wave function and the energy of the Bose-Einstein condensate as a function of the particle number, both for positive and negative scattering length. The results for the transverse and the vertical size of the cloud of atoms, as well as for the kinetic and potential energy per particle, are compared with the predictions of approximated models. We also compare the aspect ratio of the velocity distribution with experimental estimates available for 87Rb^{87}\mathrm{Rb}87Rb. Vortex states are considered and the critical angular velocity for production of vortices is calculated. We show that the presence of vortices significantly increases the stability of the condensate in the case of attractive interactions. \textcopyright{} 1996 The American Physical Society.
The nucleation of vortices and the resulting structures of vortex arrays in dilute, trapped, zero-temperature Bose-Einstein condensates are investigated numerically. Vortices are generated by rotating a three-dimensional, anisotropic harmonic atom trap. The condensate ground state is obtained by propagating the Gross-Pitaevskii equation in imaginary time. Vortices first appear at a rotation frequency significantly larger than the critical frequency for vortex stabilization, consistent with a critical velocity mechanism for vortex nucleation. At higher frequencies, the structures of the vortex arrays are strongly influenced by trap geometry.
In the present work, motivated by numerous recent experimental developments we revisit the dynamics of a single vortex ring in anisotropic harmonic traps. At the theoretical level, we start from a general Lagrangian dynamically capturing the evolution of a vortex ring and not only consider its spectrum of linearized excitations, but also explore the full nonlinear dynamical evolution of the ring as a vortical filament. The theory predicts that the ring is stable for $1\ensuremath{\le}\ensuremath{\lambda}\ensuremath{\le}2$, where $\ensuremath{\lambda}={\ensuremath{\omega}}_{z}/{\ensuremath{\omega}}_{r}$ is the ratio of the trapping frequencies along the zzz and rrr axes, i.e., for spherical to slightly oblate condensates. We compare this prediction with direct numerical simulations of the full three-dimensional Gross-Pitaevskii equation (GPE) capturing the linearization spectrum of the ring for different values of the chemical potential and as a function of the anisotropy parameter $\ensuremath{\lambda}$. We identify this result as being only asymptotically valid as the chemical potential $\ensuremath{\mu}\ensuremath{\rightarrow}\ensuremath{\infty}$, revealing how the stability interval narrows and, in particular, its upper bound decreases for finite $\ensuremath{\mu}$. Finally, we compare at the dynamical level the results of the GPE with the ones effectively capturing the ring dynamics, revealing the unstable evolution for different values of $\ensuremath{\lambda}$, as well as the good agreement between the two.