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arXiv:cond-mat/0104522 · Physical Review Letters

Bose-Einstein Condensates with Large Number of Vortices

Tin-Lun Ho

Published Jul 23, 2001• 268 citations
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Abstract

We show that as the number of vortices in a three dimensional Bose-Einstein condensate increases, the system reaches a "quantum Hall" regime where the density profile is a Gaussian in the xy plane and an inverted parabolic profile along z. The angular momentum of the system increases as the vortex lattice shrinks. However, Coriolis force prevents the unit cell of the vortex lattice from shrinking beyond a minimum size. Although the recent MIT experiment is not exactly in the quantum Hall regime, it is close enough for the present results to be used as a guide. The quantum Hall regime can be easily reached by moderate changes of the current experimental parameters.

DOI
10.1103/physrevlett.87.060403

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We consider the condensate wave function of a rapidly rotating two-component Bose gas with an equal number of particles in each component. If the interactions between like and unlike species are very similar (as occurs for two hyperfine states of (87)Rb or (23)Na) we find that the two components contain identical rectangular vortex lattices, where the unit cell has an aspect ratio of sqrt[3], and one lattice is displaced to the center of the unit cell of the other. Our results are based on an exact evaluation of the vortex lattice energy in the large angular momentum (or quantum Hall) regime.
The structure of vortices in Bose-Einstein condensed atomic gases is studied taking into account many-body correlation effects. It is shown that for excited vortices the particle density in the vortex core increases as the angular momentum of the system increases. The core density can increase by several times with only a few percent change in the angular momentum. This result provides an explanation for the observations in which the measured angular momentum is higher than the estimation based on counting the number of vortices and the visibility of the vortex cores is simultaneously reduced. The calculated density profiles for the excited vortices are in good agreement with experimental measurements of a single vortex.
When vortices are displaced in Bose-Einstein condensates (BECs), the Magnus force gives the system a momentum transverse in the direction to the displacement. We show that BECs in long channels with vortices exhibit a quantization of the current response with respect to the spatial vortex distribution. The quantization originates from the well-known topological property of the phase around a vortex; it is an integer multiple of $2\ensuremath{\pi}$. In a way similar to that of the integer quantum Hall effect, the current along the channel is related to this topological phase and can be extracted from two experimentally measurable quantities: the total momentum of the BEC and the spatial distribution. The quantization is in units of m/2hm/2hm/2h, where mmm is the mass of the atoms and hhh is Planck's constant. We derive an exact vortex momentum-displacement relation for BECs in long channels under general circumstances. Our results present the possibility that the configuration described here can be used as a novel way of measuring the mass of the atoms in the BEC using a topological invariant of the system. If an accurate determination of the plateaus are experimentally possible, this gives the possibility of a topological quantum mass standard and precise determination of the fine structure constant.
We present an analytical solution for the vortex lattice in a rapidly rotating trapped Bose-Einstein condensate in the lowest Landau level and discuss deviations from the Thomas-Fermi density profile. This solution is exact in the limit of a large number of vortices and is obtained for the cases of circularly symmetric and narrow channel geometries. The latter is realized when the trapping frequencies in the plane perpendicular to the rotation axis are different from each other and the rotation frequency is equal to the smallest of them. This leads to the cancellation of the trapping potential in the direction of the weaker confinement and makes the system infinitely elongated in this direction. For this case we calculate the phase diagram as a function of the interaction strength and rotation frequency and identify the order of quantum phase transitions between the states with a different number of vortex rows.