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arXiv:cond-mat/0212290 · Journal of Physics B Atomic Molecular and Optical Physics

Gyroscopic motion of superfluid-trapped atomic condensates

Halvor M ll Nilsen, Dermot McPeake, J F McCann

Published Apr 8, 2003• 5 citations
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Abstract

The gyroscopic motion of a trapped Bose gas containing a vortex is studied. We model the system as a classical top, as a superposition of coherent hydrodynamic states, by solution of the Bogoliubov equations, and by integration of the time-dependent Gross–Pitaevskii equation. The frequency spectrum of Bogoliubov excitations, including quantum frequency shifts, is calculated and the quantal precession frequency is found to be consistent with experimental results, although a small discrepancy does exist. The superfluid precession is found to be well described by the classical and hydrodynamic models. However, the frequency shifts and helical oscillations associated with vortex bending and twisting require a quantal treatment. In gyroscopic precession, the Kelvin modes m = ±1 are the dominant features giving a vortex kink or bend, while the m = +2 mode is found to be the dominant Kelvin wave associated with vortex twisting.

DOI
10.1088/0953-4075/36/9/303

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The gyroscopic motion of a trapped Bose gas containing a vortex is studied. We model the system as a classical top, as a superposition of coherent hydrodynamic states, by solution of the Bogoliubov equations, and by integration of the time-dependent Gross–Pitaevskii equation. The frequency spectrum of Bogoliubov excitations, including quantum frequency shifts, is calculated and the quantal precession frequency is found to be consistent with experimental results, although a small discrepancy does exist. The superfluid precession is found to be well described by the classical and hydrodynamic models. However, the frequency shifts and helical oscillations associated with vortex bending and twisting require a quantal treatment. In gyroscopic precession, the Kelvin modes m = ±1 are the dominant features giving a vortex kink or bend, while the m = +2 mode is found to be the dominant Kelvin wave associated with vortex twisting.
The gyroscopic motion of a trapped Bose gas containing a vortex is studied. We model the system as a classical top, as a superposition of coherent hydrodynamic states, by solution of the Bogoliubov equations, and by integration of the time-dependent Gross-Pitaevskii equation. The frequency spectrum of Bogoliubov excitations, including quantum frequency shifts, is calculated and the quantal precession frequency is found to be consistent with experimental results, though a small discrepancy exists. The superfluid precession is found to be well described by the classical and hydrodynamic models. However the frequency shifts and helical oscillations associated with vortex bending and twisting require a quantal treatment. In gyroscopic precession, the vortex excitation modes m = ±1 are the dominant features giving a vortex kink or bend, while the m = +2 is found to be the dominant Kelvin wave associated with vortex twisting. Superfluid atomic gyroscopes 2 1.
A hydrodynamic description is used to study the normal modes of a vortex in a zero-temperature Bose-Einstein condensate. In the Thomas-Fermi limit, the circulating superfluid velocity far from the vortex core provides a small perturbation that splits the originally degenerate normal modes of a vortex-free condensate. The relative frequency shifts are small in all cases considered (they vanish for the lowest dipole mode with ∣m∣=1|m|=1∣m∣=1), suggesting that the vortex is stable. The Bogoliubov equations serve to verify the existence of helical waves, similar to those of a vortex line in an unbounded weakly interacting Bose gas. In the large-condensate (small-core) limit, the condensate wave function reduces to that of a straight vortex in an unbounded condensate; the corresponding Bogoliubov equations have no bound-state solutions that are uniform along the symmetry axis and that decay exponentially far from the vortex core.
We have observed a three-dimensional gyroscopic effect associated with a vortex in a dilute Bose-Einstein condensed gas. A condensate with a vortex possesses a single quantum of circulation, and this causes the plane of oscillation of the scissors mode to precess around the vortex line. We have measured the precession rate of the scissors oscillation. From this we deduced the angular momentum associated with the vortex line and found a value close to Planck's over 2pi per particle, as predicted for a superfluid.
We derive a governing equation for a Kelvin wave supported on a vortex line in a Bose-Einstein condensate, in a rotating cylindrically symmetric parabolic trap, where it is assumed that the shape of the vortex line is dominated by the properties of the condensate at the center of the trap. From this solution the Kelvin wave dispersion relation is determined. In the limit of an oblate trap and in the absence of longitudinal trapping our results are consistent with previous work. We show that the derived Kelvin wave dispersion in the general case is in quantitative agreement with numerical calculations of the Bogoliubov spectrum.