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arXiv:cond-mat/0211396 · Physical Review Letters

Beliaev Damping and Kelvin Mode Spectroscopy of a Bose-Einstein Condensate in the Presence of a Vortex Line

Takeshi Mizushima, Masanori Ichioka, K. Machida

Published May 7, 2003• 48 citations
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Abstract

It is demonstrated theoretically that the counter-rotating quadrupole mode in a vortex of Bose-Einstein condensates can decay into a pair of Kelvin modes via the Beliaev process. We calculate the spectral weight of a density-response function within the Bogoliubov framework, taking account of both Beliaev and Landau processes. Good agreement with experiment on 87 Rb by Bretin et al. [Phys. Rev. Lett., 100403 (2003)]] allows us to unambiguously identify the decayed mode as the Kelvin wave propagating along a vortex line.

DOI
10.1103/physrevlett.90.180401

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It is demonstrated theoretically that the counter-rotating quadrupole mode in a vortex of Bose-Einstein condensates can decay into a pair of Kelvin modes via the Beliaev process. We calculate the spectral weight of a density-response function within the Bogoliubov framework, taking account of both Beliaev and Landau processes. Good agreement with experiment on 87 Rb by Bretin et al. [Phys. Rev. Lett., 100403 (2003)]] allows us to unambiguously identify the decayed mode as the Kelvin wave propagating along a vortex line.
We investigate the damping mechanism of two transverse quadrupole oscillations in the single-vortex state of Bose-Einstein condensates using the linear response theory. It is demonstrated theoretically that the Kelvin modes as a propagation wave along a vortex line can be excited via Beliaev damping processes for only the counter-rotating quadrupole mode. We calculate the spectral weight of the density-response function within the Bogoliubov-Popov theory in a cylindrical system taking into account both Beliaev and Landau processes. Furthermore, we extend these considerations to the full 3D cigar-shaped system, which enables us to discuss the effects of the longitudinal confinement on the damping processes. These results are compared with the recent experiment on 87 Rb by Bretin et al. [Phys. Rev. Lett. 90, 100403 (2003)].
We investigate low-energy collective excitations in Bose-Einstein condensates (BECs) with a single-vortex line or many vortices by using the microscopic theory based on the Bogoliubov-Popov formalism. The paper is divided into the following two section: (i) We first study the damping process of the quadrupole modes in a three-dimensional cigar-shaped BEC. The selective excitation of such the modes occurs during the rotation of the condensate in the opposite or same direction relative to a vortex velocity field. The dramatic change of the damping process of these modes is explained by the linear response theory. It is proposed that Kelvin modes play a leading part in the decay process of the quadrupole excitation. These results are quantitatively compared with the recent experiment on 87 Rb by Bretin et al. [Phys. Rev. Lett. 90, 100403 (2003)]. (ii) In addition, the complete spectrum of the collective excitations in vortex-lattice states is obtained by solving the Bogoliubov-de Gennes equation in a two-dimensional disk-shaped BEC. We also discuss the frequency shift of totally symmetric and twofold symmetric Tkachenko modes.
We have theoretically investigated Kelvin waves of quantized vortex lines in trapped Bose-Einstein condensates. Counterrotating perturbation induces an elliptical instability to the initially straight vortex line, driven by a parametric resonance between a quadrupole mode and a pair of Kelvin modes of opposite momenta. Subsequently, Kelvin waves rapidly decay to longer wavelengths emitting sound waves in the process. We present a modified Kelvin wave dispersion relation for trapped superfluids and propose a simple method to excite Kelvin waves of specific wave number.
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.
We study experimentally the line of a single vortex in a rotating prolate Bose-Einstein condensate confined in a harmonic potential. In agreement with predictions, we find that the vortex line is in most cases curved at the ends. We also present measurements of the quadrupole oscillation of the condensate in presence of a single vortex. A theoretical treatment to account for the short time and long time behaviors of these modes is developed. Finally, we use these measurements to infer the angular momentum per particle and relate it to the shape of the vortex line.