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Bulletin of the American Physical Society

Kelvin-Tkachenko modes of vortex arrays in trapped Bose-Einstein condensates

Tapio Simula

Published Jun 16, 2011• 0 citations
OpenAlex

Abstract

Kelvin waves are helical single-vortex displacement waves propagating in the direction of the vortex axis. Tkachenko waves are transverse shear waves of many-vortex arrays. These two types of normal modes represent certain limits of generic three-dimensional vortex-waves. We have calculated the low-lying elementary excitation modes of fully three-dimensional vortex-arrays by solving the Bogoliubov-de Gennes equations.

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Kelvin waves are helical single-vortex displacement waves propagating in the direction of the vortex axis. Tkachenko waves are transverse shear waves of many-vortex arrays. These two types of normal modes represent certain limits of generic three-dimensional vortex-waves. We have calculated the low-lying elementary excitation modes of fully three-dimensional vortex-arrays by solving the Bogoliubov-de Gennes equations.
We have calculated collective-mode spectra for three-dimensional, rotating Bose-Einstein condensates in oblate harmonic traps using the microscopic Bogoliubov--de Gennes field theory. For condensates with Nv{N}_{v}Nv​ vortices, Nv{N}_{v}Nv​ Kelvin-Tkachenko-mode branches are obtained. The features of these modes are compared with those predicted by a classical point-vortex model. We have created movies to visualize the motion of the vortices corresponding to the Kelvin-Tkachenko waves.
The complete low-energy collective-excitation spectrum of vortex lattices is discussed for rotating Bose-Einstein condensates by solving the Bogoliubov-de Gennes equation, yielding, e.g., the Tkachenko mode recently observed at JILA. The totally symmetric subset of these modes includes the transverse shear, common longitudinal, and differential longitudinal modes. We also solve the time-dependent Gross-Pitaevskii equation to simulate the actual JILA experiment, obtaining the Tkachenko mode and identifying a pair of breathing modes. Combining both approaches allows one to unambiguously identify every observed mode.
We directly image Tkachenko waves in a vortex lattice in a dilute-gas Bose-Einstein condensate. The low (sub-Hz) resonant frequencies are a consequence of the small but nonvanishing elastic shear modulus of the vortex-filled superfluid. The frequencies are measured for rotation rates as high as 98% of the centrifugal limit for the harmonically confined gas. Agreement with a hydrodynamic theory worsens with increasing rotation rate, perhaps due to the increasing fraction of the volume displaced by the vortex cores. We also observe two low-lying m=0 longitudinal modes at about 20 times higher frequency.
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 establish the general equations of motion for the modes of a vortex lattice in a rapidly rotating Bose-Einstein condensate in three dimensions, taking into account the elastic energy of the lattice and the vortex line bending energy. As in two dimensions, the vortex lattice supports Tkachenko and gapped sound modes. In contrast, in three dimensions the Tkachenko mode frequency at long wavelengths becomes linear in the wave vector for any propagation direction out of the transverse plane. We compute the correlation functions of the vortex displacements and the superfluid order parameter for a homogeneous Bose gas of bounded extent in the axial direction. At zero temperature the vortex displacement correlations are convergent at large separation, but at finite temperatures, they grow with separation. The growth of the vortex displacements should lead to observable melting of vortex lattices at higher temperatures and somewhat lower particle number and faster rotation than in current experiments. At zero temperature a system of large extent in the axial direction maintains long range order-parameter correlations for large separation, but at finite temperatures the correlations decay with separation.