Collective dynamics of vortices in trapped Bose-Einstein condensates

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 $N_v$ vortices, $N_v$ 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.

Figure 1

Ground states of a Bose-Einstein condensate in rotating harmonic traps. Each plot is a density isosurface viewed along the rotation axis. The states (a)–(f), respectively accommodating 1, 2, 3, 7, 12, and 19 vortices, are being rotated at the respective angular frequencies ${\Omega }^{\prime }=\Omega /{\omega }_{\perp }=0.45$, 0.495, 0.497, 0.58, 0.69, and 0.79. The expectation values of the orbital angular momentum per particle $L^{\prime }=\langle L\rangle /\left(\hslash N\right)$ of these ground states are 0.97, 1.8, 2.3, 4.2, 6.2, and 9.2 respectively.

Figure 2

Instantaneous density isosurfaces of the condensate with seven vortices, perturbed by the different lowest vortex-displacement modes. The view is along the rotation axis. The excitation frequency ${\omega }^{\prime }$, expressed in units of the transverse trap frequency, of each perturbing mode is indicated in the upper right-hand corner of each frame. The discrete rotation symmetries of these modes are inherited from the sixfold symmetry of the triangular equilibrium vortex lattice. The central vortices in (f) and (g) gyrate in the opposite directions [see Supplemental Material [58]]. The number in the bottom of each frame corresponds to the mode label in Table .

Figure 3

Instantaneous density isosurface of a condensate excited in a first axial $n_z=1$ Kelvin-Tkachenko mode corresponding to the mode branch 1 of the seven-vortex system in Table . The orbital vortices are symmetrically tilted toward the central vortex. For the full dynamics of this mode we refer to the Supplemental Material [58].

Figure 4

Snapshots of the time evolution of the condensate with 19 vortices and perturbed by the rational mode 14 in Table . The times $\tau$ marked in the frames (a)–(f) are in units of $2\pi /{\omega }_q$.

Figure 5

Two different decompositions of the same collective mode of a 19-vortex array into collective modes of subarrays: (a) six three-vortex modes; (b) three seven-vortex modes.

Figure 6

Calculated low-lying excitation spectra of the respective ground states displayed in Fig. 1. The vertical intervals mark the energy range of the lowest vortex-displacement modes. The center-of-mass (Kohn) mode is singled out using a square marker. For every frame there exists a zero mode corresponding to the condensate of atoms. In (d) there is an additional zero mode related to the broken continuous rotation symmetry of the ground state.

Figure 7

Excitation frequencies ${\omega }_q$ of the lowest Tkachenko modes (filled circles) and the lowest common modes (squares) as functions of the external trap rotation frequency $\Omega$. The inset shows the energy per vortex associated with the rotation of the trap as a function of the trap rotation frequency.