Dynamic stability of a doubly quantized vortex in a three-dimensional condensate

Abstract

The Bogoliubov equations are solved for a three-dimensional Bose-Einstein condensate containing a doubly quantized vortex, trapped in a harmonic potential. Complex frequencies, signifying dynamical instability, are found for certain ranges of parameter values. The existence of alternating windows of stability and instability, respectively, is explained qualitatively and quantitatively using variational calculus and direct numerical solutions. It is seen that the windows of stability disappear in the limit of a cigar-shaped condensate, which is consistent with recent experimental results on the lifetime of a doubly quantized vortex in that regime.͑r 2 + 2 z 2 ͒. ͑2͒The anisotropy of the trap is governed by the ratio between the axial and radial trapping strengths, so that when is larger than unity, the atomic cloud resembles a pancake and

Is a Doubly Quantized Vortex Dynamically Unstable in Uniform Superfluids?

“…[7][8][9] The MQV in trapped systems can be dynamically unstable, and split into vortices with smaller winding numbers according to the Bogoliubovde Gennes (BdG) analysis at zero temperature. [10][11][12][13][14][15][16][17] Dynamic instability may occur when the excitation modes have complex frequencies as a result of coupling or "mixing" between two modes with positive and negative excitation energies. The negative energy mode, called the core mode, is localized at the vortex core and decreases the angular momentum of the system by −l in the direction along the core.…”

“…The positive energy mode is a collective mode of the condensate. The instability depends on the atomic interaction strength in a complicated * hirotake@sci.osaka-cu.ac.jp manner, [10][11][12][13][14][15] obfuscating the underlying physics. Lundh and Nilsen made progress in understanding the splitting instability by employing a perturbation theory; however, no quantitative evaluation was carried out because of the complicated behavior of the imaginary part of excitation frequency (see Fig.…”

“…This is partly because long-time numerical simulations with high-spec computers are required for investigating the dynamic stability more precisely. According to the previous studies on trapped BECs, [10][11][12][13][14][15][16][17] it is not easy to answer this question, because the finite-size-effect is essential there. Although Aranson and Steinberg 18) concluded that the lifetime of an MQV may become infinite without a trap in their numerical simulation, its system-size dependence has not been clarified systematically.…”

“…A similar behavior can be seen in the numerical results for trapped BECs. [10][11][12][13][14][15] In these studies, the excitation frequency was parameterized by the interaction strength or particle number, which made the analysis of the problem complicated. Figure 1(b) shows that the peak values ofω I are proportional toR −1/2 forR 500, whileω I is asymptotic to a finite (7), presumed from the overall R-dependence ofω I .…”

“…15) We parameterize the chemical potential and the interaction coefficient with a perturbation parameter λ(≪ 1) as µ = µ 0 (1 + λ) and g → g λ ≡ g 0 (1 + λ), respectively. Then, the effective system size is represented asR…”