Perturbative behavior of a vortex in a trapped Bose-Einstein condensate

We derive a set of equations that describes the shape and behavior of a single perturbed vortex line in a Bose-Einstein condensate. Through the use of a matched asymptotic expansion and a unique coordinate transform, a relation for a vortex's velocity, anywhere along the line, is found in terms of the trapping, rotation, and distortion of the line at that location. This relation is then used to find a set of differential equations that give the line's specific shape and motion. This work extends a previous similar derivation by Svidzinsky and Fetter [Phys. Rev. A 62, 063617 (2000)], and enables a comparison with recent numerical results.

Figure 1

Depiction of the local coordinate vectors $\widehat{\mathbf{t}}$, $\widehat{\mathbf{n}}$, and $\widehat{\mathbf{b}}$ of the curve $\mathbf{c}\left(T\right)=\{T^2/2,T^2/2,T\}$, with the coordinate vectors plotted for $T=0$ (red, lower set) and $T=1$ (green, upper set).

Figure 2

Plot of the pseudoparametrization used (dimensionless) [Eq. (10) with $a=1$, $k=2$, and $b=5$].

Figure 3

Plots of the dispersion relation, given by Eq. (85), for different particle number ($N$) for a condensate of ${}^{87}$Rb atoms, with trapping frequencies ${\omega }_z=2\pi \times 11.8\text{Hz}$ and ${\omega }_{\perp }=2\pi \times 98.5\text{Hz}$, and vortex core radius $r_c=0.13\mu$m. The upper line in the plot corresponds to $N=5\times {10}^5$, the middle line corresponds to $N=5\times {10}^4$, and lower line corresponds to $N={10}^4$.