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

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.

Figure 1

Energy levels in two dimensions for $m=2$ in dimensionless units. The ∗ represent the analytical approximation to the crossing of the core mode with the trap modes shown in Eq. (20).

Figure 2

Effective potentials entering the Bogoliubov equation for a doubly quantized vortex, in dimensionless units. Full line represents the diagonal potential $V_d$, the dashed line represents the diagonal potential with the addition of a centrifugal term, $V_d+4∕r^2$, and the dotted line is the off-diagonal potential $V_c$. The potentials are defined in Eq. (10).

Figure 3

Unstable regions for a doubly quantized vortex in the parameter space of trap anisotropy $\lambda$ and shifted chemical potential ${\mu }_s=\mu -\lambda ∕2-3$, in dimensionless units. The shading indicates the largest of the imaginary parts of the Bogoliubov eigenvalues. White is zero which means that the condensate is stable.

Figure 4

Real parts of the Bogoliubov eigenenergies as functions of the shifted chemical potential ${\mu }_s=\mu -\lambda ∕2-3$ for the trap anisotropy $\lambda =1.2$, i.e., an almost isotropic trapping potential. Dimensionless units are employed.

Figure 5

Real part of the Bogoliubov energy spectrum for the case $\lambda =0.2$, i.e., in the limit of a cigar-shaped condensate as a function of ${\mu }_s=\mu -\lambda ∕2-3$, in dimensionless units.

Figure 6

Imaginary parts of the Bogoliubov eigenvalues, $\mathrm{Im}\left({\omega }_i\right)$, in the limit of weak interaction and strongly cigar-shaped geometry, calculated using the one-dimensional model defined in Eq. (26) as a function of ${\mu }_s=\mu -\lambda ∕2-3$. Dimensionless units are employed.

Figure 7

The maximal imaginary part of the complex eigenvalues for a condensate with a doubly quantized vortex in the cigar-shaped limit with trap anisotropy $\lambda =0.2$, as a function of dimensionless two-dimensional coupling strength. The two large peaks are termed 2D instabilities since they do not depend on the coordinate along the vortex line, while the unstable modes connected with the smaller peaks have nodes in the axial direction.

Figure 8

Time scale for splitting of a doubly quantized vortex in a cigar-shaped condensate with anisotropy $\lambda =0.2$, defined as the reciprocal of the largest imaginary part of the Bogoliubov eigenvalues. Units on the axes are chosen to correspond with the experiment of Ref. 3.

Figure 9

Real part of the energy spectrum for the two-dimensional case in an anharmonic potential with anharmonicity parameter $\alpha =0.92$, in dimensionless units.

Figure 10

Imaginary part of the energy spectrum in dimensionless units for the two-dimensional case in an anharmonic potential with $\alpha =0.92$.

Figure 11

Dynamical stability regions for a doubly quantized vortex in a two-dimensional anharmonic trap, as a function of coupling strength $C$ and anharmonicity $\alpha$, in dimensionless units.