Splitting instability of a multiply charged vortex in a Bose-Einstein condensate

We consider the splitting mechanism of a multiply charged vortex into singly charged vortices in a Bose-Einstein condensate confined in a harmonic potential at zero temperature. The Bogoliubov equations support unstable modes with complex eigenfrequencies (CE modes), which cause the splitting instability without the influence of thermal atoms. The investigation of the excitation spectra shows that the negative-energy (NE) mode plays an important role in the appearance of the CE modes. The configuration of vortices in splitting is determined by the angular momentum of the associated NE mode. This structure has also been confirmed by the numerical simulation of the time-dependent Gross-Pitaevskii equation.

Figure 1

Excitation spectra of a quadruply charged vortex state at the noninteracting limit. The symbols $\circ$ and $\times$ represent ${\mathcal{E}}_{ln}^{\left(\mathsf{u}\right)}$ and ${\mathcal{E}}_{ln}^{\left(\mathsf{v}\right)}$, respectively.

Figure 2

Imaginary parts of eigenfrequencies for (a) $l=5,6$ and (b) $l=2,3,4$ as functions of the interaction strength $\eta$. The result for $l=6$ is magnified 100 times. The CE mode at $\mathsf{P}$ differs from others with $l=4$ as to the associated NE mode (see text).

Figure 3

Excitation spectra for $l=2$. The solid lines correspond to excitation energies with (a) positive angular momenta and (b) negative angular momenta. In both figures, the negative-energy eigenvalues and the real part of the complex eigenfrequency are plotted by broken and dotted lines, respectively.

Figure 4

Excitation spectra for $l=4$. The description of lines is in Fig. 3a. There are two NE modes and both turn into CE excitations. The point $\mathsf{P}$ corresponds to $\mathsf{P}$ in Fig. 2 (see text).

Figure 5

Time development of elementary excitations. (a) Initial changes. The broken lines correspond to the amplitude of the CE modes (plus and minus modes), which coincide with the solid lines corresponding to the functions $\exp \left(\pm \mathrm{Im}{\omega }_{c}t\right)$. The amplitudes of other modes plotted with the dotted lines are constant. (b) Further time development. The plus and minus modes (the solid and broken lines, respectively) interact with each other and oscillate. The amplitudes of other modes (dotted lines) are also affected by the oscillation and increase more and more.

Figure 6

Contour plots of density profiles at the points $\mathsf{A}$, $\mathsf{B}$, and $\mathsf{C}$ in Fig. 5. (a) A multiply charged vortex just after the distortion. (b), (c) Vortices arranged in threefold symmetry. Vortex cores move outward (b) and into the center (c) in the oscillation.

Figure 7

Images of splitting patterns at $\eta =340$ after trap distortion with (a) $l_{\mathsf{v}}=2$, (b) $l_{\mathsf{v}}=3$, and (c) $l_{\mathsf{v}}=4$. The top panels are density profiles and the bottom panels are the corresponding phase profiles. Three CE modes are selectively excited by choosing $l_{\mathsf{v}}$.