Spontaneous Splitting of a Quadruply Charged Vortex

We studied the splitting instability of a quadruply charged vortex both experimentally and theoretically. The density defect, which is a signature of the vortex core, is experimentally observed to deform into a linear shape. The deformed defect is theoretically confirmed to be an array of four linearly aligned singly charged vortices. The array of vortices rotates and precesses simultaneously with different angular velocities. The initial state of the system is not rotationally symmetric, which enables spontaneous splitting without external perturbations.

Figure 1

Schematic plot of conservation of angular momentum. The curve shows the angular momentum $L_i$ as a function of the vortex position $r_i$. Initially a quadruply charged vortex is at radius $r_{\mathrm{init}}$. After the splitting, two of the vortices, represented by open and solid circles, move to $r_1^{\prime }$ and $r_2^{\prime }$, for example. In this process, the total angular momentum $L=\sum _{}^{}{L}_i$ must be conserved. Therefore, if $r_1^{\prime }<r_{\mathrm{init}}$, then $r_2^{\prime }>r_{\mathrm{init}}$.

Figure 2

Schematic plots of vortex dynamics after the splitting. (a) There are four vortices (circles) in the condensate. Sum of the velocity fields (arrows) of each of the vortices results in rotation of the array. (b) The gray area is the condensate. For each of the vortices in the condensate, an antivortex outside the condensate is considered as an image vortex. The original vortices (image vortices) are shown with small circles with counterclockwise (clockwise) arrows. The velocity field of the antivortices makes the original vortices precess as a whole (gray arrow).

Figure 3

(a)–(d) Axial absorption images of the BEC for the holding time of 4.0, 4.5, 5.5, and 6.5 ms. $\mathrm{TOF}=15\mathrm{ms}$. (e)–(i) 3D simulation using the time-dependent GP equation. Unit of length is $\mu \mathrm{m}$. (e)–(h) Density profile, axially integrated over a central region (from $-6.9$ to $6.9\mu \mathrm{m}$) at 3.5, 4.5, 5, and 5.5 ms. Dotted lines are operationally defined axes of the density defects. Crosses are the averages of positions of the vortices. (i) Density isosurface of the condensate at 3.5 ms for 5% of the maximum density. Contour lines at $z=0$, $\pm 10\mu \mathrm{m}$ are also plotted. Four split vortices are seen in the condensate.

Figure 4

(a) Quantum numbers $l=2$, 3, and 4 show the existence of a complex excitation that causes the splitting in a two-, three-, or fourfold symmetric pattern at the linear density $a{n}_z$, which is the same as that in Ref. 18. The arrow A (B) is the range of $a{n}_z$ within a condensate of $N=5\times {10}^4$ ($2\times {10}^5$) atoms. Splitting with a twofold symmetry is dominant when $N=5\times {10}^4$. (b) Splitting patterns of $l$-fold rotational symmetry. Circles represent singly charged vortices. Dotted lines are guides of symmetries.