Core sizes and dynamical instabilities of giant vortices in dilute Bose-Einstein condensates

Motivated by a recent demonstration of cyclic addition of quantized vorticity into a Bose-Einstein condensate, the vortex pump, we study dynamical instabilities and core sizes of giant vortices. The core size is found to increase roughly as a square-root function of the quantum number of the vortex, whereas the strength of the dynamical instability either saturates to a fairly low value or increases extremely slowly for large quantum numbers. Our studies suggest that giant vortices of very high angular momenta may be achieved by gradually increasing the operation frequency of the vortex pump.

Figure 1

Numerical solution (solid curve) and the TF approximation (dashed curve) for $\mathrm{f̃}(\mathrm{r̃})$ as a function of $\mathrm{r̃}$ for $\mathrm{g̃}=1000$. The curves correspond to the winding numbers $\kappa =1,10,20,40$ as indicated.

Figure 2

Radius of the vortex core as a function of the winding number $\kappa$ for $\mathrm{g̃}=1$ and $\mathrm{g̃}={10}^4$. The solid curve shows the numerically obtained radius ${\mathrm{r̃}}_{\mathrm{c}}$, the dotted curve is the TF approximation ${\mathrm{r̃}}_{\mathrm{c}}^{\mathrm{TF}}$ given by Eqs. (13) and (25), and the dashed curve corresponds to the approximation ${\mathrm{\xi ̃}}_{\mathrm{c}}$, Eq. (27). For $\mathrm{g̃}={10}^4$, the solid curve lies on top of the dotted curve.

Figure 3

Radius of the vortex core as a function of the interaction strength $\mathrm{g̃}$. The solid curve shows the numerically obtained radius ${\mathrm{r̃}}_{\mathrm{c}}$, the dotted curve is the TF approximation ${\mathrm{r̃}}_{\mathrm{c}}^{\mathrm{TF}}$, and the dashed curve corresponds to the approximation${\mathrm{\xi ̃}}_{\mathrm{c}}$. The curves correspond to the winding numbers $\kappa =10,30,50,100$ as indicated.

Figure 4

Dimensionless dynamical instability parameter ${\Gamma }_{\kappa }$ of a giant vortex state as a function of its winding number $\kappa$. The dashed vertical lines separate regions where the angular momentum quantum numbers of the dominant complex-frequency modes, $l_{\mathrm{dom}}$, are different.

Figure 5

Values of the dimensionless interaction strength $\mathrm{g̃}$ at which the largest dynamical instabilities occur. The angular momentum quantum number of the dominant complex-frequency mode, $l_{\mathrm{dom}}$, is also indicated.

Figure 6

Dots show the angular momentum quantum numbers $l_{\mathrm{dom}}^0$ yielding the largest dynamical instability in the noninteracting limit. The triangles denote the critical interaction strength, ${\mathrm{g̃}}_{\mathrm{cr}}^0$, above which the noninteracting result for $l_{\mathrm{dom}}$ fails. The dashed line is a guide to the eye.