Dynamics of vortices in weakly interacting Bose-Einstein condensates

We study the dynamics of vortices in ideal and weakly interacting Bose-Einstein condensates using a Ritz minimization method to solve the two-dimensional Gross-Pitaevskii equation. For different initial vortex configurations we calculate the trajectories of the vortices. We find conditions under which a vortex-antivortex pair annihilates and is created again. For the case of three vortices we show that at certain times two additional vortices may be created, which move through the condensate and annihilate each other again. For a noninteracting condensate this process is periodic, whereas for small interactions the essential features persist, but the periodicity is lost. The results are compared to exact numerical solutions of the Gross-Pitaevskii equation confirming our analytical findings.

Figure 1

(a) Trajectories for a vortex dipole for different initial positions $(\pm x_0,0)$. From inside to outside $x_0=0.4$, 0.6, 0.8, 1, 1.2. For $x_0=0.4$, 0.6, the two vortices annihilate each other at certain times and reappear again, indicated by the closed lines. For all other cases shown, there are always two vortices present apart from times where $t=(2n+1)\pi ∕2$, with $n$ an integer number. (b) Trajectories for the more general initial condition ${\mathbf{r}}_1=(0.9,0.1)$, ${\mathbf{r}}_2=(-1.1,0)$.

Figure 2

(Color online) Comparison of vortex trajectories for different initial states for a vortex dipole with initial $x_0=1$ [(a) and (b)] and $x_0=0.5$ [(c) and (d)]. For $x<0$ the trajectories for a vortex dipole with initial $p_q$ for $\beta =1$ is shown, whereas for $x>0$ the trajectories with initial $p_q$ for $\beta =0$ [(a) and (c)] and $\beta =2$ [(b) and (d)] are plotted. All trajectories are mirror symmetric to the axis where $x=0$ (dashed line). The vortices were evolved for a time interval $[0,20]$ and a condensate with $\beta =1$.

Figure 3

(Color online) (a) Precession frequency of a single vortex versus the interaction strength $\beta$. Stars show ${\omega }_p$ for a different initial distance of the vortex from the center, namely (from bottom to top) $x_0=0.1$, 0.5, 1.0, 1.5, 2.0. The circles show the analytical result given in Eq. (13), lines are guides to the eye. (b) The coefficients $c\left(x_0\right)$ for different $x_0$, see text.

Figure 4

Trajectories of a vortex pair deduced from the analytical formula Eq. (14) for $\beta =2$, $x_0=1.2$, and a time interval of $t∊[0,20]$. The trajectories are no longer exact circular lines, indicating that both vortices influence each other.

Figure 5

(Color online) Trajectories of the two vortices in a vortex dipole for (a) $x_0=0.6$, (b) $x_0=0.8$, (c) $x_0=1$, and (d) $x_0=1.2$. Analytical results calculated with Eq. (15) are only shown for $x<0$, whereas numerical results are only shown for $x>0$. Both results are mirror symmetric with respect to the axis where $x=0$ (dashed line). The interaction is chosen as $\beta =1$, and the time interval for which the trajectories are shown is given by $0\le t\le 20$.

Figure 6

(Color online) (a) Number of vortices $N$ present in the BEC vs time, with initially $x_0=1.5$. (b) Average number of vortices $⟨N⟩$ present in the BEC during the time interval $[0,20]$ for different initial $x_0$. The solid line shows the results derived from the analytical expression, Eq. (16), whereas the pluses show numerical results. In all cases we have chosen $\beta =1$, apart from the dashed line, which shows the results for $\beta =0$.

Figure 7

(Color online) (a) Trajectories of the two additional vortices (black lines) for the initial state of a vortex tripole with $x_0=\sqrt{2}$, details see text. The three stationary vortices are indicated by plus signs. (b) Density plot for $t=0.2\pi$. Plus signs indicate the position of the five vortices.