Vortex dipole in a trapped two-dimensional Bose-Einstein condensate

We study the conservative dynamics and stationary configurations of a vortex-antivortex pair in a harmonically trapped two-dimensional Bose-Einstein condensate. We establish the conceptual framework for understanding the stationary states and the topological defect trajectories, through considerations of different mechanisms of vortex motion and the bifurcation of solitonlike stationary solutions. Our insights are based on Lagrangian-based variational calculations, numerical solutions of both the time-dependent and time-independent Gross-Pitaevskii equations, and exact solutions for the noninteracting case.

Figure 1

(Color online) Initial pair velocities versus dipole separation. Thick-solid: Gaussian ansatz $g=0.1$. Dashed: a single vortex, same ansatz, same $g=0.1$. Dotted line: noninteracting condensate $\frac{1}{2}{x}_d-2/x_d$. Inset: Initial velocities versus $x_d$, Solid, dashed, and dotted-dashed lines are from Thomas-Fermi ansatz, with $g=1$, 10, and 100. Dotted line is from Gaussian ansatz $g=1$. All quantities are plotted in trap units.

Figure 2

(Color online) Half-distance between pair for stationary vortex dipole configuration $x_s$ as a function of $g$. Solid line: from Thomas-Fermi ansatz; dashed line from Gaussian ansatz; filled circles from time-independent GPE. The dotted line shows the Thomas-Fermi boundary of condensate. Lengths are plotted in trap units.

Figure 3

(Color online) Vortex and antivortex trajectories, calculated using the variational formalism with wave function (A3). The four pictures show trajectories resulting from initial positions $x_1\left(0\right)=0.3$, 0.6, 0.9, 1.2. Full and dashed lines show $g=6$ and $g=40$, respectively. Lengths are plotted in trap units.

Figure 4

(Color online) Vortex position $\left(x_1,y_1\right)$ as a function of time. Left panel: variational calculation with Thomas-Fermi ansatz $\left(g=6\right)$. Right panel: noninteracting bosons solved in terms of harmonic oscillator wave functions. All quantities are plotted in trap units.

Figure 5

(Color online) The stationary soliton solution bifurcates at $g\approx 18$, producing a stationary vortex dipole solution. Dashed (full) line shows energy of a stationary soliton (stationary vortex dipole).

Figure 6

Density profiles of stationary states: (a) soliton at $g=11$; (b) vortex dipole at $g=25$; (c) vortex dipole at $g=60$.

Figure 7

(Color online) Vortex pair trajectories from numerical solution of time-dependent GPE, $g=150$. Starting positions are $x_1\left(0\right)=0.8$, 1.1, 1.3, and 1.5. Successive vortex and antivortex positions have been joined by straight lines. The condensate radius (Thomas-Fermi) is $R_{\text{TF}}\simeq 3.72$, not visible in these plots. Lengths are plotted in trap units.

Figure 8

(Color online) Vortex pair trajectories from numerical evolution of time-dependent GPE, $g=10$ and $g=50$. Starting positions are $x_{1,2}\left(0\right)=\pm 0.8$. Successive positions have been joined by straight lines. There are some gaps in time (widely separated points joined by a straight line). These correspond to intervals where the presence of several vortex-antivortex pairs made it difficult to meaningfully track a single pair. The Thomas-Fermi value for the condensate radius are $R_{\text{TF}}\simeq 1.89$ and $R_{\text{TF}}=\simeq 2.825$ in the two cases, shown with dotted lines. Lengths are plotted in trap units.