Vortex pairs in a spin-orbit-coupled Bose-Einstein condensate

Static and dynamic properties of vortices in a two-component Bose-Einstein condensate with Rashba spin-orbit coupling are investigated. The mass current around a vortex core in the plane-wave phase is found to be deformed by the spin-orbit coupling, and this makes the dynamics of the vortex pairs quite different from those in a scalar Bose-Einstein condensate. The velocity of a vortex-antivortex pair is much smaller than that without spin-orbit coupling, and there exist stationary states. Two vortices with the same circulation move away from each other or unite to form a stationary state.

Figure 1

(a)–(c) Stable stationary state of a single vortex with counterclockwise circulation for $\kappa =1$ and $\gamma =0.8$. Panels (a) and (b) show the density and phase profiles of each component, where the unit of density is $n_0$. In (b), $\delta$ is the distance between the phase defects in the two components. Panel (c) shows the spin distribution $\mathit{S}\left(\mathit{r}\right)$ defined in Eq. (7). The arrows indicate the transverse direction of the spin vector, and the background color indicates the value of $S_z$. The dashed square region in (a) is shown magnified in (c). (d) $\kappa$ dependence of the vortex shift $\delta$. The solid curve shows $1/\kappa$ for comparison.

Figure 2

Velocity field $\mathit{v}\left(\mathit{r}\right)$ of the single-vortex state for $\gamma =0.8$ and (a) $\kappa =0.5$ and (b) $\kappa =1$. The arrows indicate the directions of the velocity, and the background color indicates the value of $\left|\mathit{v}\right|$. The regions in the dashed squares are magnified in the right-hand panels. The red arrows in the dashed squares indicate the direction of the vortex.

Figure 3

Schematic illustration of the vortex pair for $\langle 1,-1\rangle$. The black points are the vortex positions $(x_1,y_1)$ and $(x_2,y_2)$, which are defined as the midpoints between the phase defects in the two components. The open circle is the center of the vortex pair $(x_c,y_c)=[(x_1+x_2)/2,(y_1+y_2)/2]$.

Figure 4

Stable stationary states of vortex-antivortex pairs for $\kappa =1$ and $\gamma =0.8$. The winding combinations are (a) $\langle 1,-1\rangle$ and (b) $\langle -1,1\rangle$. Panels (a1) and (b1) show the density profiles, and (a2) and (b2) show the spin distributions. The regions indicated by dashed squares in (a1) and (b1) correspond to (a2) and (b2), respectively. The arrows indicate the directions of the transverse spin vector, and the background color indicates the value of $S_z$.

Figure 5

Variational energy of a vortex-antivortex pair in Eqs. (15) and (16) as a function of the distance between vortices $d_y$ for (a) $\kappa =0$, (b) $\kappa =0.5$, (c) $\kappa =1$, and (d) $\kappa =1.5$. In (b)–(d), local energy minima appear; these correspond to the stationary states shown in Fig. 4.

Figure 6

Trajectories of vortex-antivortex pairs for $\kappa =1$ and $\gamma =0.8$. The initial vortex distance is (a) $d=9.5$, (b) $d=10.3$, (c) $d=9.4$, and (d) $d=9.8$. Red (light gray) and blue (dark gray) circles indicate the positions of the vortex cores in ${\psi }_1$ and ${\psi }_2$, respectively. The directions of the circulations of the vortices are indicated by black arrows. The initial distance between vortices is $d=10$. Black circles indicate the center of the vortex pairs $(x_c,y_c)$ at $t=0$, 400, and 800, and green arrows indicate the direction of motion. See the Supplemental Material for movies of the dynamics of $|{\psi }_1{|}^2$ [42].

Figure 7

Velocity $v_x$ versus the distance $d_y$ of a vertically aligned vortex-antivortex pair for (a) $\kappa =1$ and (b) $\kappa =0.5$ with $\gamma =0.8$. The red and blue plots are for $\langle 1,-1\rangle$ and $\langle -1,1\rangle$, respectively. The configurations of the vortex pairs are illustrated in the insets. In (b), there is an unstable window (see text).

Figure 8

Time evolution of the (a) density and (b) phase of the unstable vortex-antivortex pair for $\kappa =0.5$ and $\gamma =0.8$, where the vertical gauges indicate the distance $d_y$ between the vortex cores. See the Supplemental Material for a movie of the dynamics of $|{\psi }_1{|}^2$ [42].

Figure 9

Trajectories of vortex-vortex pairs for $\kappa =1$ and $\gamma =0.8$. The initial vortex distance is (a) $d=8.0$, (b) $d=11.6$, (c) $d=18.0$, and (d) $d=12.9$. Red (light gray) and blue (dark gray) circles show the positions of the vortex cores in ${\psi }_1$ and ${\psi }_2$, respectively. Black arrows indicate the direction of circulation. Black circles are the center of the vortex pairs $(x_c,y_c)$ at $t=0$, 400, and 800, and green arrows show the directions of motion. See the Supplemental Material for a movie of the dynamics of $|{\psi }_1{|}^2$ [42].

Figure 10

Schematic illustration of the dynamics of (a) vortex-antivortex pairs and (b) vortex-vortex pairs when the distance is $d=10$. The red arrows indicate the direction of the velocity of $(x_c,y_c)$, when one vortex is located at the origin. In (a), the relative position of the vortices remains nearly constant. In (b), the relative velocity (the velocity of the vortex in the moving frame in which the other vortex is fixed to the origin) is indicated by blue arrows.