Bound vortex states and exotic lattices in multicomponent Bose-Einstein condensates: The role of vortex-vortex interaction

We numerically study the vortex-vortex interaction in multicomponent homogeneous Bose-Einstein condensates within the realm of the Gross-Pitaevskii theory. We provide strong evidence that pairwise vortex interaction captures the underlying mechanisms which determine the geometric configuration of the vortices, such as different lattices in many-vortex states, as well as the bound vortex states with two (dimer) or three (trimer) vortices. Specifically, we discuss and apply our theoretical approach to investigate intra- and intercomponent vortex-vortex interactions in two- and three-component Bose-Einstein condensates, thereby shedding light on the formation of the exotic vortex configurations. These results correlate with current experimental efforts in multicomponent Bose-Einstein condensates and the understanding of the role of vortex interactions in multiband superconductors.

Figure 1

Vortex-vortex interaction potentials, considering $M_{12}=2.0$, for (a) and (b) two vortices in the same component and (c) vortices in different components. Symbols denote different intercomponent coupling strengths: ${\gamma }_{12}=-0.20$ (blue circles), ${\gamma }_{12}=-0.45$ (yellow squares), ${\gamma }_{12}=-0.75$ (red triangles), and ${\gamma }_{12}=-0.98$ (green downward triangles).

Figure 2

The occupation number density contour plots for (a) the first component with two vortices positioned at $(-d/2,0)$ and $(d/2,0)$ and (b) the second component with a single vortex in the center of the mesh. Notice the depleted density in one component in the positions where the other component features a vortex. The investigation of the system energy with respect to $d$ for this conformation enables us to identify possible bound states in a two-component Bose-Einstein condensate with mass ratio $M_{12}=2.0$.

Figure 3

(a) Total energy for the three-vortex configuration $U_{21}$, characterizing a binary system with $M_{12}=2$. The vortex in the second component is fixed in the center of the mesh, whereas vortices in the first component are placed at symmetric positions separated by distance $d$. (b) A magnification of the case with ${\gamma }_{12}=-0.98$ shows that in the favored configuration, the two vortices in the first component are located on top of each other and on top of the vortex in the second component. (c)–(f) Occupation number density of both components, considering vortices separated by $100\xi$ for all cases considered in (a), where the solid (dashed) line stands for the first (second) component.

Figure 4

(a) Total energy of a three-vortex structure in a two-component BEC for ${\gamma }_{12}=-0.90$ (blue circles) and $-0.92$ (red squares). The curve bumps may lead to nontriangular lattices, despite the overall repulsive behavior. (b) A triple-vortex structure with ${\gamma }_{12}=-0.95$. The minimum far from the origin allows for the formation of dimers for specific vortex densities. (c) Dimer configuration profile for ${\gamma }_{12}=-0.95$.

Figure 5

Vortex core size as a function of the intercomponent coupling for $M_{12}=2.0$. The blue solid line (red dashed line) represents the core radius of a vortex in the first (second) component.

Figure 6

Two-vortex intercomponent interaction potential in the case of competing contact and Rabi coupling, $\omega =2.1\times {10}^{-5}$ and $M_{12}=1$, for several values of ${\gamma }_{12}$.

Figure 7

The occupation number density contour plots of (a) the first component with a vortex positioned at $(0,\sqrt{3}d/4)$, (b) the second component with a vortex positioned at $(-d/2,-\sqrt{3}d/4)$, and (c) the third component with a vortex positioned at $(d/2,-\sqrt{3}d/4)$.

Figure 8

(a) Total energy $U_{111}$ for the equilateral-triangle vortex configuration as a function of the side $d$ for $\gamma =0.45$ and for several values of $\omega$. (b) Plot of the optimal side of the equilateral vortex triangle vs $\omega$ for $\gamma =0.45$. The red line represents a power law with an exponent of $-0.3214$ and a coefficient of $0.9265$. (c) Same as (b) but for $\gamma =0.60$. The red line represents a power law with an exponent of $-0.3186$ and a coefficient of $1.0746$.

Figure 9

(a) Total energy $U_{111}$ as a function of the displacement of the first component vortex $x_{v1}$ from the $S_1$ position in the $x$ direction. (b) Total energy $U_{111}$ as a function of the displacement of the first component vortex $y_{v1}$ from the origin in the $y$ direction. For both cases, we have considered $\gamma =0.45$ and $\omega =1.7\times {10}^{-4}$. The vertical dashed line represents the coordinates of the first component vortex for the minimum-energy equilateral-triangle configuration.

Figure 10

(a) Total energy $U_{111}$, with vortices of the first and second components pinned in the center of the mesh, as a function of the third component vortex distance $d$ from the origin for different values of the third component extra phase ${\varphi }_3$: 0 (solid blue line), $\pi /6$ (green dash-dotted line), $\pi /3$ (yellow dashed line), and $\pi /2$ (red dotted line), with ${\omega }_{12}={\omega }_{13}=-{\omega }_{23}=1\times {10}^{-2}$. (b) Total energy $U_{111}$ for the same conformation as in (a), keeping ${\varphi }_3=0$ and assuming different values of $\omega$: $\omega =1\times {10}^{-2}$ (blue circles), $\omega =2\times {10}^{-2}$ (green squares), $\omega =3\times {10}^{-2}$ (yellow upward triangles), and $\omega =4\times {10}^{-2}$ (red downward triangles).