Multiply quantized and fractional skyrmions in a binary dipolar Bose-Einstein condensate under rotation

We consider a binary dipolar Bose-Einstein condensate with repulsive contact and dipolar interactions under rotation. Our results show that the interplay among short-range interaction, long-range interaction, and rotation can give rise to a rich variety of topological configurations, including giant skyrmions with multiply topological charges and skyrmion-vortex lattices. In particular, we find that for fixed rotation frequencies, tuning the short- and long-range interactions can derive novel ground-state phases, such as a meron pair composed of two fractional skyrmions and a skyrmion with topological charge $Q=2$ centered in giant skyrmions.

Figure 1

(a) Atoms of the first component have magnetic dipole moments. The trapping geometry is realized by the crossed optical dipole trap (red areas) [24], and the quasi-2D BEC (green disk) is rotated with the trap beam in the $z=0$ plane. (b) Mixture of dipolar (red spheres) and nondipolar (green spheres) atoms in a harmonic trap. Black arrows across the balls describe the pseudospin states $|\uparrow \rangle$ and $|\downarrow \rangle$. Top: Polarized atoms interact via dipole-dipole interaction separated by a distance $|\mathbf{r}-{\mathbf{r}}^{\prime }|$, and the angle between the direction of polarization and the relative position of atoms is $\theta$. (c) A skyrmion configuration is induced in the effective magnetic field which is generated by DDI in (a) [17].

Figure 2

Ground-state density and phase distributions of the two components for varied inter-component and dipole-dipole interactions under the rotation frequency $\mathrm{\Omega }=0.3$. From left to right in each column: density ${\rho }_1={\left|{\psi }_1\right|}^2$, density ${\rho }_2={\left|{\psi }_2\right|}^2$, and phase ${\varphi }_1=arg{\psi }_1$. The density is indicated by a color map from red (high) to blue (low), and the phase ranges from $-\pi$ (blue) to $\pi$ (red). $\beta$ is the ratio between dipolar and intracomponent contact interactions, $\gamma$ is the ratio between inter- and intracomponent interactions, and $\theta$ is the angle between the direction of polarization and the relative position of atoms.

Figure 3

Vortex necklaces conjoin with giant skyrmions in rotating BECs as intercomponent interactions $\gamma$ are increased. The color map and parameters are the same as in the rightmost column in Fig. 2, but for the rotation frequency $\mathrm{\Omega }=0.5$ (left column) and $\mathrm{\Omega }=0.6$ (right column). Meron pairs emerge when $\gamma \sim 1$ for $\mathrm{\Omega }=0.6$, and the topological charge of a meron for $\gamma \in [0.85,1.3)$ is $Q=\frac{1}{2}$.

Figure 4

Density and phase profiles of giant skyrmions with different quantized numbers in a dipolar BEC. A doubly quantized skyrmion ($Q=2$) is formed in the center when $\beta =2$. Here the parameters are chosen as $\mathrm{\Omega }=0.6$ and $\gamma =3$. Color map same as for Fig. 2.

Figure 5

Ground-state density distributions of skyrmion-vortex lattices ($\gamma =1, \beta =3$ or 4), an isolate skyrmion rounded by a vortex ring ($\gamma =3, \beta =3$), skyrmion pairs rounded by vortex rings ($\gamma =2, \beta =3$ or $\gamma =3, \beta =4$), and triangular-type skyrmions rounded by a vortex ring ($\gamma =2, \beta =4$). The rotation frequency is fixed at $\mathrm{\Omega }=0.7$. Color map of the density same as for Fig. 2.

Figure 6

Topological and rotationally symmetric ground-state structures of the total density profiles $\rho =|{\psi }_1{|}^2+{\left|{\psi }_2\right|}^2$ and their corresponding spin textures. Color map of the density same as for Fig. 2. Values of the pseudospin density are from $-1$ (blue) to 1 (red). The parameters ($\mathrm{\Omega },\beta ,\gamma$) are chosen as (a1) ($0.3,1,0.5$), (b1) ($0.3,1,1.5$), (c1) ($0.3,1,2$), (d1) ($0.3,1,3$), (a2) ($0.6,1,0.5$), (b2) ($0.6,1,1.5$), (c2) ($0.6,1,2$), (d2) ($0.6,1,3$), (a3) ($0.7,3,1$), (b3) ($0.7,4,1$), (c3) ($0.7,3,3$), and (d3) ($0.7,4,3$), respectively.

Figure 7

Examples of spin textures of different kinds of skyrmions with different topological charges. Systems with the same parameters as in the left column in Fig. 3 ($\mathrm{\Omega }=0.5,\beta =1,\gamma =2$), the right column in Fig. 3 ($\mathrm{\Omega }=0.6,\beta =1,\gamma =1.29$), the fourth column in Fig. 4 ($\mathrm{\Omega }=0.6,\beta =2,\gamma =3$), and the right column in Fig. 5 ($\mathrm{\Omega }=0.7,\beta =4,\gamma =2$). Lower panels are zoom-ins of the regions in the rectangles the upper panels. Values of the pseudospin density are from $-1$ (blue) to 1 (red).