Spin textures in slowly rotating Bose-Einstein condensates

Slowly rotating spin-1 Bose-Einstein condensates are studied through a variational approach based upon lowest Landau level calculus. The author finds that in a gas with ferromagnetic interactions, such as ${}^{87}\mathrm{Rb}$, angular momentum is predominantly carried by clusters of two different types of skyrmion textures in the spin-vector order parameter. Conversely, in a gas with antiferromagnetic interactions, such as ${}^{23}\mathrm{Na}$, angular momentum is carried by $\pi$ disclinations in the nematic order parameter which arises from spin fluctuations. For experimentally relevant parameters, the cores of these $\pi$ disclinations are ferromagnetic, and can be imaged with polarized light.

Figure 1

Top view and perspective view of a $4\pi$ skyrmion. At the center (edge) the spins point out of (into) the page, while half way between the center and edge the spins lie in the plane of the page, rotating once as one circles the center. The spins are located on a two-dimensional plane, but point in three dimensions.

Figure 2

Top view and perspective view of an $8\pi$ skyrmion. This texture differs from Fig. 1 in that the planar projection of the spins rotates twice as one circles the origin.

Figure 3

Local vorticity $\widehat{z}·\mathbf{\nabla }\times v$, for the $4\pi$ (a) and $8\pi$ (b) skyrmion. Darker shades represent larger vorticity.

Figure 4

(Color online) top view and perspective view of an $8\pi$ skyrmion. This is the same texture as Fig. 2, except all spins have been rotated by 90° about the $\widehat{\mathbf{y}}$ axis. This texture can be interpreted as a composite of four $4\pi$ skyrmions, whose cores have been marked with circles, and labeled by letters (A) through (D). The spins in (A) and (C) point out of the page and the planar projections of the spins wrap the equator of the order-parameter sphere in a positive sense, while (B) and (D) point into the page and the projected spins wrap the equator in a negative sense.

Figure 5

Representations of a $\pi$ disclination in a nematic. Lines represent the local orientation of the nematic director.

Figure 6

Structure of the nematic ring texture (K). Horizontal axes represent distance from the center of trap along the $x$ axis, measured in units of the trap length. The top panel shows the density $\rho$. At each position in the middle panel, an arrow is drawn which represents the direction and strength of the local spin. At each position in the bottom panel, a rod points in the direction of the nematic director $\mathbf{n}$, corresponding to the largest eigenvalue of $Q^{\left(s\right)}$. The lengths of these directors are scaled by the total density, so that their lengths are related to the amount of local nematic order. To construct the full three-dimensional spin texture, one rotates this picture around the origin, so that the spins near the center look like the skyrmion in Fig. 1, and the nematic order away from the center forms a crown texture.

Figure 7

Scaled energies $\overline{\mathcal{E}}$ as a funtion of rotation speed, parametrized by $\eta$ defined in Eq. (21). Larger $\eta$ corresponds to faster rotation. Top∕bottom panels show ferromagnetic∕antiferromagnetic interaction with $c_2=\mp 0.05c_0$. Each curve represents a state of different symmetry, as described in the text and Figs. 10, 11. More detailed images of these states can be found in the EPAPS archive [23]. Energy scalings (top, $\overline{\mathcal{E}}={\mathcal{E}}^2-1.7125\eta$; bottom, $\overline{\mathcal{E}}={\mathcal{E}}^2-1.8084\eta$) are chosen to aid in comparing these different curves.

Figure 8

Angular momentum per particle $L$ and spin polarization per particle $\sigma =\mid \int d^{2}r\mathbf{S}\left(\mathbf{r}\right)\mid$ of ferromagnetic spin textures as a function of rotation speed, parametrized by $\eta$. Both are measured in units of $\hslash$. Note the discontinuities when the ground-state symmetry changes.

Figure 9

Angular momentum per particle $L$ and spin polarization per particle $\sigma$ of nematic spin textures as a function of rotation speed, parametrized by $\eta$.

Figure 10

Spatial structure of spin textures in a gas with ferromagnetic interactions. From left to right, the columns represent density $\rho$, vorticity $\widehat{\mathbf{z}}·\mathbf{\nabla }\times \mathbf{v}$, spin density ${\mid S\mid }^2$, and nematic order ${\overline{Q}}_{ab}^{\left(2\right)}{\overline{Q}}_{ba}^{\left(2\right)}={\rho }^2{Q}_{ab}^{\left(2\right)}{Q}_{ba}^{\left(2\right)}-{\rho }^2∕6$. Darker shades corresponds to larger magnitudes. Although not universally true, $4\pi \left(8\pi \right)$-skyrmions tend to show up as black (white) dots in column 2 and white (black) dots in column 3.

Figure 11

Spatial structure of spin textures in a gas with antiferromagnetic interactions. With the exception of texture K, all black dots in the third column (which also coincide with white dots in the fourth column) can be identified as the cores of $\pi$ disclinations.

Figure 12

Densities ${\mid {\psi }_1\mid }^2$, ${\mid {\psi }_0\mid }^2$, and ${\mid {\psi }_{-1}\mid }^2$ of the three components of texture (H), as seen from two different quantization axes. Darker shades represent higher density. (a) uses the natural quantization axis where the skyrmion axes are aligned with the $\widehat{\mathbf{z}}$ direction, while (b) uses an arbitrary axis.