Quantum Elliptic Vortex in a Nematic-Spin Bose-Einstein Condensate

We find a novel topological defect in a spin-nematic superfluid theoretically. A quantized vortex spontaneously breaks its axisymmetry, leading to an elliptic vortex in nematic-spin Bose-Einstein condensates with small positive quadratic Zeeman effect. The new vortex is considered the Joukowski transform of a conventional vortex. Its oblateness grows when the Zeeman length exceeds the spin healing length. This structure is sustained by balancing the hydrodynamic potential and the elasticity of a soliton connecting two spin spots, which are observable by in situ magnetization imaging. The theoretical analysis clearly defines the difference between half quantum vortices of the polar and antiferromagnetic phases in spin-1 condensates.

Figure 1

The cross-sectional profile of an elliptic vortex for $q/\mu =2^{-17}\approx 7.6\times {10}^{-6}$. (a) The left and right sides show the profiles of $|{\mathrm{\Phi }}_0{|}^2/n_P$ and $|{\mathrm{\Phi }}_1{|}^2/(2n_P)$, respectively. The density $|{\mathrm{\Phi }}_{-1}{|}^2$ (not shown) is the same as $|{\mathrm{\Phi }}_1{|}^2$. (b) The vector field on the left shows ${\mathit{v}}_0=(\hslash /M)\nabla {\mathrm{\Theta }}_0$, with the background plot of ${\mathrm{\Theta }}_0=\arg {\mathrm{\Phi }}_0$. The phase $\arg {\mathrm{\Phi }}_{\pm 1}$ is homogeneous inside the core (not shown). The spin density $s_y$ is plotted on the right, while $s_x=s_z=0$. (c) Left: the texture of the unit vector $\mathit{g}/|\mathit{g}|$ (arrow) in Eq. (3). The color of the arrows and the surface correspond to ${\mathrm{\Theta }}_0$ and $s_y$ in (b), respectively. Right: schematic of the three-dimensional structure of the vortex core. The distance $l_{\mathrm{spin}}$ between the two spin spots with opposite transverse magnetization (blue and red) is determined by the balance between the hydrodynamic potential and the elastic potential by the AF soliton (green). The width ($\sim l_{\mathrm{spin}}$) and thickness of the core are represented by black and white arrows, respectively.

Figure 2

(a) The $q$ dependence of $l_{\mathrm{spin}}$. The solid curve represents the evaluation by Eq. (12). All lengths are rescaled by ${\xi }_n$. (b) The $q$ dependence of the spin interaction $E_{\mathrm{spin}}$, the maximum spin density $s_{\perp }^{\max }=\max (s_y)$, and the core density $n_{\pm 1}(0)$. The solid curve tracing the data corresponds to an analytic formula (see text).

Figure 3

(a) The $q$ dependence of the width $l_{\mathrm{spin}}^{\mathrm{\Omega }}$ of a rotating elliptic vortex with angular velocity $\mathrm{\Omega }$. The single vortex is unstable for larger $|\mathrm{\Omega }|$ and smaller $q$ due the boundary effect. Inset: an elliptic-vortex lattice obtained after the instability due to the boundary effect for $\mathrm{\Omega }\hslash /\mu =0.001$ and $q/\mu =2^{-11}$. (b) The three-dimensional solution of an elliptic vortex in a harmonic trap for $\mathrm{\Omega }\hslash /\mu =0.0005$. The isovolume plot shows the region $|{\mathrm{\Psi }}_0{|}^2{c}_0/\mu \le 0.3$ for $x>0$ with its $x=0$ cross-sectional profile. A translucent surface along the $z$ axis represents the isosurface of $|{\mathrm{\Psi }}_{\pm 1}{|}^2{c}_0/\mu =0.15$, to which the two poles of the isosurfaces $s_y{c}_0/\mu =\pm 0.7$ are attached (red for positive and blue for negative).