Magnetization of a half-quantum vortex in a spinor Bose-Einstein condensate

Magnetization dynamics of a half-quantum vortex in a spin-1 Bose-Einstein condensate with a ferromagnetic interaction are investigated by mean-field and Bogoliubov analyses. The transverse magnetization is shown to break the axisymmetry and form threefold domains. This phenomenon originates from the topological structure of the half-quantum vortex and spin conservation.

Figure 1

(Color) (a) Time evolution of the normalized wave function ${\tilde{\psi }}_m$ and transverse magnetization ${\tilde{F}}_{+}={\tilde{F}}_x+i{\tilde{F}}_y$, where ${\tilde{F}}_x$ and ${\tilde{F}}_y$ are defined by Eq. (2) with ${\psi }_m$ replaced by ${\tilde{\psi }}_m$. The size of each panel is $10\times 10$ in units of $\sqrt{\hslash ∕\left(m{\omega }_{\perp }\right)}$. (b) Time evolution of the population in each component. The interaction coefficients are ${\tilde{c}}_0=3000$ and ${\tilde{c}}_1=-{\tilde{c}}_0∕216$ (corresponding to spin-1 ${}^{87}\mathrm{Rb}$).

Figure 2

Imaginary part of the Bogoliubov spectrum for the half-quantum vortex state of a spin-1 ${}^{87}\mathrm{Rb}$ BEC $({\tilde{c}}_1=-{\tilde{c}}_0∕216)$. The solid line is obtained by diagonalizing Eq. (13a), where the mode function has the form (15) with $\ell =2$ and $-1$, corresponding to the threefold domain formation. The dashed line is obtained from Eq. (13b), where the mode functions have the form (16) with $\ell =0$ and 2, corresponding to the dipole deformation.

Figure 3

Lines indicating the directions of maximum fluctuations in the transverse magnetization for the half-quantum vortex state in Eq. (26), where (a) and (b) correspond to the + and − signs, respectively. The dashed circles show schematics of the threefold domains and the arrows show the directions of magnetization in the domains.