Fractional-charge vortex in a spinor Bose-Einstein condensate

We classify all possible fractional charge vortices of charge less than unity in spin-1 and spin-2 polar and cyclic Bose-Einstein condensates (BECs) with zero magnetization. Statics and dynamics of these vortices in quasi-two-dimensional spinor BECs are studied employing an accurate numerical solution and a Lagrange variational approximation. The results for density and collective-mode oscillation are illustrated using the fractional-charge BEC vortex of ${}^{23}\mathrm{Na}$ and ${}^{87}\mathrm{Rb}$ atoms with realistic interaction and trapping potential parameters.

Figure 1

(a) Numerical (lines) and variational (symbols) radial densities for 1/2-1/2 fractional-charge vortices of 1000, 5000, and 50 000 ${}^{23}\mathrm{Na}$ atoms for components ${\varphi }_{\pm 1}$ of a spin-1 polar BEC. The fractional charge of the vortex is 1/2. The quoted values of $c_j$ are for $N=1000.$ Variables plotted in this and other figures are dimensionless.

Figure 2

Numerical (lines) and variational (symbols) radial densities for (a) 1/2-1/2 and (b) 1/2-1/4 vortices in a polar spin-2 BEC of 1000, 5000, and 50 000 ${}^{23}\mathrm{Na}$ atoms. The same densities for ${}^{87}\mathrm{Rb}$ atoms are shown in (c) and (d), respectively. In the case of 1/2-1/2 vortices (a) and (c), the densities correspond to components ${\varphi }_{\pm 1}$ of spin-2 polar BECs, whereas for 1/2-1/4 vortices, they correspond to components ${\varphi }_{\pm 2}$. The fractional charge of all vortices is 1/2. The quoted values of $c_j$ are for $N=1000.$

Figure 3

Numerical (lines) and variational (symbols) radial densities of 1000 and 5000 ${}^{23}\mathrm{Na}$ atoms corresponding to components ${\varphi }_{+2}$ and ${\varphi }_{-1}$ of a cyclic spin-2 BEC generating (a) a 1/3-2/3 vortex, (b) a 1/3-1/3 vortex, (c) a 2/3-2/3 vortex, and (d) a 2/3-1/3 vortex. In (a) and (b) the fractional charge of the vortices is 1/3, and in (c) and (d) it is 2/3. The quoted values of $c_j$ refer to $N=1000.$

Figure 4

Numerical (lines) and variational (symbols) results of collective-mode oscillation of the rms sizes $\langle r\rangle$ of components (a) $m_f=+2$ and (b) $m_f=-2$ in a spin-2 polar ${}^{23}\mathrm{Na}$ BEC of 50 000 atoms hosting a 1/2-1/4 vortex. Numerically, the mode was excited by applying the perturbed potential, (42). The stationary density of this vortex is shown in Fig. 2. The equivalent binary system has a singly charged vortex in one of the components, $m_f=+2$.

Figure 5

Numerical (lines) and variational (symbols) results of collective-mode oscillation of the rms sizes $\langle r\rangle$ of components $m_f=+2,-1$ in a spin-2 cyclic ${}^{23}\mathrm{Na}$ BEC of 50 000 atoms hosting a 1/3-2/3 vortex. Numerically, the mode was excited by multiplying the nonlinearities $c_j$ by the constant factor 1.095 in real-time simulation. The stationary density of this vortex is shown in Fig. 3. The equivalent binary system has a singly charged vortex in both components, $m_f=+2,-1$.