Topological vortex formation in a Bose-Einstein condensate under gravitational field

Topological phase imprinting is a unique technique for vortex formation in a Bose-Einstein condensate (BEC) of an alkali-metal gas, in that it does not involve rotation: the BEC is trapped in a quadrupole field with a uniform bias field which is reversed adiabatically leading to vortex formation at the center of the magnetic trap. The scenario has been experimentally verified by Leanhardt et al. employing ${}^{23}\mathrm{Na}$ atoms. Recently similar experiments have been conducted by Hirotani et al. in which a BEC of ${}^{87}\mathrm{Rb}$ atoms was used. In the latter experiments the authors found that fine-tuning of the field reverse time $T_{\mathrm{rev}}$ is required to achieve stable vortex formation. Otherwise, they often observed vortex fragmentation or a condensate without a vortex. It is shown in this paper that this behavior can be attributed to the heavy mass of the $\mathrm{Rb}$ atom. The confining potential, which depends on the eigenvalue $m_B$ of the hyperfine spin $\mathit{F}$ along the magnetic field, is now shifted by the gravitational field perpendicular to the vortex line. Then the positions of two weak-field-seeking states with $m_B=1$ and 2 deviate from each other. This effect is more prominent for BECs with a heavy atomic mass, for which the deviation is greater and, moreover, the Thomas-Fermi radius is smaller. We found, by solving the Gross-Pitaevskii equation numerically, that two condensates interact in a very complicated way leading to fragmentation of vortices, unless $T_{\mathrm{rev}}$ is properly tuned.

Figure 1

(Color) Time development of the condensate with $T_{\mathrm{rev}}=4\mathrm{ms}$. The white lines show the coordinate system, the origin of which is the center of the quadrupole field where the vortex formation takes place. Upper panel shows the total density of the condensate. The number in the panel shows the normalized time $t∕T_{\mathrm{rev}}$. The middle (lower) panel shows the density of the $m_B=2\left(m_B=1\right)$ component. The cross $(\times )$ shows the minimum of the potential for the respective component. Observe that the center of each $m_B$ component follows the minimum of the corresponding potential and accordingly the center of the $m_B=1$ component is below that of the $m_B=2$ component at $t=T_{\mathrm{rev}}$.

Figure 2

(Color) Time development of the condensate with $T_{\mathrm{rev}}=2\mathrm{ms}$. See the caption of Fig. 1 for the definition of the symbols. The condensate is pushed up while the potential is deformed by reversing $B_z$. The $m_B=1$ component is still above the minimum of its potential at $t=T_{\mathrm{rev}}$ so that the center of the component is close to that of the $m_B=2$ component.

Figure 3

(Color) Time development of the condensate with $T_{\mathrm{rev}}=1.5\mathrm{ms}$. See the caption of Fig. 1 for the definition of the symbols. The condensate is pushed up by a quick deformation of the potential and the centers of both $m_B$ components are still above the minima of the potentials at $t=T_{\mathrm{rev}}$. This is more prominent for the $m_B=1$ component so that the center of the $m_B=1$ component is above that of the $m_B=2$ component.