Dynamics of a vortex dipole across a magnetic phase boundary in a spinor Bose-Einstein condensate

The dynamics of a vortex dipole in a spin-1 Bose-Einstein condensate in which magnetic phases are spatially distributed is investigated. When a vortex dipole travels from the ferromagnetic phase to the polar phase, or vice versa, it penetrates the phase boundary and transforms into one of the various spin vortex dipoles, such as a leapfrogging ferromagnetic-core vortex dipole and a half-quantum vortex dipole. Topological connections of spin wave functions across the phase boundary are discussed.

Figure 1

Time evolution of a vortex dipole traveling from the ferromagnetic phase to the polar phase, where the initial distance between the vortices is $\Delta =1.6{\xi }_s$. (a)–(f) show $|{\psi }_1{|}^2$, $|{\psi }_{-1}{|}^2$, and ${\sum }_m{\left|{\psi }_m\right|}^2$, and (g) shows $|{\psi }_1{|}^2$ and $|{\psi }_{-1}{|}^2$. The density $|{\psi }_0{|}^2$ is always negligible. The arrows in (a) indicate the directions in which the vortices circulate. The windows of each panel are (a)–(e) $-4{\xi }_s<x<4{\xi }_s$ and $-3{\xi }_s<y<3{\xi }_s$, (f)$10{\xi }_s<x<18{\xi }_s$ and $-3{\xi }_s<y<3{\xi }_s$, and (g) $2{\xi }_s<x<4{\xi }_s$ and $-2{\xi }_s<y<2{\xi }_s$. Note that the scale of (f) is the same as it is for (a)–(e). See the Supplemental Material for a movie of the dynamics of $|{\psi }_1{|}^2$ [25].

Figure 2

Spatiotemporal image of the magnetization density $\left|\mathit{F}\right|$ during $10<t/\tau <16$. The holes indicated by the arrows correspond to the vortex cores in Fig. 1. The parameters are the same as those in Fig. 1.

Figure 3

The spin states in Figs. 1 to 1 are rotated by $\pi /2$ about the $y$ axis. The upper and lower panels show the density profiles $|{\psi }_1{|}^2$ and $|{\psi }_0{|}^2$, respectively. The density $|{\psi }_{-1}{|}^2$ is similar to $|{\psi }_1{|}^2$. See the Supplemental Material for a movie of the dynamics of $|{\psi }_1{|}^2$ [25].

Figure 4

Time evolution of a vortex dipole traveling from the ferromagnetic phase to the polar phase, where the initial distance between the vortices is $\Delta =3.2{\xi }_s$. The upper and lower panels show the density profiles $|{\psi }_1{|}^2$ and $|{\psi }_{-1}{|}^2$, respectively. The density $|{\psi }_0{|}^2$ is always negligible. The arrows indicate the directions in which the vortices circulate. The window of each panel is $-5{\xi }_s<x<5{\xi }_s$ and $-10{\xi }_s<y<10{\xi }_s$. See the Supplemental Material for a movie of the dynamics of $|{\psi }_1{|}^2$ [25].

Figure 5

Time evolution of a vortex dipole traveling from the polar phase to the ferromagnetic phase, where the initial distance between the vortices is $\Delta =1.6{\xi }_s$. (a)–(f) show $|{\psi }_1{|}^2$, $|{\psi }_{-1}{|}^2$, and ${\sum }_m{\left|{\psi }_m\right|}^2$, where $|{\psi }_0{|}^2$ is always negligible. The arrows in (a) indicate the directions in which the vortices circulate. (g) Time evolution of the density profiles $|{\psi }_1{|}^2$ and $|{\psi }_0{|}^2$, where the spin quantization axis is rotated by $\pi /2$ about the $y$ axis. The windows of each panel are (a)–(f) $-4{\xi }_s<x<4{\xi }_s$ and $-3{\xi }_s<y<3{\xi }_s$, and (g) $-4.5{\xi }_s<x<-1.5{\xi }_s$ and $-3{\xi }_s<y<3{\xi }_s$. See the Supplemental Material for a movie of the dynamics of $|{\psi }_{-1}{|}^2$ in (a)–(f) and that of $|{\psi }_1{|}^2$ in (g) [25].

Figure 6

Time evolution of a system in which a half-quantum vortex dipole is imprinted to the polar side of the initial state. In (a), singly quantized vortices are imprinted only to the $m=1$ wave function, and in (b), they are imprinted only to the $m=-1$ wave function. The distances between the vortices in the initial vortex dipole are (a) $\Delta =6.4{\xi }_s$ and (b) $\Delta =3.2{\xi }_s$. The arrows indicate the directions in which the vortices circulate. The window of each panel is $-5{\xi }_s<x<5{\xi }_s$ and $-10{\xi }_s<y<10{\xi }_s$. See the Supplemental Material for movies of the dynamics of $|{\psi }_1{|}^2$ in (a) and that of $|{\psi }_{-1}{|}^2$ in (b) [25].