Dynamics of a coupled spin-vortex pair in dipolar spinor Bose-Einstein condensates

The collisional and magnetic field quench dynamics of a coupled spin-vortex pair in dipolar spinor Bose-Einstein condensates in a double-well potential are numerically investigated in the mean-field theory. Upon a sudden release of the potential barrier the two layers of condensates collide with each other in the trap center with the chirality of the vortex pair exchanged after each collision, showing the typical signature of in-phase collision for the parallel spin-vortex phase, and out-of-phase collision for the antiparallel phase. When quenching the transverse magnetic field, the vortex center in the single-layered condensate starts to make a helical motion with oval-shaped trajectories and the displacement of the center position is found to exhibit a damped simple harmonic oscillation with an intrinsic frequency and damping rate. The oscillation mode of the spin-vortex pair may be tuned by the initial magnetic field and the height of the Gaussian barrier; e.g., the gyrotropic motions for a parallel spin-vortex pair are out of sync with each other in the two layers, while those for the antiparallel pair exhibit a double-helix structure with the vortex centers moving opposite to each other with the same amplitude.

Figure 1

Collision dynamics of a coupled spin-vortex pair in (a) PSVP $\left(A=100\hslash {\omega }_{\perp }\right)$ and (b) ASVP $\left(A=300\hslash {\omega }_{\perp }\right)$ phases of the condensate when the Gaussian barrier is turned off suddenly. Each image denotes the cloud density in the $x\text{−}z$ plane after integrating along the $y$ axis.

Figure 2

Spin-vortex pair in the three collisional moments for the ASVP phase. Spin structures in the initial potential minima are shown in the upper and lower layers with the chiralities exchanged after each collision; the chiralities are inverted at $t=\tau /2$ and restored at $t=\tau$.

Figure 3

Time-resolved trajectories of (a) the vortex centers in the $z=0$ plane and (b) the displacement of the vortex center in the $y$ axis when the transverse magnetic field $(B_x=25$ $\mu \mathrm{G})$ is removed suddenly after initially preparing the system in the ground state of a harmonic trap. (c) The magnetization $M_x$ shows similar oscillation behavior with the same characteristic frequency as can be seen from (d) the spectral analysis for the two signals. The solid lines in (b) and (c) are fitted with a damped simple harmonic oscillation.

Figure 4

Quench dynamics of the spin structure in a single-layered condensate staring from a polarized ground state with a transverse magnetic field $B_x=70$ $\mu \mathrm{G}$. The top panel shows the time dependence of the magnetization along the $x$ axis. The middle panels show the formation and stabilization of the spin vortex in the $z=0$ plane at four evolution times $t=0{\omega }_{\perp }^{-1},10{\omega }_{\perp }^{-1},60{\omega }_{\perp }^{-1}$, and $110{\omega }_{\perp }^{-1}$ (magenta dotted vertical lines in the upper panel), respectively. The solid circles in the lower panel show the displacement of the vortex center in the $y$ axis after the stabilization of the spin vortex, and the open circles show the corresponding long time oscillation of the vortex center for $B_x=25$ $\mu \mathrm{G}$ in Fig. 3.

Figure 5

Time-resolved trajectories of the vortex centers in the $z=z_{\text{min}}$ planes (first row) and the displacement of the vortex center on the $y$ axis (second row) when the transverse magnetic field $(B_x=25$ $\mu \mathrm{G})$ is removed suddenly after initially preparing the system in the PSVP (left) and ASVP (right) phases, respectively. Panels in the third row show the time evolution of the corresponding magnetizations $M_x$ in the upper $\left(z>0\right)$ layer and lower layer $\left(z<0\right)$, both of which oscillate in the same characteristic frequency as the motion of the vortex centers. The spectral analyses for the two signals are plotted in the bottom panels while the insets show the situation for a wider separation of the two layers, ${\sigma }_0=1.2{\ell }_{\perp }$. The red (blue) lines are for the upper (lower) layer and the green lines in the second row plot the average position of the two vortex centers.

Figure 6

Dynamics of a single vortex and a polarized state in the PSVP and ASVP phases. The time dependencies of the magnetization along the $x$ axis are shown in (a), (b), (e), and (f) and the displacement of the vortex center on $y$ axis in (c), (d), (g), and (h) for the PSVP case (left) and ASVP case (right), respectively, when the magnetic field is turned off suddenly. The corresponding initial transverse magnetic fields are $B_x=60$ $\mu \mathrm{G}$ for (a) and (c), $50\mu \mathrm{G}$ for (b) and (d), and 70 $\mu \mathrm{G}$ for (e)–(h).