Collisional Dynamics of Half-Quantum Vortices in a Spinor Bose-Einstein Condensate

We present an experimental study on the interaction and dynamics of half-quantum vortices (HQVs) in an antiferromagnetic spinor Bose-Einstein condensate. By exploiting the orbit motion of a vortex dipole in a trapped condensate, we perform a collision experiment of two HQV pairs, and observe that the scattering motions of the HQVs is consistent with the short-range vortex interaction that arises from nonsingular magnetized vortex cores. We also investigate the relaxation dynamics of turbulent condensates containing many HQVs, and demonstrate that spin wave excitations are generated by the collisional motions of the HQVs. The short-range vortex interaction and the HQV-magnon coupling represent two characteristics of the HQV dynamics in the spinor superfluid.

Figure 1

HQV pairs in an antiferromagnetic spinor BEC in the easy-plane polar (EPP) phase. The diagrams show the spatial structures of four HQV pairs in terms of the superfluid phase $\theta$ (color) and the spin orientation $\overrightarrow{d}$ (arrow). The magnetization $M_z$ in the core region is indicated by brightness. The topological charges of the left and right HQVs are $(q_{n1},q_{s1})=(\frac{1}{2},\frac{1}{2})$ and $(|q_{n2}|,|q_{s2}|)=(\frac{1}{2},\frac{1}{2})$, respectively, where $q_n$ and $q_s$ correspond to the winding numbers of $\theta$ and $\overrightarrow{d}$, respectively. $\kappa =q_{n1}{q}_{n2}+q_{s1}{q}_{s2}$. The dashed line between the two HQVs denotes a sudden change of $\theta \to \theta +\pi$ and $\overrightarrow{d}\to -\overrightarrow{d}$. The superfluid order parameter is continuous across the dashed line due to the $Z_2$ symmetry (see text). The HQV pair with $\kappa =-1/2$ is a HQV dipole with zero net charges.

Figure 2

Collision of HQV pairs. (a) A vortex dipole consisting of two singly charged vortices with opposite circulations is generated in the center region of the condensate [23]. The two vortices show an orbit motion depicted by the solid lines. (b)–(d) Optical density (OD) images of BECs in the easy-axis polar (EAP) phase, taken at (b) $t=1\mathrm{s}$, (c) 2 s, and (d) 4 s after generating a vortex dipole. (e) When the BEC is transmuted to the EPP phase after the vortex generation, each singly charged vortex is split into a pair of HQVs with opposite core magnetizations, and the HQV pair moves along the trajectory of its original singly charged vortex. (f)–(h) Magnetization images of BECs in the EPP phase at (f) $t=1\mathrm{s}$, (g) 2 s, (h) 4 s, and (i)–(l) 5 s. At $t\approx 4\mathrm{s}$, two HQV pairs collide in the upper right region of the BEC and scatter into two HQV dipoles. (m)–(p) Descriptions of the vortex states in (i)–(l). The dashed lines indicate the propagation lines of the HQV dipoles. In (l), only one HQV dipole is identified. All images were taken after a free expansion of the samples for 24 ms.

Figure 3

Relaxation of quantum turbulence with HQVs. OD images of (a)–(d) the $m_z=1$ and (e)–(h) $m_z=-1$ spin components of turbulent BECs at various hold times, $t$, in the EPP phase. The two spin components were imaged simultaneously with Stern-Gerlach spin separation and 24 ms time of flight [23]. The vortex positions in the two spin components coincide initially and become uncorrelated in the subsequent evolution, indicating the development of HQVs. The cloud shapes of the two spin components are slightly mismatched due to the inhomogeneity of the spin-separating field gradient. (i)–(l) In situ magnetization images of the BECs for the corresponding hold times. Spin wave excitations are rapidly generated as HQVs develop. Temporal evolutions of (m) the HQV number, $N_v$, and (n) the variance of spin fluctuations, $⟨\delta M_z^2⟩$, for various sample conditions. $N_v$ was measured from the OD images of the two spin components and $⟨\delta M_z^2⟩$ was determined from the central $150\mu \mathrm{m}\times 150\mu \mathrm{m}$ region of the in situ magnetization image. The data points were obtained by averaging ten measurements of the same experiment, and the error bars indicate the standard deviation of the measurements. The inset in (m) shows the atom number evolution of the BEC for initial vortex number $N_v\approx 80$ (solid) and 20 (open). Over 12 s evolution, the condensate fraction slightly decreased from $>90\%$ to $\approx 85\%$.

Figure 4

Relaxation trajectories of turbulent BECs in the plane of the dimensionless HQV density, $n_v{\xi }_s^2$, and $⟨\delta M_z^2⟩$. The arrows represent the evolution direction of the trajectories. The dashed line indicates the thermal equilibrium value of $⟨\delta M_z^2⟩$ for the stationary condensate, which are obtained by averaging the data points marked by open squares in Fig. 3 for long hold times $t\ge 6\mathrm{s}$. The solid line shows the estimated contribution of magnetized HQV cores, $⟨\delta M_z^2{⟩}_{\mathrm{HQV}}$ [23], with the thermal equilibrium offset. The dotted line is a guide for the eyes. The variation of the atom number of the condensate was taken into account in the calculation of $n_v{\xi }_s^2$ [Fig. 3, inset]. The inset shows the universal decay curve of the vortex number, constructed by joining the data in Fig. 3 for $t\ge 1\mathrm{s}$ with adjusted hold time offsets.