Soliton magnetization dynamics in spin-orbit-coupled Bose-Einstein condensates

Ring-trapped Bose-Einstein condensates subject to spin-orbit coupling support localized dark-soliton excitations that show periodic density dynamics in real space. In addition to the density feature, the solitons also carry a localized pseudospin magnetization that exhibits a rich and tunable dynamics. Analytical results for Rashba-type spin-orbit coupling and spin-invariant interactions predict a conserved magnitude and precessional motion for the soliton magnetization that allows for the simulation of spin-related geometric phases recently seen in electronic transport measurements.

Figure 1

Time evolution of a gray-bright soliton's magnetization in a ring-trapped BEC with Rashba spin-orbit coupling. During a full cycle of the soliton's motion on the ring, the magnetization vector follows a trajectory on the surface of a sphere. The magnetization vectors at the beginning and the end of a cycle (indicated by arrows) differ by an angle $\vartheta$ that is related to a spin-related geometric phase. Soliton parameters (see text): $v_{\mathrm{s}}/c=0.5$, $\tan \eta =2$, $g=100$, $\kappa =-0.01$.

Figure 2

Magnetization dynamics of a gray-bright soliton (a) and a gray-gray soliton with zero background magnetization in the local spin frame (b). Parameters used are $g=100$ (100), $\tan \eta =0.2$ ($0.5$), $v_{\mathrm{s}}/c=0.5$ (0.2), and $\kappa =-0.5$ for the gray-bright (gray-gray) case. (c) Angles used to describe the two-step precessional motion of $\mathrm{S}$. The angle ${\alpha }^{\prime }$ is measured with respect to an $x^{\prime }$ axis that is perpendicular to both the $z$ and $z^{\prime }$ axes.

Figure 3

Laboratory-frame spin densities (a),(b) and $z$ component of the magnetization density (c),(d) for a stationary gray-bright soliton (a),(c) and a stationary gray-gray soliton (b),(d). (a) and (b) show the total density (solid yellow curve) and individual-spin (dashed blue $=$ $\uparrow$, dotted red $=$ $\downarrow$) densities normalized to the background-density value. In (c) and (d), the total density is plotted again for reference as the solid yellow curve, together with the magnetization profiles $s_z^{\left(\mathrm{s}\right)}$ (dashed blue curve), and the magnetization density $s_{\mathrm{b}z}^{\left(\mathrm{s}\right)}-s_z^{\left(\mathrm{s}\right)}$ associated with the soliton excitations only (dotted red curve). Parameters are $\tan \eta =1.0$, $g=100$, and (for the gray-bright soliton) $\kappa =-0.5$.