Einstein–de Haas Effect in Dipolar Bose-Einstein Condensates

The general properties of the order parameter for a dipolar spinor Bose-Einstein condensate are discussed based on symmetries of interactions. An initially spin-polarized dipolar condensate is shown to dynamically generate a nonsingular vortex via spin-orbit interactions—a phenomenon reminiscent of the Einstein–de Haas effect in ferromagnets.

Figure 1

(a) Relative population $N_{\alpha }/N=\int d\mathit{r}|{\psi }_{\alpha }{|}^2/N$ of each magnetic sublevel $\alpha$ versus $\omega t$ for $B_{\mathrm{ext}}=0$ and $\lambda =1$. (b)–(d) Isopycnic surfaces of (b) ${\psi }_{-3}$, (c) ${\psi }_{-2}$, and (d) ${\psi }_{-1}$ at $\omega t=2$, where $|{\psi }_{\alpha }{|}^2{a}_{\mathrm{ho}}^3/N=0.0001$ for (b) and (c) and $5\times {10}^{-5}$ for (d). The color on the surfaces represents the phase of the order parameter (refer to scale at right).

Figure 2

(a) Dipole field at $t=0$ for $\lambda =1$. The solid line shows the isopycnic curve at $|{\psi }_{-3}{|}^2{a}_{\mathrm{ho}}^3/N=0.0001$. The color of the arrows denotes the magnitude of the field. (b),(c) Spin configurations on the (b) $z=2\mu \mathrm{m}$ plane and (c) $z=-2\mu \mathrm{m}$ plane at $\omega t=2$ for $B_{\mathrm{ext}}=0$ and $\lambda =0$. The length of the arrows represents the magnitude of the spin vector projected on the $xy$ plane and the color shows $|\mathit{S}|a_{\mathrm{ho}}^3/N$. Note that the spins tilt in a direction perpendicular to ${\mathit{B}}_{\mathrm{eff}}(\mathit{r})$.

Figure 3

(a) Population of $\alpha =-2$ sublevel versus $\omega t$ for $B_{\mathrm{ext}}=\pm 1\mathrm{mG}$ and $\lambda =1$ and 0.5. (b),(c) Cross sections of $|{\psi }_{-2}{|}^2{a}_{\mathrm{ho}}^3/N$ with $\lambda =1$ at $\omega t=8$ for (b) $B_{\mathrm{ext}}=1\mathrm{mG}$ and (c) $B_{\mathrm{ext}}=-1\mathrm{mG}$. The cross sections are at $z=0.7\mu \mathrm{m}$ (left) and $y=0\mu \mathrm{m}$ (right). The size of each panel is $8.2\times 8.2\mu \mathrm{m}$.

Figure 4

Dipole field at $t=0$ for $\lambda =0.5$. The solid line shows the isopycnic curve at $|{\psi }_{-3}{|}^2{a}_{\mathrm{ho}}^3/N=0.0005$. Note that the sign of $B_{\mathrm{eff}}^z(\mathit{r})$ in the condensate is opposite to that in Fig. 2a.