Twisted spin vortices in a spin-1 Bose-Einstein condensate with Rashba spin-orbit coupling and dipole-dipole interaction

We consider a spin-1 Bose-Einstein condensate with Rashba spin-orbit coupling and dipole-dipole interaction confined in a cigar-shaped trap. Due to the combined effects of spin-orbit coupling, dipole-dipole interaction, and trap geometry, the system exhibits a rich variety of ground-state spin structures, including twisted spin vortices. The ground-state phase diagram is determined with respect to the strengths of the spin-orbit coupling and dipole-dipole interaction.

Figure 1

Ground-state phase diagram of the spin-orbit coupled dipolar BEC with respect to ${\mathrm{g}}_{\mathrm{soc}}$ and ${\mathrm{g}}_{\mathrm{dd}}$ for ${\mathrm{g}}_0=4000,{\mathrm{g}}_1=0$, and $\lambda =0.2$. There are eight different phases marked by A-H. The white region represents instability against dipolar collapse.

Figure 2

Typical density distribution and spin texture of the system for (a) ${\mathrm{g}}_{\mathrm{dd}}=0.0$ and ${\mathrm{g}}_{\mathrm{soc}}=2.4$ and (b) ${\mathrm{g}}_{\mathrm{dd}}=100$ and ${\mathrm{g}}_{\mathrm{soc}}=2.4$, which correspond to the states represented in the A and B phases of Fig. 1, respectively. The arrows in the spin texture represent the transverse spin vector $(F_{\mathrm{x}},F_{\mathrm{y}})$ with the background color representing $F_{\mathrm{z}}$.

Figure 3

Typical density distribution and spin texture of the system for (a) ${\mathrm{g}}_{\mathrm{dd}}=250$ and ${\mathrm{g}}_{\mathrm{soc}}=2.0$ and (b) ${\mathrm{g}}_{\mathrm{dd}}=300$ and ${\mathrm{g}}_{\mathrm{soc}}=1.5$, which correspond to the C and D phases in Fig. 1, respectively. The arrows in the spin texture represent the transverse spin vector $(F_{\mathrm{x}},F_{\mathrm{y}})$ with the background color representing $F_{\mathrm{z}}$.

Figure 4

Isodensity surfaces of the three components $|{\psi }_1{|}^2=0.001$ (red), $|{\psi }_0{|}^2=0.0007$ (green), and $|{\psi }_{-1}{|}^2=0.001$ (blue), which correspond to the C phase shown in Fig. 3. See [42] for a movie showing the 3D structure.

Figure 5

Isodensity surfaces of the three components $|{\psi }_1{|}^2=0.001$ (red), $|{\psi }_0{|}^2=0.0007$ (green), and $|{\psi }_{-1}{|}^2=0.0004$ (blue), which correspond to the D phase shown in Fig. 3. See [42] for a movie showing the 3D structure.

Figure 6

Typical density distribution and spin texture of the system for (a) ${\mathrm{g}}_{\mathrm{dd}}=300$ and ${\mathrm{g}}_{\mathrm{soc}}=0.6$, (b) ${\mathrm{g}}_{\mathrm{dd}}=700$ and ${\mathrm{g}}_{\mathrm{soc}}=0.1$, and (c) ${\mathrm{g}}_{\mathrm{dd}}=700$ and ${\mathrm{g}}_{\mathrm{soc}}=0$, which correspond to the states represented by the E, G, and H phases in Fig. 1, respectively. The arrows in the spin texture represent the transverse spin vector $(F_{\mathrm{x}},F_{\mathrm{y}})$ with the background color representing $F_{\mathrm{z}}$.

Figure 7

Isodensity surfaces of the three components $|{\psi }_1{|}^2=0.001$ (red), $|{\psi }_0{|}^2=0.001$ (green), and $|{\psi }_{-1}{|}^2=0.001$ (blue), which correspond to the G phase shown in Fig. 6. See [42] for a movie showing the 3D structure.

Figure 8

Orbital angular momentum $〈L_{\mathrm{z}}〉$ as a function of ${\mathrm{g}}_{\mathrm{soc}}$ for ${\mathrm{g}}_{\mathrm{dd}}=700$. The vertical lines separate the phases and the solid curve indicates the trend.