Spontaneous Spin Textures in Dipolar Spinor Condensates

We have mapped out a detailed phase diagram that shows the ground state structure of a spin-1 condensate with magnetic dipole-dipole interactions. We show that the interplay between the dipolar and the spin-exchange interactions induces a rich variety of quantum phases that exhibit spontaneous magnetic ordering in the form of intricate spin textures.

Figure 1

Phase diagram of dipolar spin-1 condensates of (a) ferromagnetic and (b) antiferromagnetic spin-exchange interaction. $\mathbf{s}=\mathbf{S}/n$ is the normalized local spin vector.

Figure 2

Ground state of the P phase for ferromagnetic coupling with $\lambda =2$ and $q=1$. (a) Density of each spin component along the $x$ axis. The total density is denoted by the dashed line. (b) and (c) are the respective phase images for ${\varphi }_1$ and ${\varphi }_{-1}$ in the $z=0$ plane. The phase of ${\varphi }_0$ is a constant and is not shown here. The phases in the $z\ne 0$ plane are similar. (d) The vector plot of $\mathbf{S}$ in the $xy$ plane with the color map corresponding to the total density. Here, as well as in Figs. 3, 4, we adopt $a_{\mathrm{ho}}=(\hslash /m{\omega }_0{)}^{1/2}$ and $N/a_{\mathrm{ho}}^3$ as the units for length and density, respectively.

Figure 3

(a) The streamline plot of the spins for ferromagnetic coupling in the C phase with $\lambda =0.4$ and $q=1.6$. The color map represents $|{\mathbf{S}}_{\perp }|$ on these cross sections. (b) The vector plot of ${\mathbf{s}}_{\parallel }$ in the $xz$ plane. Same parameters as in (a). (c) The same as (b) except for $q=6$.

Figure 4

The spin structure in the S phase for antiferromagnetic coupling. (a) Spin projection in the $z=2.29$ plane. $\lambda =0.4$ and $q=1$. (b) Spin projection in the $y=0$ plane. $\lambda =0.8$ and $q=1$. (c) The spin structure for $\lambda =0.8$ and $q=1$. (d)–(f) The dipolar strength dependence of the toroidal moments for $\lambda =1.2$, 0.8, and 0.4, respectively.