Spin textures in condensates with large dipole moments

We have solved numerically the ground states of a Bose-Einstein condensate in the presence of dipolar interparticle forces using a semiclassical approach. Our motivation is to model, in particular, the spontaneous spin textures emerging in quantum gases with large dipole moments, such as ${}^{52}\mathrm{Cr}$ or $\mathrm{Dy}$ condensates, or ultracold gases consisting of polar molecules. For a pancake-shaped harmonic (optical) potential, we present the ground-state phase diagram spanned by the strength of the nonlinear coupling and dipolar interactions. In an elongated harmonic potential, we observe a helical spin texture. The textures calculated according to the semiclassical model in the absence of external polarizing fields are predominantly analogous to previously reported results for a ferromagnetic $F=1$ spinor Bose-Einstein condensate, suggesting that the spin textures arising from the dipolar forces are largely independent of the value of the quantum number $F$ or the origin of the dipolar interactions.

Figure 1

Magnetization $\mathbf{M}(\mathbf{r})$ (arrows) and density $n(\mathbf{r})$ (color) of the (a) flare and (b) spin vortex states for $g=100$, $g_d/g=0.15$ in a trap with aspect ratio $\lambda =10$. Both quantities are shown in the $z=0$ plane, the density being nearly Gaussian in the axial direction and $M_z$ small. For the chosen parameter values, the flare state in (a) is the energetically favored configuration. Panel (c) illustrates a spin vortex state with opposite spin winding compared to (b). Such a state is not energetically favorable for the parameter values considered in this work. Each panel has dimensions $8\hspace{0.5em}{a}_r\times 8\hspace{0.5em}{a}_r$.

Figure 2

Ground-state phase diagram of a dipolar condensate in a harmonic trap with aspect ratio $\lambda =10$. The effective contact interaction coupling constant $g$ is represented in logarithmic scale on the vertical axis. The horizontal axis, measuring the strength of dipolar interactions through the ratio $g_d/g$, has a linear scale. The phase diagram is divided into three regions: flare [Fig. 1a], spin vortex [Fig. 1b], and the region where the spin vortex becomes unstable against collapse.

Figure 3

Second derivative of the total energy with respect to a scaling transformation of the form given in Eq. (8) as a function of $g_d/g$ shown in units of the corresponding quantity at $g_d/g=0.10$. The curves correspond to the parameter values $g={10}^4$ (solid), $g={10}^3$ (dashed), and $g={10}^2$ (dash-dotted). The value of $g_d/g$ for which the second derivative vanishes indicates the critical strength beyond which the spin vortex state becomes unstable against collapse. The inset shows the ratio of axial and radial scaling for which the minimal value of the bracketed expression in the main figure is obtained.

Figure 4

Spin textures in the flare (a)–(c), spin vortex (d)–(f), and spin helix (g)–(i) states in a cigar-shaped trapping geometry with aspect ratio $\lambda =0.20$ and dipolar interaction strengths $g_d/g=0.030,$ $0.080$, and $0.030$, respectively. The arrows illustrate the magnetization within a given radial cross section projected onto the $x$-$y$ plane, whereas the color refers to the axial magnetization $M_z$ normalized with respect to maximal magnetization within each state separately. Each panel has dimensions $12\hspace{0.5em}{a}_r\times 12\hspace{0.5em}{a}_r$.

Figure 5

The twisting angle $\alpha (z)$ (a), tilting angle $\beta (z)$ (b), and the axial column density $n_z(z)$ (c) for three values of the dipolar interaction strength $g_d/g=(0.075,0.15,0.235)$ shown with solid, dashed, and dash-dotted lines, respectively. The total magnetization ${\mathbf{M}}_{\mathrm{tot}}$, which is directed along the $z$ axis by symmetry, is shown in (d) for the aspect ratios $\lambda =0.10$ (dashed), $0.20$ (solid), and $0.50$ (dash-dotted). The dots in (d) refer to the values of $g_d/g$ used in (a), (b), and (c). Units are shown in brackets.

Figure 6

(a) $M_x$ (solid) and $M_y$ (dashed) on the $z$ axis as a function of $z$ in the spin helix state for $\lambda =0.10$ and $g_d/g=0.20$. The dash-dotted curve in the inset shows $M_z$ along the $y$ axis in the range $[-5,5]a_r$. The vertical axis spans the interval $[-0.0028,0.0028]N{\mu }_M/a_r^3$ both in the main figure and the inset. (b) The column density $n(x,z)=\int n(\mathbf{r})\mathit{dy}$ illustrating density oscillations characteristic of the spin helix state for strong dipolar interactions. The parameters are as in (a), and the field of view is $8\hspace{0.5em}{a}_r\times 60\hspace{0.5em}{a}_r$. (c) Wave vector of the spin helix state for $g_d/g=(0.080,0.20,0.23)$ shown with solid, dashed, and dash-dotted curves, respectively. The peaks at the ends of the cloud are finite-size effects. Units are shown in brackets.