Spontaneous magnetic ordering in a ferromagnetic spinor dipolar Bose-Einstein condensate

We study the spin dynamics of a spin-1 ferromagnetic Bose-Einstein condensate with magnetic dipole-dipole interaction (MDDI) based on the Gross-Pitaevskii and Bogoliubov theories. We find that various magnetic structures such as checkerboards and stripes emerge in the course of the dynamics due to the combined effects of spin-exchange interaction, MDDI, quadratic Zeeman and finite-size effects, and nonstationary initial conditions. However, the short-range magnetic order observed by the Berkeley group [Phys. Rev. Lett. 100, 170403 (2008)] is not fully reproduced in our calculations; the periodicity of the order differs by a factor of 3 and the checkerboard pattern eventually dissolves in the course of time. Possible reasons for the discrepancy are discussed.

Figure 1

Phase diagram of a spin-1 BEC in the absence of the MDDI, where the total longitudinal magnetization is fixed to be zero.

Figure 2

Quadratic Zeeman energy $q$ and helical pitch $\lambda$ dependences of the dynamical and Landau instabilities. The abscissa and ordinate show $k_x/2\pi$ and $k_z/2\pi$, respectively. Shown are $\left|\mathrm{Im}\hspace{0.5em}{\epsilon }_{\mathit{k}}\right|$ (red) and $-\mathrm{Re}\hspace{0.5em}{\epsilon }_{\mathit{k}}$ (blue) which correspond to the dynamical and Landau instabilities, respectively; they are calculated for $({\stackrel{\wedge }{\mathit{e}}}_1,{\stackrel{\wedge }{\mathit{e}}}_2,{\stackrel{\wedge }{\mathit{e}}}_3)=(\stackrel{\wedge }{\mathit{z}},\stackrel{\wedge }{\mathit{x}},\stackrel{\wedge }{\mathit{y}})$, $\mathit{\kappa }=(2\pi /\lambda )\stackrel{\wedge }{\mathit{z}}$, $n(x_3=0)=2.3\times {10}^{14}$ cm${}^{-3}$ and $d=1.0$ $\mu$m. No instability is found for (c), (f), and (i).

Figure 3

Direction $\theta$ of the magnetic fluctuation defined in Eq. (65) for the lowest-energy modes. Each panel corresponds to that in Fig. 2. The parameters are the same as those in Fig. 2.

Figure 4

Time evolution of (a) the transverse magnetization, (b) the longitudinal magnetization, and (c) the spin correlation function (70) in the absence of the MDDI. The initial spin configuration is uniform and $q/h=2.9$ Hz. The solid circles in (a) and (b) represent the TF radius (140 $\mu$m) at $y=0$. The size of each panel is $310\times 310$ $\mu$m. The peak-to-peak distance of the correlation function at $t=1.8$ s is about 70 $\mu$m.

Figure 5

Spin dynamics in the absence of the MDDI starting with a spin helix of $\lambda =60$ $\mu$m. Shown are the snapshots of (a) the transverse magnetization and (b) the longitudinal magnetization for $q/h=1.8$ Hz, and (c) the time evolution of $M_z$ for $q/h=0.46$ Hz (red), 1.8 Hz (green), and 2.9 Hz (blue). The size of each panel in (a) and (b) is $310\times 310$ $\mu$m.

Figure 6

Spin dynamics in the presence of the MDDI for $q/h=0.46$ Hz starting with a uniform spin structure. The corresponding unstable mode is characterized in Figs. 2j and 3j. Shown are snapshots of (a) transverse magnetization, (b) longitudinal magnetization, and (c) amplitude of the magnetization and (d) time developments of $M_{\perp }$ (red) and $M_z$ (green). The size of each panel is $310\times 310$ $\mu$m in (a) and (b), and $155\times 155$ $\mu$m in (c).

Figure 7

Spin dynamics in the presence of the MDDI for $q/h=2.9$ Hz starting with a uniform spin structure. The corresponding unstable mode is shown in Figs. 2l and 3l. Shown are snapshots of (a) transverse magnetization and (b) longitudinal magnetization and (c) time evolution of $M_{\perp }$ (red) and $M_z$ (green). The size of each panel in (a) and (b) is $310\times 310$ $\mu$m.

Figure 8

Spin dynamics in the presence of the MDDI for $q/h=0.46$ Hz starting with a spin helix with a pitch of $\lambda =120\hspace{0.5em}\mu$m in (a) and (b) and that of $\lambda =60\hspace{0.5em}\mu$m in (c) and (d). Shown are snapshots of (a) and (c) transverse magnetization and (b) and (d) longitudinal magnetization. (e) Time evolution of $M_{\perp }$ and $M_z$. The size of each panel in (a)–(d) is $310\times 310$ $\mu$m. Examples for the Mermin-Ho vortex (MHV) and polar-core vortex (PCV) are enlarged at $t=2.3$ s in (a) and (b) and then (c) and (d), respectively, where the vortex core of the MH vortex is magnetized and that of the polar-core vortex is unmagnetized. In both cases of MH and polar-core vortices, the transverse magnetization changes by $2\pi$ around the vortex core.

Figure 9

Spin dynamics in the presence of the MDDI for $q/h=2.9$ Hz starting with a spin helix with a pitch of $\lambda =120\hspace{0.5em}\mu$m in (a) and (b) and that of $\lambda =60\hspace{0.5em}\mu$m in (c) and (d). Shown are snapshots of transverse magnetization [(a) and (c)] and longitudinal magnetization [(b) and (d)]. (e) Time evolution of $M_{\perp }$ and $M_z$. The size of each panel in (a)–(d) is $310\times 310$ $\mu$m. The sense of twisting of the spin helix dose not change in (a), while it is reversed in (c) at $t=2.3$ s.

Figure 10

(a) Snapshots of spin configuration at $t=0.82$ s in the absence of the MDDI starting with a uniform spin configuration for $q/h=2.9$ Hz, where each panel shows the distribution of transverse magnetization $\arg ({\mathrm{f̄}}_{+})$ (left), longitudinal magnetization ${\mathrm{f̄}}_z/{\mathrm{n̄}}_{\mathrm{tot}}(0)$ (middle), and spin correlation function $g$ (right). The peak-to-peak distance of the correlation function is about $25\hspace{0.5em}\mu$m. (b) Time development of transverse magnetization in the absence of the MDDI starting with a spin helix of pitch $\lambda =60$ $\mu$m for $q/h=1.8$ Hz. The domain size of the checkerboard pattern in (b) is about 20 $\mu$m. The solid curves in the distribution of longitudinal and transverse magnetizations represent the TF boundaries at $y=0$. The size of each panel is $45\times 359$ $\mu$m. Note that the color scales of $\arg ({\mathrm{f̄}}_{+})$ in (a) and (b) are different.

Figure 11

Magnetic structures developed from a uniform spin configuration in the presence of the MDDI at $q/h=0.46$ Hz [(a) and (b)] and 2.9 Hz [(c)], where (a) and the left panel in (c) show the distributions of the longitudinal magnetization, while (b) and the right panel in (c) show the direction of the transverse magnetization. The longitudinal magnetization is scaled so that ${\mathrm{f̄}}_z^{\max }=1.0$ in (a), and ${\mathrm{f̄}}_z^{\max }=0.6$ in the left panel of (c). The transverse magnetization forms a helical structure in both cases for $q/h=0.46$ and 2.9 Hz. The size of each panel is $45\times 359$ $\mu$m.

Figure 12

(a) Time development of the direction of the transverse magnetization with the MDDI starting with a spin helix of pitch $\lambda =60\hspace{0.5em}\mu$m for $q/h=1.8$ Hz. (b) Snapshot of the longitudinal magnetization at $t=4.0$ s. The size of each panel is $45\times 359$ $\mu$m. The solid circles and dotted circles in (a) at 4.0 s and (b) indicate the locations of the polar-core vortex (PCV) and Mermin-Ho vortex (MHV), respectively.

Figure 13

Time evolution of spin fluctuations of (a), (c), (d) $M_z\equiv \int d\mathit{r}|f_z(\mathit{r})|/N$ and (b) $M_y\equiv |\int d\mathit{r}{f}_y(\mathit{r})|/N$, where ${\sigma }_{\alpha }=0.1$ means ${\sigma }_{\alpha }=0.1$ and other components vanish, etc. In (c) and (d), $\sigma =0$ and 0.1 mean ${\sigma }_{\alpha }={\sigma }_{\beta }={\sigma }_{\gamma }={\sigma }_{\delta }=0$ and 0.1, respectively.

Figure 14

Magnetic structure at $t=5.3$ s starting from a uniform spin structure for $q/h=0.46$ Hz with the initial noise of (a) ${\sigma }_{\alpha }=0.03$ and ${\sigma }_{\beta }={\sigma }_{\gamma }={\sigma }_{\delta }=0$ and (b) ${\sigma }_{\beta }=0.03$ and ${\sigma }_{\alpha }={\sigma }_{\gamma }={\sigma }_{\delta }=0$. The length scale of the fluctuation in the transverse magnetization is smaller (larger) than that in the longitudinal magnetization in (a) [(b)]. The size of each panel is $45\times 359$ $\mu$m.

Figure 15

Stationary configuration of (a) longitudinal magnetization ${\mathrm{f̄}}_z/{\mathrm{n̄}}_{\mathrm{tot}}(0)$ and (b) transverse magnetizations $\arg ({\mathrm{f̄}}_{+})$ for the trap geometry and the number of atoms used in Ref. [2]. The condensate is almost fully magnetized in all cases (see Fig. 16). The size of each panel is $45\times 359$ $\mu$m.

Figure 16

$q$ dependence of the amplitude of the transverse magnetization $M_{\perp }=\int d\mathit{r}\sqrt{f_x^2+f_y^2}/N$, longitudinal magnetization $M_z=\int d\mathit{r}\left|f_z\right|/N$, and total magnetization $M=\int d\mathit{r}\sqrt{f_x^2+f_y^2+f_z^2}/N$, of the stationary state for the trap geometry and the number of atoms used in Ref. [2].

Figure 17

(a) Spin configurations around vortices in Eqs. (77, 78, 79), where the arrows show the direction of the transverse magnetization, P indicates the polar core, and $\odot$ and $\otimes$ mean that the magnetization at the vortex core points $+\stackrel{\wedge }{\mathit{z}}$ and $-\stackrel{\wedge }{\mathit{z}}$ directions, respectively. (b) Two configurations are topologically equivalent if the core structure is the same. (c) Unit of a vortex lattice that exhibits a periodic structure of the transverse magnetization. (d)–(e) Possible sets of vortices to form a vortex lattice.

Figure 18

Spin dynamics starting with the spin-vortex lattice shown in Fig. 17e at $q/h=1.8$ Hz for the trap geometry and the number of atoms used in Ref. [2]. (a)–(c) Transverse magnetization $\arg ({\mathrm{f̄}}_{+})$ (left), longitudinal magnetization ${\mathrm{f̄}}_z$ (middle), and spin correlation function $g$ (right) calculated in the presence of the MDDI. (d) Spin correlation function $g$ calculated in the absence of the MDDI. Note that the periodic pattern in (d) is quite similar to that in the right panel of (b). The size of each panel is $45\times 359$ $\mu$m.