Symmetry-breaking magnetization dynamics of spinor dipolar Bose-Einstein condensates

The symmetry-breaking magnetization dynamics of a spin-1 Bose-Einstein condensate (BEC) due to the dipole-dipole interaction are investigated using the mean-field and Bogoliubov theories. When a magnetic field is applied along the symmetry axis of a pancake-shaped BEC in the $m=0$ hyperfine sublevel, transverse magnetization develops breaking the chiral or axial symmetry. A variety of magnetization patterns are formed depending on the strength of the applied magnetic field. The proposed phenomena can be observed in ${}^{87}\mathrm{Rb}$ and ${}^{23}\mathrm{Na}$ condensates.

Figure 1

Imaginary part of Bogoliubov frequency $\omega$ for magnon excitation from the $m=0$ state of ${}^{87}\mathrm{Rb}$ atoms as a function of $P-Q$ with $Q=50\hslash {\omega }_{\perp }$. The excitation mode has $2(L-1)$-fold symmetry around the $z$ axis, where $L$ is defined in Eq. (6). The number of atoms is $N={10}^5$, and the radial and axial trap frequencies are ${\omega }_{\perp }=2\pi \times 100$ Hz and ${\omega }_z=2\pi \times 400$ Hz. The values of $P$ indicated by the arrows are used in Figs. 2 and 3.

Figure 2

Time evolution of the population of the $m=1$ state $\int \left|{\psi }_1\right|{}^{2}d\mathit{r}/N$ for the values of $p\equiv (P-Q)/(\hslash {\omega }_{\perp })$ indicated by the arrows in Fig. 1. The parameters are the same as those in Fig. 1. The populations of the $m=-1$ state are too small to be discerned at this scale of the ordinate.

Figure 3

Magnitude of the integrated transverse magnetization $|\int F_{+}\mathit{dz}|$ (left panels) and its direction arg $(\int F_{+}\mathit{dz})$ (right panels) at the first peaks of the curves in Fig. 2. The values of $p\equiv (P-Q)/(\hslash {\omega }_{\perp })$ used are indicated by the arrows in Fig. 1. The unit of $\int F_{+}\mathit{dz}$ is $NM{\omega }_{\perp }/\hslash$. The length of the vector is proportional to $|\int F_{+}\mathit{dz}|$. The field of view is $9.7\times 9.7$ $\mu \mathrm{m}$.

Figure 4

Imaginary part of the Bogoliubov frequency $\omega$ for magnon excitation from the $m=0$ state of ${}^{23}\mathrm{Na}$ atoms as a function of the linear Zeeman energy $P$. The range of $P$ is (a) $0⩽P/(\hslash {\omega }_{\perp })⩽0.5$ and (b) $1.5⩽P/(\hslash {\omega }_{\perp })⩽2$. The excitation mode has $2(L-1)$-fold symmetry around the $z$ axis, where $L$ is defined in Eq. (6). The number of atoms is $N={10}^6$, and the trap frequencies are the same as those in Fig. 1. The values of $P$ indicated by the arrows in (b) are used in Fig. 5.

Figure 5

Integrated transverse magnetization $|\int F_{+}\mathit{dz}|$ at the time when $\int \left|{\psi }_1\right|{}^{2}d\mathit{r}$ becomes the first maximum in time evolution for the values of $P$ indicated by the arrows in Fig. 4b. The unit of $|\int F_{+}\mathit{dz}|$ is $NM{\omega }_{\perp }/\hslash$. The field of view is $21\times 21$ $\mu \mathrm{m}$.