Spin Hall effect in a spinor dipolar Bose-Einstein condensate

We theoretically show that the spin Hall effect arises in a Bose-Einstein condensate (BEC) of neutral atoms interacting via the magnetic dipole-dipole interactions (MDDIs). Since the MDDI couples the total spin angular momentum and the relative orbital angular momentum of two colliding atoms, it works as a spin-orbit coupling. Thus, when we prepare a BEC in a magnetic sublevel $m=0$, thermally and quantum-mechanically excited atoms in the $m=1$ and $-1$ states feel the Lorentz-like forces in the opposite directions. This is the origin for the emergence of the spin Hall effect. We define the mass-current and spin-current operators from the equations of continuity and calculate the spin Hall conductivity from the off-diagonal current-current correlation function within the Bogoliubov approximation. We find that the correction of the current operators due to the MDDI significantly contributes to the spin Hall conductivity. A possible experimental situation is also discussed.

Figure 1

Schematic picture of the spin-density waves in the $\uparrow /\downarrow$ modes propagating along the $x$ direction. In the $\uparrow$ mode ($\downarrow$ mode), the spin density arises parallel (perpendicular) to the momentum $\mathbf{k}\parallel \widehat{x}$.

Figure 2

The directions of spin-dependent Lorenz-like force $F_{+}\left(\mathbf{k}\right)$ acting on spin-up atoms in the momentum space.

Figure 3

(a) Spin Hall conductivity ${\sigma }_{xy}^{\mathrm{SH}}$ as a function of temperature $c_{1}n/\left(k_{\mathrm{B}}T\right)$ and the MDDI strength $c_{\mathrm{dd}}/c_1$ at the quadratic Zeeman energy of $q_{\mathrm{Z}}/\left(c_{1}n\right)=0.01$, where $c_1$ is the strength of the spin-exchange interaction and $n$ is the number density of the system. The conductivity is scaled in units of ${\left(\hslash \xi \right)}^{-1}$, where $\xi \equiv \hslash /\sqrt{2M{c}_{1}n}$ is the spin healing length of the condensate. The system of spin-1 ${}^{23}\mathrm{Na}$ atoms traces the yellow solid line in (a) at $c_{\mathrm{dd}}/c_1=1.3\times {10}^{-2}$. (b) The same as (a) but for fixed $c_{\mathrm{dd}}/c_1=1.3\times {10}^{-2}$.

Figure 4

The spin Hall conductivity ${\sigma }_{xy}^{\mathrm{SH}}$ as a function of the quadratic Zeeman energy $q_{\mathrm{Z}}/\left(c_{1}n\right)$ at $c_{1}n/\left(k_{\mathrm{B}}T\right)=0.01$ and $c_{\mathrm{dd}}/c_1=1.3\times {10}^{-2}$. The conductivity diverges at $q_{\mathrm{Z}}=0$ because the magnon modes become gapless (see text).

Figure 5

The transverse velocity $\langle J_x^M\rangle /N^{\mathrm{ex}}$ of the excited atoms in response to the magnetic field gradient of $B^{\prime }$=$1.0\times {10}^{-2}$ mG/mm. Calculated for a spin-1 ${}^{23}\mathrm{Na}$ BEC with the number density $n=2.3\times {10}^{20}{\mathrm{m}}^{-3}$. Here, $N^{\mathrm{ex}}$ is the number of atoms excited in the $m=\pm 1$ states, and the red (solid), purple (dashed), and blue (dash-dotted) curves show the results for $c_{1}n/\left(k_{\mathrm{B}}T\right)=0.1$, 0.01, and 0.001, respectively.