Spin-Orbit Coupling and Berry Phase with Ultracold Atoms in 2D Optical Lattices

We show how spin-orbit coupling and Berry phase can appear in two-dimensional optical lattices by coupling atoms’ internal degrees of freedom to radiation. The Rashba Hamiltonian, a standard description of spin-orbit coupling for two-dimensional electrons, is obtained for the atoms under certain circumstances. We discuss the possibility of observing associated phenomena, such as the anomalous Hall and spin Hall effects, with cold atoms in optical lattices.

Figure 1

Spatial profile of the different terms of the potential (6): (a) $V_0$ term, (b) $V_1$ term, (c) $V_2$ term, and (d) $V_3$ term. The amplitude of a vector component is proportional to the length of the arrow. The inset in panel (a) shows the configuration of the laser wave vectors ${\mathbf{q}}_i$ (dashed arrows) and the lattice vectors ${\mathbf{k}}_i$ (solid arrows). Dots indicate wells A, triangles indicate wells B. Dashed lines show a unit cell.

Figure 2

(a) Dispersion of the lowest four bands in the potential (6). The dashed line corresponds to the case in which only a scalar potential is present, $V_0=1$. The solid line is for the case $V_0=1$, $V_1=V_3=0.1$, and $V_2=0$ [17]. (b) Berry curvature for the lowest band in the presence of an external field $h_0=0.005$. The band structure in the presence of $h_0$ is essentially the same as shown in (a), except for a splitting by $\approx 0.01$ at the degeneracy points $\Gamma$, K, and ${\mathrm{K}}^{\mathrm{\prime }}$.

Figure 3

(a) AHE. Trajectory of the wave packet center for a force $\mathbf{f}$ in the positive $y$ direction. Small dots are period-averaged positions. The initial position is indicated by the large dot in the origin. (b) SHE. Distribution of the spin vertical component as a function of momentum when finite force $\mathbf{f}=f_x\widehat{x}$ is applied. Spin is positive for $k_y>0$ and negative for $k_y<0$. The arrows show effective momentum dependent field ${\Delta }_{\mathbf{k}}$ close to the $\Gamma$ point.