Spin-Orbit-Coupled Bose-Einstein Condensates in a One-Dimensional Optical Lattice

We investigate a spin-orbit-coupled Bose-Einstein condensate loaded into a translating optical lattice. We experimentally demonstrate the lack of Galilean invariance in the spin-orbit-coupled system, which leads to anisotropic behavior of the condensate depending on the direction of translation of the lattice. The anisotropy is theoretically understood by an effective dispersion relation. We experimentally confirm this theoretical picture by probing the dynamical instability of the system.

Figure 1

Spin-orbit-coupled ${}^{87}\mathrm{Rb}$ BEC in a one-dimensional optical lattice. (a) Experimental geometry. The BEC (yellow hashed) is confined in an optical dipole trap (solid green). Two sets of laser beams intersect the BEC at a 45° angle, generating the spin-orbit coupling (white arrow) and a translating optical lattice (striped arrows). (b) Raman coupling scheme in the $F=1$ manifold of ${}^{87}\mathrm{Rb}$ with detuning $\delta$. (c) Typical band structure $E_{\pm }(k_z)$ of $H_{\text{SOC}}$ with the color (gray scale) indicating the spin polarization, defined as the relative population difference of the bare spin components $(|{\psi }_{\uparrow }{|}^2-|{\psi }_{\downarrow }{|}^2)/(|{\psi }_{\uparrow }{|}^2+|{\psi }_{\downarrow }{|}^2)$. The BEC is prepared at the minimum of the lower band (circle). The arrows indicate a possible two-photon coupling due to the lattice translating with negative (dashed) or positive (solid) velocity. (d) Bloch spectrum of a stationary optical lattice in the presence of spin-orbit coupling. The lines correspond to $E_{\pm }(k_z)$ and $E_{\pm }(k_z+2n{k}_{\text{lat}})$, where $n$ is an integer. The spin composition is encoded in the line color (gray scale). The parameters used for (c) and (d) are $\hslash \delta =1.6E_{\text{Ram}}$, $\hslash \mathrm{\Omega }=2E_{\text{Ram}}$ with the additional parameters $U_0=-1.4E_{\text{lat}}$ and $v=0$ for (d).

Figure 2

Effective band structure as a function of the lattice velocity. The thick green lines indicate the position at which the BEC is placed in the experiments. (a) BEC with spin-orbit coupling and $\hslash \delta =1.6E_{\text{Ram}}$ as shown in Fig. 3. (b) BEC without spin-orbit coupling as in Fig. 4. The numbers in the graphs indicate the order of the associated multiphoton resonances.

Figure 3

Dynamical instability of the spin-orbit-coupled BEC as a function of lattice speed with (a)–(d) $\hslash \delta /E_{\text{Ram}}=\{3.2,1.6,0.8,0.4\}$, respectively. The strength of the dynamical instability is measured experimentally by the loss rate of atoms in the BEC (upper panels), while theoretically it is represented by the largest growth rate of Bogoliubov excitations (lower panels). Each resonance (vertical line) is labeled with the number of photons generating the band edge, with underlined integers denoting resonances between the upper and lower spin-orbit bands. The solid red triangles (open blue circles) indicate the positive (negative) direction of the lattice motion.

Figure 4

Dynamical instability of the BEC without spin-orbit coupling as a function of lattice velocity. The strength of the dynamical instability is measured experimentally by the loss rate of atoms in the BEC (upper panels), while theoretically, it is represented by the largest growth rate of any Bogoliubov excitation (lower panels). Each resonance (vertical line) is labeled with the number of photons generating the band edge.