Spin-exchange-induced spin-orbit coupling in a superfluid mixture

We investigate the ground-state properties of a dual-species spin-1/2 Bose-Einstein condensate. One of the species is subjected to a pair of Raman laser beams that induces spin-orbit (SO) coupling, whereas the other species is not coupled to the Raman laser. In certain limits, analytical results can be obtained. It is clearly shown that, through the interspecies spin-exchange interaction, the second species also exhibits SO coupling. This mixture system displays a very rich phase diagram, with many of the phases not present in an SO-coupled single-species condensate. Our work provides a way of creating SO coupling in atomic quantum gases, and opens up an avenue of research in SO-coupled superfluid mixtures. From a practical point of view, the spin-exchange-induced SO coupling may overcome the heating issue for certain atomic species when subjected to Raman beams.

Figure 1

(a) Schematic representation of the dual-species BEC system subjected to a pair of Raman beams. (b) Atomic level structure for either species. We assume in this work that the Raman coupling strength ${\mathrm{\Omega }}_B$ for species $B$ is negligible [25].

Figure 2

(a), (b) Ground-state phase diagram obtained analytically and numerically, respectively, in the case of $N_A\gg N_B$. (c) Typical density profiles for various phases. ${\mathrm{\Omega }}_A$ is in units of the photon recoil energy $E_L=k_L^2/2m_A$. In each subplot in (c), we plot only the left (right) half of the density profiles for species $A$ ($B$), as the full profile is mirror symmetric about $x=0$. In this calculation, we take $m_A$ and $m_B$ to be the mass of ${}^{23}\mathrm{Na}$ and ${}^{87}\mathrm{Rb}$, respectively; and $N_A=2.5\times {10}^4=25N_B$. The box potential has a length of $L_A=100k_L^{-1}\approx 27\mu \text{m}$ for species $A$. To minimize the edge effects, we choose the box length for $B$ to be slightly smaller with $L_B=0.9L_A$. The interaction strengths are $g^A=6.48\times {10}^{-3}{E}_L/k_L$, $g^B=3.44\times {10}^{-3}{E}_L/k_L$, $g_{\uparrow \downarrow }^i=0.8g^i$, $\gamma =2.87\times {10}^{-3}{E}_L/k_L$. The densities in (c) are in units of $k_L$, and are renormalized such that $\int dx{\rho }_i\left(x\right)=1$.

Figure 3

(a) Ground-state phase diagram, and (b) typical density profiles for various phases, in the case of $N_A=N_B=1\times {10}^4$. Both species are confined in a box potential with length $L_A=L_B=100k_L^{-1}$. Other parameters used are the same as in Fig. 2. Note that the PST/$\mathcal{ST}$ phase has two degenerate ground states. The one not shown has the same density profiles in species $B$, but reversed density distributions for ${\rho }_{A,\uparrow }$ and ${\rho }_{A,\downarrow }$ as plotted in the first panel of (b).