Bose-Einstein Condensates with Cavity-Mediated Spin-Orbit Coupling

We propose a novel scheme to generate the spin-orbit coupling for a condensate placed inside an optical cavity by using a standing wave and a traveling wave. It is shown that the interplay of the laser lights and the cavity gives rise to rich quantum phases. Our scheme also generates an nontrivial lattice of the magnetic flux, which may facilitate the study of the exotic quantum phases.