Many-body phases of a planar Bose-Einstein condensate with cavity-induced spin-orbit coupling

We explore the many-body phases of a two-dimensional Bose-Einstein condensate with cavity-mediated dynamic spin-orbit coupling. By help two transverse non-interfering, counterpropagating pump lasers and single standing-wave cavity mode, degenerate Zeeman sub-levels quantum gas are Raman coupled in double-$\Lambda$-configuration. Beyond critical strength mode is populated via coherent superradiant scattering from lasers, leading to appearance dynamical coupling for atoms. identify three distinct atomic photonic properties: normal ``homogeneous'' phase, ``spin-helix'' ``supersolid spin-density-wave'' phase. The latter exhibits an emergent periodic density distribution orthorhombic centered rectangular-lattice structure due interplay between photon into resonator collision-induced momentum lattice spacing crystal set by