Title: Self-Ordered Supersolid in Spinor Condensates with Cavity-Mediated Spin-Momentum-Mixing Interactions

Abstract: Ultracold atoms with cavity-mediated long-range interactions offer a promising platform for investing novel quantum phenomena. Exploiting recent experimental advancements, we propose an experimental scheme to create self-ordered supersolid in spin-$1/2$ condensates confined within an optical cavity. The interplay of cavity and pump fields gives rise to supersolid square and plane wave phases, comprehensively described by the two-component Tavis-Cummings model. We show that the self-ordered supersolid phase exhibits an undamped gapless Goldstone mode over a wide parameter range. This proposal, achievable with current experimental setups utilizing identical laser configurations, is in contrast to the realization of checkerboard supersolidity, which hinges on constructing a $U(1)$ symmetry by utilizing two ${\cal Z}_2$ symmetries with precisely matched atom-cavity coupling in multimode resonators. By employing the superradiant photon-exchange process, we realize for the first time cavity-mediated spin-momentum-mixing interactions between highly correlated spin and momentum modes, analogous to that observed spin-mixing in spin-1 condensates. Our scheme provides a unique platform for realizing spin-momentum squeezing and spatially distributed multipartite entanglement.