Dark Superradiance in Cavity-Coupled Polar Molecular Bose-Einstein Condensates

We propose an experimental scheme to realize phase transition from {\it dark superradiance} to conventional superradiance in a microwave cavity coupled to polar molecules. The competition between cavity-mediated infinite-range repulsions and finite-range attractive dipolar interactions stabilizes a variety of exotic quantum phases, including vortex, vortex anti-vortex pairs, and superradiant phase, all emerging without external driving fields. In vortex phase associated with {\it dark superradiance}, cavity remains in vacuum state while profoundly reshaping the condensate's ground-state wave functions. In particular, the spin configuration locally parallel but globally anti-parallel is a direct consequence of competing for two nonlocal interactions. Beyond Dicke paradigm, dipolar dressing of condensate enables access to an unexplored regime of repulsion-dominated superradiance. A Bogoliubov analysis of low-energy excitation spectrum confirms that the condensate remains stable, avoiding roton-maxon induced collapse even in strongly dipolar regime.