Simulation of topological superconductors and their competing orders using photon-mediated interactions

Realizing and controlling the unconventional pairing featured by topological superconductors remains a central challenge. We introduce a cavity QED quantum simulator that engineers competing chiral $p_x+i{p}_y$ and $d_{x^2-y^2}+i{d}_{xy}$ orders by tailoring cavity-mediated couplings between atomic pseudospins that emulate momentum-dependent pairing channels. The desired spatially inhomogeneous cavity-mediated couplings can be engineered in a 2D optical lattice using incommensurate cavity-lattice wavelengths naturally occurring in cavity QED systems. This minimal and fully tunable platform enables controlled state preparation and continuous measurement of superconducting order parameters, revealing phases in both equilibrium and sudden-quench settings with a single dominant pairing channel, as well as coexistence regimes with competing pairing channels. Crucially, our implementation allows direct observation of topological transitions in and out of equilibrium, providing a powerful route to the quantum simulation of competing topological superconducting phases that remain elusive in solid-state and ultracold-atom systems.