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We utilize multilevel atoms trapped in a driven resonant optical cavity toproduce scalable multi-mode squeezed states for quantum sensing and metrology.While superradiance or collective dissipative emission by itself has beentypically a detrimental effect for entanglement generation in optical cavities,in the presence of additional drives it can also be used as an entanglementresource. In a recent work [Phys. Rev. Lett. 132, 033601 (2024)], we describeda protocol for the dissipative generation of two-mode squeezing in the darkstate of a six-level system with only one relevant polarization. There weshowed that up to two quadratures can be squeezed. Here, we develop ageneralized analytic treatment to calculate the squeezing in any multilevelsystem where atoms can collectively decay by emitting light into twopolarization modes in a cavity. We show that in this more general system up tofour spin squeezed quadratures can be obtained. We study how finite-sizeeffects constrain the reachable squeezing, and analytically compute the scalingwith $N$. Our findings are readily testable in current optical cavityexperiments with alkaline-earth-like atoms.

Driven-dissipative four-mode squeezing of multilevel atoms in an optical cavity