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