  We analyze the generation of spin-squeezed states via coupling of three-level
atoms to an optical cavity and continuous quantum measurement of the
transmitted cavity field in order to monitor the evolution of the atomic
ensemble. Using analytical treatment and microscopic simulations of the
dynamics, we show that one can achieve significant spin squeezing, favorably
scaling with the number of atoms $N$. However, contrary to some previous
literature, we clarify that it is not possible to obtain Heisenberg scaling
without the continuous feedback that is proposed in optimal approaches. In
fact, in the adiabatic cavity removal approximation and large $N$ limit, we
find the scaling behavior $N^{-2/3}$ for spin squeezing and $N^{-1/3}$ for the
corresponding protocol duration. These results can be obtained only by
considering the curvature of the Bloch sphere, since linearizing the collective
spin operators tangentially to its equator yields inaccurate predictions. With
full simulations, we characterize how spin-squeezing generation depends on the
system parameters and departs from the bad cavity regime, by gradually mixing
with cavity-filling dynamics until metrological advantage is lost. Finally, we
discuss the relevance of this spin-squeezing protocol to state-of-the-art
optical clocks.
