  We determine the incoherent spectrum, squeezing properties and second-order
correlation function of the collective atomic degrees of freedom in absorptive
optical bistability. This is accomplished via a linearized Fokker-Planck
equation in the positive P representation, guided by the analysis of [H. J.
Carmichael, Phys. Rev. A 33, 3262 (1986)] which does not resort to adiabatic
elimination. We focus on the regimes of weak and strong intracavity excitation,
addressing the good-cavity and bad-cavity limits as well as the limit of
collective strong coupling. Adiabatic elimination of the intracavity field
sustained by an auxiliary resonator coupled to the ensemble is used to probe
the atomic correlations via the formation of a collective emission channel. We
compare to the corresponding expressions for the forwards-scattered light with
reference to experimental results, discussing key differences between the lower
and upper branch of the steady-state semiclassical bistability curve. Our
analysis is carried out around the stable states situated far away from the
turning points, where analytical expressions can be obtained self-consistently,
demonstrating a clear departure from classical behavior.
