  We present a unifying theoretical framework that describes recently observed
many-body effects during the interrogation of an optical lattice clock operated
with thousands of fermionic alkaline earth atoms. The framework is based on a
many-body master equation that accounts for the interplay between elastic and
inelastic p-wave and s-wave interactions, finite temperature effects and
excitation inhomogeneity during the quantum dynamics of the interrogated atoms.
Solutions of the master equation in different parameter regimes are presented
and compared. It is shown that a general solution can be obtained by using the
so called Truncated Wigner Approximation which is applied in our case in the
context of an open quantum system. We use the developed framework to model the
density shift and decay of the fringes observed during Ramsey spectroscopy in
the JILA 87Sr and NIST 171Yb optical lattice clocks. The developed framework
opens a suitable path for dealing with a variety of strongly-correlated and
driven open-quantum spin systems.
