  Exploring operational regimes of many-body cavity QED with multi-level atoms
remains an exciting research frontier for their enhanced storage capabilities
of intra-level quantum correlations. In this work, we consider an
experimentally feasible many-body cavity QED model describing a four-level
system, where each of those levels is formed from a combination of different
spin and momentum states of ultra-cold atoms in a cavity. The resulting model
comprises a pair of Dicke Hamiltonians constructed from pseudo-spin operators,
effectively capturing two intertwined superradiant phase transitions. The phase
diagram reveals regions featuring weak and strong entangled states of spin and
momentum atomic degrees of freedom. These states exhibit different dynamical
responses, ranging from slow to fast relaxation, with the added option of
persistent entanglement temporal oscillations. We discuss the role of cavity
losses in steering the system's dynamics into such entangled states and propose
a readout scheme that leverages different light polarizations within the
cavity. Our work paves the way to connect the rich variety of non-equilibrium
phase transitions that occur in many-body cavity QED to the buildup of quantum
correlations in systems with multi-level atom descriptions.
