  We investigate two key aspects of quantum systems by using the Tavis-Cummings
dimer system as a platform. The first aspect involves unraveling the
relationship between the phenomenon of self-trapping (or lack thereof) and
integrability (or quantum chaos). Secondly, we uncover {the possibility of}
mixed behavior in this quantum system using diagnostics based on random matrix
theory and make an in-depth study of classical-quantum correspondence. The
setup chosen for the study is precisely suited as it (i) enables a transition
from delocalized to self-trapped states and (ii) has a well-defined classical
limit, thereby amenable to studies involving classical-quantum conjectures. The
obtained classical model in itself has rich chaotic and ergodic properties
which were probed via maximal Lyapunov exponents. Furthermore, we present
aspects of chaos in the corresponding open quantum system and make connections
with non-Hermitian random matrix theory.
