  In this work, we study the $ U(1)/Z_2 $ Dicke model at a finite $ N $ by
using the $ 1/J $ expansion and exact diagonization. This model includes the
four standard quantum optics model as its various special limits. The $ 1/J $
expansions is complementary to the strong coupling expansion used by the
authors in arXiv:1512.08581 to study the same model in its dual $ Z_2/U(1) $
representation. We identify 3 regimes of the system's energy levels: the
normal, $ U(1) $ and quantum tunneling (QT) regime. The system's energy levels
are grouped into doublets which consist of scattering states and Schrodinger
Cats with even ( e ) and odd ( o ) parities in the $ U(1) $ and quantum
tunneling (QT) regime respectively. In the QT regime, by the WKB method, we
find the emergencies of bound states one by one as the interaction strength
increases, then investigate a new class of quantum tunneling processes through
the instantons between the two bound states in the compact photon phase. It is
the Berry phase interference effects in the instanton tunneling event which
leads to Schrodinger Cats oscillating with even and odd parities in both ground
and higher energy bound states. We map out the energy level evolution from the
$ U(1) $ to the QT regime and also discuss some duality relations between the
energy levels in the two regimes. We also compute the photon correlation
functions, squeezing spectrum, number correlation functions in both regimes
which can be measured by various experimental techniques. The combinations of
the results achieved here by $ 1/J $ expansion and those in arXiv:1512.08581 by
strong coupling method lead to rather complete understandings of the $ U(1)/Z_2
$ Dicke model at a finite $ N $ and any anisotropy parameter $ \beta $.
