  We study the phenomenon of localization and delocalization in a circuit-QED
network with connectivity varying from finite-range to all-to-all coupling. We
find a fascinating interplay between interactions and connectivity. In
particular, we consider (i) Harmonic (ii) Jaynes-Cummings and (iii)
Bose-Hubbard networks. We start with the initial condition where one of the
nodes in the network is populated and then let it evolve in time. The time
dynamics and steady state characterize the features of localization
(self-trapping) in these large-scale networks. For the case of Harmonic
networks, exact analytical results are obtained and we demonstrate that
all-to-all connection shows self-trapping whereas the finite-ranged
connectivity shows delocalization. The interacting cases (Jaynes-Cummings,
Bose-Hubbard networks) are investigated both via exact quantum dynamics and
semi-classical approach. We obtain an interesting phase diagram when one varies
the range of connectivity and the strength of the interaction. We investigate
the consequence of imperfections in the cavity/qubit and the role of inevitable
disorder. Our results are relevant especially given recent experimental
progress in engineering systems with long-range connectivity.
