  We propose a hybrid quantum architecture for engineering a photonicMott
insulator-superfluid phase transition in a two-dimensional (2D) square lattice
of a superconducting transmission line resonator (TLR) coupled to a single
nitrogen-vacancy (NV) center encircled by a persistent current qubit. The
localization-delocalization transition results from the interplay between the
on-site repulsion and the nonlocal tunneling. The phase boundary in the case of
photon hopping with real-valued and complex-valued amplitudes can be obtained
using the mean-field approach. Also, the quantum jump technique is employed to
describe the phase diagram when the dissipative effects are considered. The
unique feature of our architecture is the good tunability of effective on-site
repulsion and photon-hopping rate, and the local statistical property of TLRs
which can be analyzed readily using presentmicrowave techniques. Our work opens
new perspectives in quantum simulation of condensed-matter and many-body
physics using a hybrid spin circuit-QED system. The experimental challenges are
realizable using currently available technologies.
