  The Bose Hubbard model (BHM) of interacting bosons in a lattice has been a
paradigm in many-body physics, and it exhibits a Mott insulator (MI)-superfluid
(SF) transition at integer filling. Here a quantum simulator of the BHM using a
superconducting circuit is proposed. Specifically, a superconducting
transmission line resonator supporting microwave photons is coupled to a charge
qubit to form one site of the BHM, and adjacent sites are connected by a
tunable coupler. To obtain a mapping from the superconducting circuit to the
BHM, we focus on the dispersive regime where the excitations remain
photon-like. Standard perturbation theory is implemented to locate the
parameter range where the MI-SF transition may be simulated. This simulator
allows single-site manipulations and we illustrate this feature by considering
two scenarios where a single-site manipulation can drive a MI-SF transition.
The transition can be analyzed by mean-field analyses, and the exact
diagonalization was implemented to provide accurate results. The variance of
the photon density and the fidelity metric clearly show signatures of the
transition. Experimental realizations and other possible applications of this
simulator are also discussed.
