In this paper, we propose to directly detect Mott lobes, i.e. the order parameter 〈a〉, describing the Mott insulator (MI) to superfluid (SF) quantum phase transition of the Bose–Hubbard (BH) model. By weakly coupling the system to an environment in the SF phase, the U(1) symmetry breaking of the system is simulated, and the order parameter can be read from the AC Josephson current between the system and the environment. The relation between the order parameter and the Josephson current is obtained from both the mean-field theory approach and an exact numerical simulation of a finite-size example. Our numerical simulations show that the profile of the order parameter read from the Josephson current is different from it predicted by the mean-field theory, but similar to it in a system whose U(1) symmetry is broken by a weak field proportional to a + a†. This proposal is feasible in optical lattices.
Isolated hybrid normal/superconducting ring in a magnetic flux: From persistent current to Josephson current
