  The competition between short-range and cavity-mediated infinite-range
interactions in a cavity-boson system leads to the existence of a superfluid
phase and a Mott-insulator phase within the self-organized regime. We
quantitatively compare the steady-state phase boundaries of this transition
measured in experiments and simulated using the Multiconfigurational
Time-Dependent Hartree Method for Indistinguishable Particles. To make the
problem computationally viable, we represent the full system by the exact
many-body wave function of a two-dimensional four-well potential. We argue that
the validity of this representation comes from the nature of both the
cavity-atomic system and the Bose-Hubbard physics, and verify that it only
induces small systematic errors. The experimentally measured and theoretically
predicted phase boundaries agree reasonably. We thus propose a new approach for
the quantiative numerical determination of the superfluid--Mott-insulator phase
boundary.
