  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. In this
work, 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 feasible, 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. Additionally we show that
the chosen representation only induces small systematic errors, and that the
experimentally measured and theoretically predicted phase boundaries agree
reasonably. We thus demonstrate a new approach for the quantitative numerical
determination of the superfluid--Mott-insulator phase boundary.
