  Ultracold atomic gases in optical lattices have proven to be a controllable,
tunable and clean implementation of strongly interacting quantum many-body
systems. An essential prospect for such quantum simulators is their ability to
map out the phase diagram of fundamental many-body model Hamiltonians. However,
the results need to be validated first for representative benchmark problems
via state-of-the-art numerical methods of quantum many-body theory. Here we
present the first ab-initio comparison between experiments and quantum Monte
Carlo simulations for strongly interacting Bose gases on a lattice for large
systems (up to N = 3e5 particles). The comparison has enabled us to perform
thermometry for the interacting quantum gas and to experimentally determine the
finite temperature phase diagram for bosonic superfluids in an optical lattice.
Our results reveal a downshift of the critical temperature as the transition to
the Mott insulator is approached.
