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We investigate harmonically-trapped, laser-pumped bosons with infinite-rangeinteractions induced by a dissipative high-finesse red-detuned optical cavitywith numerical and analytical methods. We obtain multiple cavity and atomicobservables as well as the full phase diagram of the system using themulticonfigurational time-dependent Hartree method for indistinguishableparticles (MCTDH-X) approach. Besides the transition from an unorganized normalphase to a superradiant phase where atoms self-organize, we focus on anin-depth investigation of the self-organized superfluid to self-organized Mottinsulator phase transition in the superradiant phase as a function of thecavity-atom coupling. The numerical results are substantiated by an analyticalstudy of an effective Bose-Hubbard model. We numerically analyze cavityfluctuations and emergent strong correlations between atoms in the many-bodystate across the Mott transition via the atomic density distributions andGlauber correlation functions. Unexpectedly, the weak harmonic trap leads tofeatures like a lattice switching between the two symmetry-broken$\mathbb{Z}_2$ configurations of the untrapped system and a reentrance ofsuperfluidity in the Mott insulating phase. Our analytical considerationsquantitatively explain the numerically observed correlation features.

Superfluid--Mott insulator transition of ultracold superradiant bosons in a cavity