  Understanding the Hubbard model is crucial for investigating various quantum
many-body states and its fermionic and bosonic versions have been largely
realized separately. Recently, transition metal dichalcogenides heterobilayers
have emerged as a promising platform for simulating the rich physics of the
Hubbard model. In this work, we explore the interplay between fermionic and
bosonic populations, using a $\rm{WS}_2$/$\rm{WSe}_2$ heterobilayer device that
hosts this hybrid particle density. We independently tune the fermionic and
bosonic populations by electronic doping and optical injection of electron-hole
pairs, respectively. This enables us to form strongly interacting excitons that
are manifested in a large energy gap in the photoluminescence spectrum. The
incompressibility of excitons is further corroborated by measuring exciton
diffusion, which remains constant upon increasing pumping intensity, as opposed
to the expected behavior of a weakly interacting gas of bosons, suggesting the
formation of a bosonic Mott insulator. We explain our observations using a
two-band model including phase space filling. Our system provides a
controllable approach to the exploration of quantum many-body effects in the
generalized Bose-Fermi-Hubbard model.
