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Multimode optical cavities can be used to implement interatomic interactionswhich are highly tunable in strength and range. For bosonic atoms trapped in anoptical lattice, cavity-mediated interactions compete with the short-rangeinteratomic repulsion, which we study using an extended Bose-Hubbard model.Already in a single-mode cavity, where the corresponding interaction has aninfinite range, a rich phase diagram has been experimentally observed,featuring density-wave and supersolid self-organized phases in addition to theusual superfluid and Mott insulator. Here we show that, for any finite range ofthe cavity-mediated interaction, quantum self-bound droplets dominate theground state phase diagram. Their size and in turn density is not externallyfixed but rather emerges from the competition between local repulsion andfinite-range attraction. Therefore, the phase diagram becomes very rich,featuring both compressible superfluid/supersolid as well as incompressibleMott and density-wave droplets. Additionally, we observe droplets with acompressible core and incompressible outer shells.

Light-induced quantum droplet phases of lattice bosons in multimode cavities