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Ultracold atoms trapped in laser-generated optical lattices serve as aversatile platform for quantum simulations. However, as these lattices areinfinitely stiff, they do not allow to emulate phonon degrees of freedom. Thisrestriction can be lifted in emerged optical lattices inside multimodecavities. Motivated by recent experimental progress in multimode cavity QED, wepropose a scheme to implement and study supersolid and droplet states withphonon-like lattice excitations by coupling a Bose gas to many longitudinalmodes of a ring cavity. The interplay between contact collisional andtunable-range cavity-mediated interactions leads to a rich phase diagram, whichincludes elastic supersolid as well as insulating droplet phases exhibitingroton-type mode softening for a continuous range of momenta across thesuperradiant phase transition. The non-trivial dynamic response of the systemto local density perturbations further proves the existence of phonon-likemodes.

Tuning photon-mediated interactions in a multimode cavity: from supersolid to insulating droplets hosting phononic excitations