  Insights into complex phenomena in quantum matter can be gained from
simulation experiments with ultracold atoms, especially in cases where
theoretical characterization is challenging. However these experiments are
mostly limited to short-range collisional interactions. Recently observed
perturbative effects of long-range interactions were too weak to reach novel
quantum phases. Here we experimentally realize a bosonic lattice model with
competing short- and infinite-range interactions, and observe the appearance of
four distinct phases - a superfluid, a supersolid, a Mott insulator and a
charge density wave. Our system is based on an atomic quantum gas trapped in an
optical lattice inside a high finesse optical cavity. The strength of the
short-ranged on-site interactions is controlled by means of the optical lattice
depth. The infinite-range interaction potential is mediated by a vacuum mode of
the cavity and is independently controlled by tuning the cavity resonance. When
probing the phase transition between the Mott insulator and the charge density
wave in real-time, we discovered a behaviour characteristic of a first order
phase transition. Our measurements have accessed a regime for quantum
simulation of many-body systems, where the physics is determined by the
intricate competition between two different types of interactions and the zero
point motion of the particles.
