  Confining ultracold gases in cavities creates a paradigm of quantum trapping
potentials. We show that this allows to bridge models with global collective
and short-range interactions as novel quantum phases possess properties of
both. Some phases appear solely due to quantum light-matter correlations. Due
to global, but spatially structured, interaction, the competition between
quantum matter and light waves leads to multimode structures even in
single-mode cavities, including delocalized dimers of matter-field coherences
(bonds), beyond density orders as supersolids and density waves.
