  We show how bond order emerges due to light mediated synthetic interactions
in ultracold atoms in optical lattices in an optical cavity. This is a
consequence of the competition between both short- and long-range interactions
designed by choosing the optical geometry. Light induces effective many-body
interactions that modify the landscape of quantum phases supported by the
typical Bose-Hubbard model. Using exact diagonalization of small system sizes
in one dimension, we present the many-body quantum phases the system can
support via the interplay between the density and bond (or matter-wave
coherence) interactions. We find numerical evidence to support that dimer
phases due to bond order are analogous to valence bond states. Different
possibilities of light-induced atomic interactions are considered that go
beyond the typical atomic system with dipolar and other intrinsic interactions.
This will broaden the Hamiltonian toolbox available for quantum simulation of
condensed matter physics via atomic systems.
