  Ultracold atoms uniformly filling an optical lattice can be treated like an
artificial crystal. An implementation including the atomic occupation of a
single excited atomic state can be represented by a two-component Bose-Hubbard
model. Its phase diagram exhibits a quantum phase transition from a superfluid
to a Mott insulator phase. The dynamics of electronic excitations governed by
electrostatic dipole-dipole interactions in the ordered regime can be well
described by wave-like collective excitations called excitons. Here we present
an extensive study of such excitons for a wide range of geometries and
dimensionality. Their lifetimes can vary over many orders of magnitude from
metastable propagation to superradiant decay. Particularly strong effects occur
in one dimensional atomic chains coupled to tapered optical fibers. For an
optical lattice within a cavity the excitons are coupled to cavity photons and
the resulting collective cavity QED model can be efficiently formulated in
terms of polaritons. Their properties are explicitly calculated for different
lattices and they constitute a non-destructive monitoring tool for important
system properties. Even the formation of molecules in optical lattices
manifests itself in modified polariton properties as e.g. an anisotropic
optical spectrum. Partial dissipation of the exciton energy in the lattice
leads to heating, which can be microscopically understood through a mechanism
transferring atoms into higher Bloch bands via a resonant excitation transfer
among neighboring lattice sites. The presence of lattice defects like vacancies
in the Mott insulator induces a characteristic scattering of polaritons, which
can be optically observed to monitor the lattice integrity. Our models can be
applied to simulate and understand corresponding collective phenomena in solid
crystals, where many effects are often masked by noise and disorder.
