  We calculate the dynamics of the one and two body correlation functions in a
homogeneous Bose gas at zero temperature following a sudden change in the
interaction strength, with and without an underlying lattice. We focus on
conceptually simple examples that highlight the features arising separately
from interactions and band structure. In the continuum, we show that the
Bogoliubov spectrum leads to a diffusive evolution of density correlations for
short times, and ballistic at long times. In the lattice we find that the
correlation functions develop additional oscillations. Moreover, the lattice
dispersion induces an additional velocity scale, and some features instead
propagate with that velocity. Finally, we discuss the time-evolution of the
contact following a quench. Our predictions, which can be readily tested in
experiments, serve as a benchmark for considering the dynamics of more
complicated systems.
