  We study the temporal expansion of an ultracold Bose gas in two-dimensional,
square optical lattices. The gas is described by the Bose-Hubbard model deep in
the superfluid regime, with initially all bosons condensed in the central site
of the lattice. We use the previously developed nonequilibrium propagator
method for capturing the time evolution of an interacting bosonic system, where
the many-body Hamiltonian is represented in an appropriate local basis and the
corresponding field operators are separated into the classical [Bose-Einstein
condensate (BEC)] part and quantum mechanical fluctuations. After a quench,
i.e. after a sudden switch of the lattice nearest-neighbor hopping, the
expanding, bosonic cloud separates spatially into a fast, ballistic forerunner
and a slowly expanding central part controlled by selftrapping. We show that
the forerunner expansion is driven by the coherent dynamics of the BEC and that
its velocity is consistent with the Lieb-Robinson bound. For smaller lattices
we analyze how quasiparticle collisions lead to enhanced condensate depletion
and oscillation damping.
