  We study nonequilibrium dynamics in the one-dimensional Bose-Hubbard model
starting from an initial product state with one boson per site and a single
hole defect. We find that for parameters close to the quantum critical point,
the hole splits into a core showing a very slow diffusive dynamics, and a fast
mode which propagates ballistically. Using an effective fermionic model at
large Hubbard interactions U, we show that the ballistic mode is a consequence
of an interference between slow holon and fast doublon dynamics, which occurs
once the hole defect starts propagating into the bosonic background at unit
filling. Within this model, the signal velocity of the fast ballistic mode is
given by the maximum slope of the dispersion of the doublon quasiparticle in
good agreement with the numerical data. Furthermore, we contrast this global
quench with the dynamics of a single hole defect in the ground state of the
Bose-Hubbard model and show that the dynamics in the latter case is very
different even for large values of U. This can also be seen in the entanglement
entropy of the time evolved states which grows much more rapidly in time in the
global quench case.
