  Lieb-Robinson and related bounds set an upper limit on the rate of spreading
of information in non-relativistic quantum systems. Experimentally, they have
been observed in the spreading of correlations in the Bose-Hubbard model after
a quantum quench. Using a recently developed two particle irreducible (2PI)
strong coupling approach to out-of-equilibrium dynamics in the Bose-Hubbard
model we calculate both the group and phase velocities for the spreading of
single-particle correlations in one, two and three dimensions as a function of
interaction strength. Our results are in quantitative agreement with
measurements of the velocities for the spreading of single particle
correlations in both the one and two dimensional Bose-Hubbard model realized
with ultra-cold atoms. They also agree with the claim that the phase velocity
rather than the group velocity was observed in recent experiments in two
dimensions. We demonstrate that there can be large differences between the
phase and group velocities for the spreading of correlations and also explore
the variation of the anisotropy in the velocity at which correlations spread
across the phase diagram of the Bose-Hubbard model. Our results establish the
2PI strong coupling approach as a powerful tool to study out-of-equilibrium
dynamics in the Bose-Hubbard model in dimensions greater than one.
