  Quantum simulations of the Bose-Hubbard model (BHM) at commensurate filling
can follow spreading of correlations after a sudden quench for times long
enough to estimate their propagation velocities. In this work we perform tensor
network simulation of the quantum Kibble-Zurek (KZ) ramp from the Mott towards
the superfluid phase in the square lattice BHM and demonstrate that even
relatively short ramp/quench times allow one to test the power laws predicted
by the KZ mechanism (KZM). They can be verified for the correlation length and
the excitation energy but the most reliable test is based on the KZM scaling
hypothesis for the single particle correlation function: the correlation
functions for different quench times evaluated at the same scaled time collapse
to the same scaling function of the scaled distance. The scaling of the space
and time variables is done according to the KZ power laws.
