  We revisit here the Kibble-Zurek mechanism for superfluid bosons slowly
driven across the transition towards the Mott-insulating phase. By means of a
combination of the Time-Dependent Variational Principle and a Tree-Tensor
Network, we characterize the current flowing during annealing in a ring-shaped
one-dimensional Bose-Hubbard model with artificial classical gauge field on up
to 32 lattice sites. We find that the superfluid current shows, after an
initial decrease, persistent oscillations which survive even when the system is
well inside the Mott insulating phase. We demonstrate that the amplitude of
such oscillations is connected to the residual energy, characterizing the
creation of defects while crossing the quantum critical point, while their
frequency matches the spectral gap in the Mott insulating phase. Our
predictions can be verified in future atomtronics experiments with neutral
atoms in ring shaped traps. We believe that the proposed setup provides an
interesting but simple platform to study the non-equilibrium quantum dynamics
of persistent currents experimentally.
