  We study the time evolution of a one-dimensional system of strongly
correlated electrons (a 'sample') that is suddenly coupled to a smaller,
initially empty system (a 'nanoprobe'), which can subsequently move along the
system. Our purpose here is to study the role of interactions in this model
system when it is far from equilibrium. We therefore take both the sample and
the nanoprobe to be described by a Hubbard model with on-site repulsive
interactions and nearest-neighbor hopping. We compute the behavior of the local
particle density and the local density of states (LDOS) as a function of time
using time-dependent matrix product states at quarter and at half filling,
fillings at which the chain realizes a Luttinger liquid or a Mott insulator,
respectively. This allows us to study in detail the oscillation of the
particles between the sample and the nanoprobe. While, for noninteracting
systems, the LDOS is time-independent, in the presence of interactions, the
backflow of electrons to the sample will lead to nontrivial dynamics in the
LDOS. In particular, studying the time-dependent LDOS allows us to study how
the Mott gap closes locally and how this melting of the Mott insulator
propagates through the system in time after such a local perturbation -- a
behavior that we envisage can be investigated in future experiments on
ultrashort time scales or on optical lattices using microscopy setups.
