  We investigate the ground state properties and the nonequilibrium dynamics of
a lattice trapped bosonic mixture consisting of an impurity species and a
finite-sized medium. For the case of one as well as two impurities we observe
that, depending on the lattice depth and the interspecies interaction strength,
a transition from a strongly delocalized to a localized impurity distribution
occurs. In the latter regime the two species phase separate, thereby forming a
particle-hole pair. For two impurities we find that below a critical lattice
depth they are delocalized among two neighboring outer lattice wells and are
two-body correlated. This transition is characterized by a crossover from
strong to a suppressed interspecies entanglement for increasing impurity-medium
repulsion. Turning to the dynamical response of the mixture, upon quenching the
interspecies repulsion to smaller values, we reveal that the predominant
tunneling process for a single impurity corresponds to that of a particle-hole
pair, whose dynamical stability depends strongly on the quench amplitude.
During the time-evolution a significant increase of the interspecies
entanglement is observed, caused by the build-up of a superposition of states
and thus possesses a many-body nature. In the case of two bosonic impurities
the particle-hole pair process becomes unstable in the course of the dynamics
with the impurities aggregating in adjacent lattice sites while being strongly
correlated.
