  Using near-exact numerical simulations we study the propagation of an
impurity through a one-dimensional Bose lattice gas for varying bosonic
interaction strengths and filling factors at zero temperature. The impurity is
coupled to the Bose gas and confined to a separate tilted lattice. The precise
nature of the transport of the impurity is specific to the excitation spectrum
of the Bose gas which allows one to measure properties of the Bose gas
non-destructively, in principle, by observing the impurity; here we focus on
the spatial and momentum distributions of the impurity as well as its reduced
density matrix. For instance we show it is possible to determine whether the
Bose gas is commensurately filled as well as the bandwidth and gap in its
excitation spectrum. Moreover, we show that the impurity acts as a witness to
the cross-over of its environment from the weakly to the strongly interacting
regime, i.e., from a superfluid to a Mott insulator or Tonks-Girardeau lattice
gas and the effects on the impurity in both of these strongly-interacting
regimes are clearly distinguishable. Finally, we find that the spatial
coherence of the impurity is related to its propagation through the Bose gas,
giving an experimentally controllable example of noise-enhanced quantum
transport.
