  We consider a model for the motion of an impurity interacting with two
parallel, one-dimensional (bosonized) fermionic baths. The impurity is able to
move along any of the baths, and to jump from one to the other. We provide a
perturbative expression for the state evolution of the system when the impurity
is injected in one of the baths, with a given wave packet. The nontrivial
choice of the unperturbed dynamics makes the approximation formally
infinite-order in the impurity-bath coupling, allowing us to reproduce the
orthogonality catastrophe. We employ the result for the state evolution to
observe the dynamics of the impurity and its effect on the baths, in particular
in the case when the wave packet is Gaussian. We observe and characterize the
propagation of the impurity along the baths and the hopping between them. We
also analyze the dynamics of the bath density and momentum density (i.e. the
particle current), and show that fits an intuitive semi-classical
interpretation. We also quantify the correlation that is established between
the baths by calculating the inter-bath, equal-time spatial correlation
functions of both bath density and momentum, finding a complex pattern. We show
that this pattern contains information on both the impurity motion and on the
baths themselves, and that these can be unveiled by taking appropriate "slices"
of the time evolution.
