  By means of time-dependent density-matrix renormalization-group (TDMRG) we
are able to follow the real-time dynamics of a single impurity embedded in a
one-dimensional bath of interacting bosons. We focus on the impurity breathing
mode, which is found to be well-described by a single oscillation frequency and
a damping rate. If the impurity is very weakly coupled to the bath, a
Luttinger-liquid description is valid and the impurity suffers an
Abraham-Lorentz radiation-reaction friction. For a large portion of the
explored parameter space, the TDMRG results fall well beyond the
Luttinger-liquid paradigm.
