Collective mode of an impurity and a Tonks-Girardeau gas

We investigate the quantum dynamics of an impurity immersed in a one-dimensional gas of strongly repulsive bosons, or equivalently fully-polarized fermions, interacting via a contact interaction. Using Bethe Ansatz we obtain essentially exact results at all timescales and for all couplings to the impurity. We find at strong coupling that if the impurity starts off with a momentum of the order of the Fermi momentum or higher a new type of collective mode is excited, corresponding to long lived oscillations of the impurity with respect to the background gas. We characterize this mode and discuss how it can be observed experimentally.