Momentum relaxation of a mobile impurity in a one-dimensional quantum gas

We investigate the time evolution of the momentum of an impurity atom injected into a degenerate Tonks-Girardeau gas. We establish that given an initial momentum $p_0$ the impurity relaxes to a steady state with a non-vanishing momentum $p_{\infty}$. The nature of the steady state is found to be drastically different for integrable and non-integrable impurity models, which is due to multiple coherent scattering processes leading to a resonant interaction between the impurity and the host in the integrable case. The dependence of $p_{\infty}$ on $p_0$ remains non-trivial even in the limit of vanishing interaction between the impurity and host particles. In this limit $p_{\infty}(p_0)$ is found explicitly and the case of the external force applied to the impurity is analyzed as well.