Role of thermal two-phonon scattering for impurity dynamics in a low-dimensional BEC

We numerically study the relaxation dynamics of a single, heavy impurity atom interacting with a finite one- or two-dimensional, ultracold Bose-gas. While there is a clear separation of time scales between processes resulting from single- and two-phonon scattering in three spatial dimensions, the thermalization in lower dimensions is dominated by two-phonon processes. This is due to infrared divergencies in the corresponding scattering rates in the thermodynamic limit, which are a manifestation of the Mermin-Wagner-Hohenberg theorem. It makes it necessary to include second-order phonon scattering in one-dimensional systems even at $T=0$ and above a crossover temperature $T_\textrm{2ph}$ in two spatial dimensions. $T_\textrm{2ph}$ scales inversely with the system size and is much smaller than currently experimentally accessible.