  We investigate the behavior of a finite-momentum impurity immersed in a
weakly interacting three-dimensional Bose-Einstein condensate (BEC) of
ultra-cold atoms, giving a detailed account of the dynamical quantum Cherenkov
transition discussed in Ref. [arXiv:2101.00030]. Using a time-dependent
variational approach, we identify a transition in the far-from-equilibrium
dynamics of the system after the attractive short-range impurity-boson
interaction is quenched on. The transition occurs as the impurity's velocity
crosses an interaction-dependent critical value, and manifests in the long-time
behavior of the Loschmidt echo and average impurity velocity. This behavior is
also reflected in the finite momentum ground state of the system, where the
group velocity of the interaction-dressed impurity loses it's dependence on the
total momentum of the system as the critical point is crossed. The transition
we discuss should be experimentally observable via a variety of common
protocols in ultracold atomic systems such as time-of-flight imaging, RF
spectroscopy, Ramsey interferometry, and absorption imaging.
