  We consider the dynamics of a quantum impurity after a sudden interaction
quench into a one-dimensional degenerate Bose gas. We use the Keldysh path
integral formalism to derive a truncated Wigner like approach that takes the
back action of the impurity onto the condensate into account already on the
mean-field level and further incorporates thermal and quantum effects up to
one-loop accuracy. This framework enables us not only to calculate the real
space trajectory of the impurity but also the absorption spectrum. We find that
quantum corrections and thermal effects play a crucial role for the impurity
momentum at weak to intermediate impurity-bath couplings.Furthermore, we see
the broadening of the absorption spectrum with increasing temperature.
