  We have analyzed the behavior of a mobile quantum impurity in a bath formed
by a two-leg bosonic ladder by a combination of field theory
(Tomonaga-Luttinger liquid) and numerical (Density Matrix Renormalization
Group) techniques. Computing the Green's function of the impurity as a function
of time at different momenta, we find a power law decay at zero momentum, which
signals the breakdown of any quasi-particle description of the impurity motion.
We compute the exponent both for the limits of weak and strong impurity-bath
interactions. At small impurity-bath interaction, we find that the impurity
experiences the ladder as a single channel one-dimensional bath, but effective
coupling is reduced by a factor of $\sqrt 2$, thus the impurity is less mobile
in the ladder compared to a one dimensional bath. We compared the numerical
results for the exponent at zero momentum with a semi-analytical expression
that was initially established for the chain and find excellent agreement
without adjustable parameters. We analyze the dependence of the exponent in the
transverse hopping in the bath and find surprisingly an increase of the
exponent at variance with the naive extrapolation of the single channel regime.
We study the momentum dependence of the impurity Green's function and find
that, as for the single chain, two different regime of motion exist, one
dominated by infrared metatrophy and a more conventional polaronic behavior. We
compute the critical momentum between these two regimes and compare with
prediction based on the structure factor of the bath. In the polaronic regime
we also compute numerically the lifetime of the polaron. Finally we discuss how
our results could be measured in cold atomic experiments.
