  We study the dynamics of a mobile impurity in a two-leg bosonic ladder. The
impurity moves both along and across the legs and interacts with a bath of
interacting bosonic particles present in the ladder. We use both analytical
(Tomonaga-Luttinger liquid - TLL) and numerical (Density Matrix Renormalization
Group - DMRG) methods to compute the Green's function of the impurity. We find
that for a small impurity-bath interaction, the bonding mode of the impurity
effectively couples only to the gapless mode of the bath while the anti-bonding
mode of the impurity couples to both gapped and gapless mode of the bath. We
compute the time dependence of the Green's function of the impurity, for
impurity created either in the anti-bonding or bonding mode with a given
momentum. The later case leads to a decay as a power-law below a critical
momentum and exponential above, while the former case always decays
exponentially. We compare the DMRG results with analytical results using the
linked cluster expansion and find a good agreement. In addition we use DMRG to
extract the lifetime of the quasi-particle, when the Green's function decays
exponentially. We also treat the case of an infinite bath-impurity coupling for
which both the bonding and antibonding modes are systematically affected. For
this case the impurity Green's function in the bonding mode decays as a
power-law at zero momentum.The corresponding exponent increases with increasing
transverse-tunneling of the impurity. We compare our results with the other
impurity problems for which the motion of either the impurity or the bath is
limited to a single chain. Finally we comments on the consequences of our
findings for experiments with the ultracold gasses.
