  We present a theoretical study based upon quantum Monte Carlo methods of the
Bose polaron in one-dimensional systems with contact interactions. In this
instance of the problem of a single impurity immersed in a quantum bath, the
medium is a Lieb-Liniger gas of bosons ranging from the weakly interacting to
the Tonks-Girardeau regime, whereas the impurity is coupled to the bath via a
different contact potential producing both repulsive and attractive
interactions. Both the case of a mobile impurity, having the same mass as the
particles in the medium, and of a static impurity with infinite mass are
considered. We make use of exact numerical techniques that allow us to
calculate the ground-state energy of the impurity, its effective mass as well
as the contact parameter between the impurity and the bath. These quantities
are investigated as a function of the strength of interactions between the
impurity and the bath and within the bath. In particular, we find that the
effective mass rapidly increases to very large values when the impurity gets
strongly coupled to an otherwise weakly repulsive bath. This heavy impurity
hardly moves within the medium, thereby realizing the "self-localization"
regime of the Landau-Pekar polaron. Furthermore, we compare our results with
predictions of perturbation theory valid for weak interactions and with exact
solutions available when the bosons in the medium behave as impenetrable
particles.
