Impurities in a trapped 1D Bose gas of arbitrary interaction strength: localization-delocalization transition and absence of self-localization

Abstract: We discuss impurities in a one-dimensional Bose gas with arbitrary boson-boson and boson-impurity interactions. To fully account for quantum effects, we employ numerical simulations based on the density-matrix renormalization group (DMRG) and - in the regime of strong boson-boson interactions - the mapping to weakly interacting fermions. A mean-field description of the Bose polaron based on coupled Gross-Pitaevski -- Schr\"odinger equations predicts the existence of a self-localized polaron. We here show that such a solution does not exist and is an artifact of the underlying decoupling approximation. To this end we consider a mobile impurity in a box potential. Our work demonstrates that correlations between the impurity position and the bosons are important even in the limit where mean-field approaches are expected to work well. Furthermore we derive analytical approximations for the energy of a single polaron formed by a heavy impurity for arbitrary interaction strengths and large but finite boson-boson couplings which accurately reproduce DMRG results. This demonstrates that the polaron problem of a heavy impurity in a 1D Bose gas can be accurately approximated by a proper mean-field description plus a linearized treatment of quantum fluctuations for arbitrary boson-boson and impurity-boson couplings. Finally we determine the polaron-polaron interaction potential $V(r)$ in Born-Oppenheimer approximation for small and intermediate distances $r$, which in the Tonks gas limit is oscillatory due to Friedel oscillations in the Bose gas.