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

TL;DR

This paper investigates impurities in a 1D Bose gas across various interaction strengths, revealing the absence of self-localization predicted by mean-field theories and emphasizing the importance of correlations, with analytical and numerical results supporting a refined understanding of polaron behavior.

Contribution

It demonstrates that self-localized polarons do not exist in 1D Bose gases and provides accurate analytical approximations for polaron energies across interaction regimes.

Findings

Self-localization predicted by mean-field is an artifact and does not occur.

Correlations significantly influence impurity behavior even where mean-field is expected to be valid.

Analytical approximations match DMRG results for polaron energies.

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.