Impurity-impurity interaction mediated by a one-dimensional Bose gas

The first step for understanding a Bose gas with impurity atoms is to study a single impurity, i.e. the Bose polaron problem. However, the physics of systems with many impurities can be drastically different from the Bose polaron, because the Bose gas mediates interactions between impurities. We use the flow equation approach to calculate repulsive and attractive impurity-impurity correlations mediated by a weakly interacting one-dimensional Bose gas in the Born-Oppenheimer approximation. We find that leading order perturbation theory, which leads to Yukawa-type potentials, fails when the boson-impurity interaction is stronger than the boson-boson interaction. For attractive impurity-impurity potentials, we also calculate the potential within a mean-field approximation (MFA). Our results show that the MFA is useful to study mesoscopic and large systems with two impurities. In particular, it allows one to calculate the induced impurity-impurity interaction potential beyond leading order perturbation theory in a simple manner, i.e., without including any information about beyond-mean-field boson-boson correlations.