Impurities in a one-dimensional Bose gas: the flow equation approach

A few years ago, flow equations were introduced as a technique for calculating the energies of cold Bose gases with and without impurities. In this paper, we extend this approach to compute observables other than the energy. As an example, we calculate the energies, densities, and phase fluctuations of one-dimensional Bose gases with one and two impurities. For a single mobile impurity, we show that the flow equation results agree well with the mean-field results obtained upon the Lee-Low-Pines transformation if the phase coherence length is larger than the healing length of the condensate. This agreement occurs for all values of the boson-impurity interaction strength as long as the boson-boson interaction is weak. For two static impurities, we calculate repulsive and attractive impurity-impurity correlations mediated by the Bose gas. We find that leading order perturbation theory fails when the boson-impurity interaction strength is larger than the boson-boson interaction strength. We note that the mean-field approximation reproduces the flow equation results for weak boson-boson interactions.