Mediated interactions between two impurities immersed in a Bose–Einstein condensate

We consider two pointlike static impurities without direct interaction immersed in a three-dimensional Bose–Einstein condensate (BEC) at zero temperature. By solving the Gross–Pitaevskii (GP) equation in a perturbative manner, we calculate the ground state energy in the region where the atom–impurity interaction is assumed to be weak. We obtain an analytical expression for the spatial distribution of atom number density and the effective force between these two impurities. The effective force is found to be closely related to the strength of the atom–impurity interaction and the relative distance between these two impurities. Two critical relative distances are found between the two impurities. The first one corresponds to the vanishing of the perturbed energy with impurities, although the effective force between the two impurities still exists. At the second critical value, the energy of the impurities changes linearly with the atom–impurity interaction; otherwise, it changes quadratically with the atom–impurity interaction.