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The interaction of two spatially separated impurity atoms through phonon exchange in a Bose-Einstein condensate is studied within a Bogoliubov approach. The impurity atoms are held by deep and narrow trap potentials and experience level shifts which consist of a mean-field part and vacuum contributions from the Bogoliubov-phonons. In addition there is a conditional energy shift resulting from the exchange of phonons between the impurity atoms.doi:10.1103/physreva.71.033605 fatcat:ht6l36ielveeteaits45gjbymi