This thesis reports on the implementation of the Fröhlich Hamiltonian in an ultracold atomic mixture. To this end, impurity atoms are immersed into a macroscopic Bose-Einstein condensate (BEC). The impurities are tightly trapped in one direction by a species-selective optical potential. In this scenario their coupling to the excitations (phonons) of the BEC causes energy shifts of their external states that are analogues to the electronic Lamb shift in the hydrogen atom, which originates from interaction with the vacuum. Therefore the BEC can be denoted synthetic vacuum and the energy shift termed phonon-induced Lamb shift. The energy gap between the lowest lying trap levels for the impurities is determined via motional Ramsey spectroscopy. For the detection of the energetic modifications due to interaction with the background, experiments are performed with and without BEC. The background modifies the gap by two mechanisms: the modulation of the BEC density by the optical trapping potential for the impurities, and the interaction with phonons. Both effects are observed and a quantitative description is derived. Furthermore we present the dependence of the phonon induced shift on the absolute atom number and the relative population of states for 7Li. This first observation of the Fröhlich Hamiltonian in motional degrees of freedom in an ultracold gas mixture paves the way to put theory to a test in a highly flexible environment.