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Experiments have demonstrated that atoms excited to specific Rydberg stateswithin Bose-Einstein condensates cause significant excitation of phonons. Ifthe Rydberg atom is brought into an electronic superposition state, thiscoupling to phonons leads to decoherence. We provide the theoretical basis forthe treatment of the latter. To this end, we evaluate Rydberg-phonon couplingcoefficients using a combination of analytical and numerical techniques. Fromthese coefficients, we calculate bath correlation functions, spectral densitiesand re-organisation energies. All these quantify the influence of theenvironment and form essential inputs for follow up open quantum systemtechniques. We find that the amplitude of bath correlations scales like thepower law $\nu^{-5}$ with the principal quantum number $\nu$, whilere-organisation energies scale like $\nu^{-3}$, reflecting the extremetunability of Rydberg atomic properties. Rydberg impurities in condensates thusconstitute a non-trivial open-quantum system with exceptionally controlledsystem, coupling and environmental Hamiltonians.

Tayloring Bose-Einstein condensate environments for a Rydberg impurity