  Experiments have demonstrated that atoms excited to specific Rydberg states
within Bose-Einstein condensates cause significant excitation of phonons. If
the Rydberg atom is brought into an electronic superposition state, this
coupling to phonons leads to decoherence. We provide the theoretical basis for
the treatment of the latter. To this end, we evaluate Rydberg-phonon coupling
coefficients using a combination of analytical and numerical techniques. From
these coefficients, we calculate bath correlation functions, spectral densities
and re-organisation energies. All these quantify the influence of the
environment and form essential inputs for follow up open quantum system
techniques. We find that the amplitude of bath correlations scales like the
power law $\nu^{-5}$ with the principal quantum number $\nu$, while
re-organisation energies scale like $\nu^{-3}$, reflecting the extreme
tunability of Rydberg atomic properties. Rydberg impurities in condensates thus
constitute a non-trivial open-quantum system with exceptionally controlled
system, coupling and environmental Hamiltonians.
