  We investigate a hybrid system composed of ultracold Rydberg atoms immersed
in an atomic Bose-Einstein condensate (BEC). The coupling between the Rydberg
electrons and BEC atoms leads to the excitation of phonons, the exchange of
which induces Yukawa interaction between Rydberg atoms. Due to the small
electron mass, the effective charge associated with this
quasi-particle-mediated interaction can be large, while its range is equal to
the healing length of the BEC, which can be tuned by adjusting the scattering
length of the BEC atoms. We find that for small healing lengths, the distortion
of the BEC can "image" the wave function density of the Rydberg electron, while
large healing lengths induce an attractive Yukawa potential between the two
Rydberg atoms that can form a new type of ultra-long-range molecule. We discuss
both cases for a realistic system.
