  We study the entanglement of two impurity qubits immersed in a Bose-Einstein
condensate (BEC) reservoir. This open quantum system is particularly
interesting because the reservoir and system parameters are easily controllable
and the reduced dynamics is highly non-Markovian. We show how the model allows
for interpolation between a common dephasing scenario and an independent
dephasing scenario by simply modifying the wavelength of the superlattice
superposed to the BEC, and how this influences the dynamical properties of the
impurities. We demonstrate the existence of very rich entanglement dynamics
correspondent to different values of reservoir parameters, including phenomena
such as entanglement trapping, entanglement sudden death, revivals of
entanglement, and BEC-mediated entanglement generation. In the spirit of
reservoir engineering, we present the optimal BEC parameters for entanglement
generation and trapping, showing the key role of the ultracold gas
interactions.
