  We theoretically explore the possibility of creating spin entanglement by
simultaneously coupling two electronic spins to a nuclear ensemble. By
microscopically modeling the spin ensemble with a single mode boson field, we
use the time-dependent Fr\"{o}hlich transformation (TDFT) method developed most
recently [Yong Li, C. Bruder, and C. P. Sun, Phys. Rev. A \textbf{75}, 032302
(2007)] to calculate the effective coupling between the two spins. Our
investigation shows that the total system realizes a solid state based
architecture for cavity QED. Exchanging such kind effective boson in a virtual
process can result in an effective interaction between two spins. It is
discovered that a maximum entangled state can be obtained when the velocity of
the electrons matches the initial distance between them in a suitable way.
Moreover, we also study how the number of collective excitations influences the
entanglement. It is shown that the larger the number of excitation is, the less
the two spins entangle each other.
