  We introduce an exactly solvable model to study decoherence of a central spin
interacting with a spin bath where the coupling is mediated by phonons which we
assume to be in a coherent state or thermal distribution. For the coherent
state case, we find that the decoherence factor decays in a Gaussian fashion
and it becomes independent of the phonon frequencies at short times. If the
phonon energies are much larger than spin-phonon coupling or bath spins are
fully polarized, decoherence time becomes independent of the initial phonon
state. For the thermal state case, phonons play more important role in
decoherence with increasing temperature. We also discuss possible effects of
the temperature on spin bath contribution to decoherence.
