Time scales of phonon-induced decoherence of semiconductor spin qubits

Decoherence of a shallow donor electron spin in Si and Ge caused by electron-lattice interaction is studied. We find that there are two time scales associated with the evolution of the electron spin density matrix: the fast, but incomplete decay due to the interaction with nonresonant phonons, followed by slow relaxation resulting from spin flips accompanied by resonant phonon emission. We estimate both time scales, as well as the magnitude of the initial drop of coherence for P donor in Si and Ge, and argue that the approach used in the paper is suitable for evaluation of decoherence for a general class of localized spin states in semiconductors.