Non-Markovian dynamics of a localized electron spin due to the hyperfine interaction

We review our theoretical work oil the dynamics of a localized electron spin interacting with ail environment of nuclear Spins. Our perturbative calculation is valid for arbitrary polarization p of the nuclear spin system and arbitrary nuclear spin I in a Sufficiently large magnetic field. In general, the electron spin shows rich dynamics, described by a sum of contributions with exponential decay, nonexponential decay, and undamped oscillations. We have found ail abrupt crossover in the long-time spin dynamics at a critical shape and dimensionality of the electron envelope wave function. We conclude with a discussion of our proposed scheme to measure the relevant dynamics using a standard spin-echo technique.