Hyperfine interaction in a quantum dot: Non-Markovian electron spin dynamics

We have performed a systematic calculation for the non-Markovian dynamics of a localized electron spin interacting with an environment of nuclear spins via the Fermi contact hyperfine interaction. This work applies to an electron in the $s$-type orbital ground state of a quantum dot or bound to a donor impurity, and is valid for arbitrary polarization $p$ of the nuclear spin system, and arbitrary nuclear spin $I$ in high magnetic fields. In the limit of $p=1$ and $I=\frac{1}{2}$, the Born approximation of our perturbative theory recovers the exact electron spin dynamics. We have found the form of the generalized master equation (GME) for the longitudinal and transverse components of the electron spin to all orders in the electron spin-nuclear spin flip-flop terms. Our perturbative expansion is regular, unlike standard time-dependent perturbation theory, and can be carried out to higher orders. We show this explicitly with a fourth-order calculation of the longitudinal spin dynamics. In zero magnetic field, the fraction of the electron spin that decays is bounded by the smallness parameter $\delta =1∕p^{2}N$, where $N$ is the number of nuclear spins within the extent of the electron wave function. However, the form of the decay can only be determined in a high magnetic field, much larger than the maximum Overhauser field. In general the electron spin shows rich dynamics, described by a sum of contributions with nonexponential decay, exponential decay, and undamped oscillations. There is an abrupt crossover in the electron spin asymptotics at a critical dimensionality and shape of the electron envelope wave function. We propose a scheme that could be used to measure the non-Markovian dynamics using a standard spin-echo technique, even when the fraction that undergoes non-Markovian dynamics is small.