Long-time coherence in fourth-order spin correlation functions

We study the long-time decay of fourth-order electron spin correlation functions for an isolated singly charged semi-conductor quantum dot. The electron spin dynamics is governed by the applied external magnetic field as well as the hyperfine interaction. While the long-time coherent oscillations in the correlation functions can be understood within an semi-classical approach treating the Overhauser field as frozen, the field dependent decay of its amplitude reported in different experiments cannot be explained by the central-spin model indicating the insufficiency of such a description. By incorporating the nuclear Zeeman splitting and the strain induced nuclear-electric quadrupolar interaction, we find the correct crossover from a fast decay in small magnetic fields to a slow exponential asymptotic in large magnetic fields. It originates from a competition between the quadrupolar interaction inducing an enhanced spin decay and the nuclear Zeeman term that suppressed the spin-flip processes. We are able to explain the magnetic field dependency of the characteristic long-time decay time $T_2$ depending on the experimental setups. The calculated asymptotic values of $T_2 = 3 -4\,\mu$s agree qualitatively well with the experimental data.