  We investigate experimentally and theoretically the temporal evolution of the
spin of the conduction band electron and that of the valence band heavy hole,
both confined in the same semiconductor quantum dot. In particular, the
coherence of the spin purity in the limit of a weak externally applied magnetic
field, comparable in strength to the Overhauser field due to fluctuations in
the surrounding nuclei spins. We use an all-optical pulse technique to measure
the spin evolution as a function of time after its initialization. We show for
the first time that the spin purity performs complex temporal oscillations
which we quantitatively simulate using a central spin model. Our model
encompasses the Zeeman and the hyperfine interactions between the spin and the
external and Overhauser fields, respectively. Our novel studies are essential
for the design and optimization of quantum-dot-based entangled multi-photon
sources. Specifically, cluster and graph states, which set stringent
limitations on the magnitude of the externally applied field.
