  We analyze nuclear spin dynamics in quantum dots and defect centers with a
bound electron under electron-mediated coupling between nuclear spins due to
the hyperfine interaction ("J-coupling" in NMR). Our analysis shows that the
Overhauser field generated by the nuclei at the position of the electron has
short-time dynamics quadratic in time for an initial nuclear spin state without
transverse coherence. The quadratic short-time behavior allows for an extension
of the Overhauser field lifetime through a sequence of projective measurements
(quantum Zeno effect). We analyze the requirements on the repetition rate of
measurements and the measurement accuracy to achieve such an effect. Further,
we calculate the long-time behavior of the Overhauser field for effective
electron Zeeman splittings larger than the hyperfine coupling strength and
find, both in a Dyson series expansion and a generalized master equation
approach, that for a nuclear spin system with a sufficiently smooth
polarization the electron-mediated interaction alone leads only to a partial
decay of the Overhauser field by an amount on the order of the inverse number
of nuclear spins interacting with the electron.
