  The nuclear spin bath (NSB) dynamics and its quantum control are of
importance for the storage and processing of quantum information within a
semiconductor environment. In the presence of a carrier spin, primarily it is
the hyperfine interaction that rules the high frequency NSB characteristics.
Here, we first study the overall coherence decay and rephasings in a
hyperfine-driven NSB through the temporal and spectral behaviors of the
so-called Loschmidt echo (LE). Its dependence on the NSB size, initial
polarization, and coupling inhomogeneity are separately investigated, which
leads to a simple phenomenological expression that can accommodate all of these
attributes. Unlike the prevailing emphasis on spin 1/2, the NSBs with larger
spin quantum numbers are equally considered. For this case, additionally the
effect of nuclear electric quadrupole interaction is taken into account where
its biaxiality term is influential on the decoherence. The insights gained from
model systems are then put to use for two generic realistic semiconductor
systems, namely, a donor center and a quantum dot that represent small and
large nanoscale NSB examples, respectively. The spectrum of LE for large
quantum dots can reach the 100~MHz range, whereas, for donor centers, it
reduces to a few MHz, making them readily amenable for dynamical decoupling
techniques. The effect of quadrupole interaction on LE is seen to be negligible
for large quantum dots, while it becomes significant for donor centers, most
notably in the form of depolarizing a polarized NSB.
