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The nuclear spin bath (NSB) dynamics and its quantum control are ofimportance for the storage and processing of quantum information within asemiconductor environment. In the presence of a carrier spin, primarily it isthe hyperfine interaction that rules the high frequency NSB characteristics.Here, we first study the overall coherence decay and rephasings in ahyperfine-driven NSB through the temporal and spectral behaviors of theso-called Loschmidt echo (LE). Its dependence on the NSB size, initialpolarization, and coupling inhomogeneity are separately investigated, whichleads to a simple phenomenological expression that can accommodate all of theseattributes. Unlike the prevailing emphasis on spin 1/2, the NSBs with largerspin quantum numbers are equally considered. For this case, additionally theeffect of nuclear electric quadrupole interaction is taken into account whereits biaxiality term is influential on the decoherence. The insights gained frommodel systems are then put to use for two generic realistic semiconductorsystems, namely, a donor center and a quantum dot that represent small andlarge nanoscale NSB examples, respectively. The spectrum of LE for largequantum dots can reach the 100~MHz range, whereas, for donor centers, itreduces to a few MHz, making them readily amenable for dynamical decouplingtechniques. The effect of quadrupole interaction on LE is seen to be negligiblefor large quantum dots, while it becomes significant for donor centers, mostnotably in the form of depolarizing a polarized NSB.

Loschmidt echo driven by hyperfine and electric-quadrupole interactions in nanoscale nuclear spin baths