  We study the free evolution of a superposition initialized with high fidelity
in the neutral-exciton state of a quantum dot. Readout of the state at later
times is achieved by polarized photon detection, averaged over a large number
of cycles. By controlling the fine-structure splitting (FSS) of the dot with a
dc electric field, we show a reduction in the degree of polarization of the
signal when the splitting is minimized. In analogy with the "free induction
decay" observed in nuclear magnetic resonance, we attribute this to hyperfine
interactions with nuclei in the semiconductor. We numerically model this effect
and find good agreement with experimental studies. Our findings have
implications for storage of superpositions in solid-state systems and for
entangled photon pair emission protocols that require a small value of the FSS.
