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We evaluate free-induction decay for the transverse components of a localizedelectron spin coupled to a bath of nuclear spins via the Fermi contacthyperfine interaction. Our perturbative treatment is valid for special(narrowed) bath initial conditions and when the Zeeman energy of the electron$b$ exceeds the total hyperfine coupling constant $A$: $b>A$. Using one unifiedand systematic method, we recover previous results reported at short and longtimes using different techniques. We find a new and unexpected modulation ofthe free-induction-decay envelope, which is present even for a purely isotropichyperfine interaction without spin echoes and for a single nuclear species. Wegive sub-leading corrections to the decoherence rate, and show that, ingeneral, the decoherence rate has a non-monotonic dependence on electron Zeemansplitting, leading to a pronounced maximum. These results illustrate thelimitations of methods that make use of leading-order effective Hamiltoniansand re-exponentiation of short-time expansions for a strongly-interactingsystem with non-Markovian (history-dependent) dynamics.

Free-induction decay and envelope modulations in a narrowed nuclear spin bath