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

We evaluate free-induction decay for the transverse components of a localized electron spin coupled to a bath of nuclear spins via the Fermi-contact hyperfine 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 unified and systematic method, we recover previous results reported at short and long times using different techniques. We find an unexpected modulation of the free-induction-decay envelope, which is present even for a purely isotropic hyperfine interaction without spin echoes and for a single nuclear species. We give subleading corrections to the decoherence rate, and show that, in general, the decoherence rate has a nonmonotonic dependence on electron Zeeman splitting, leading to a pronounced maximum. These results illustrate the limitations of methods that make use of leading-order effective Hamiltonians and re-exponentiation of short-time expansions for a strongly interacting system with non-Markovian (history-dependent) dynamics.