  The coherence of the electron spin in a semiconductor quantum dot is strongly
enhanced by mode locking through nuclear focusing, where the synchronization of
the electron spin to periodic pulsing is slowly transferred to the nuclear
spins of the semiconductor material, mediated by the hyperfine interaction
between these. The external magnetic field that drives the Larmor oscillations
of the electron spin also subjects the nuclear spins to a Zeeman-like coupling,
albeit a much weaker one. For typical magnetic fields used in experiments, the
energy scale of the nuclear Zeeman effect is comparable to that of the
hyperfine interaction, so that it is not negligible. In this work, we analyze
the influence of the nuclear Zeeman effect on mode locking quantitatively.
Within a perturbative framework, we calculate the Overhauser-field distribution
after a prolonged period of pulsing. We find that the nuclear Zeeman effect can
exchange resonant and non-resonant frequencies. We distinguish between models
with a single type and with multiple types of nuclei. For the latter case, the
positions of the resonances depend on the individual $g$ factors, rather than
on the average value.
