Decoherence of nuclear spins in the frozen core of an electron spin

Hybrid qubit systems combining electronic spins with nearby (proximate) nuclear spin registers offer a promising avenue towards quantum information processing, with even multispin error-correction protocols recently demonstrated in diamond. However, for the important platform offered by spins of donor atoms in cryogenically cooled silicon, decoherence mechanisms of Si-29 proximate nuclear spins are not yet well understood. The reason is partly because proximate spins lie within a so-called frozen core region where the donor electronic hyperfine interaction strongly suppresses nuclear dynamics. We investigate the decoherence of a central proximate nuclear qubit arising from quantum spin baths outside, as well as inside, the frozen core around the donor electron. We consider the effect of a very large nuclear spin bath comprisingmany (greater than or similar to 10(8)) weakly contributing pairs outside the frozen core (the far bath). We also propose that there may be an important contribution from a few (of order 100) symmetrically sited nuclear spin pairs (equivalent pairs), which were not previously considered because their effect is negligible outside the frozen core. If equivalent pairs represent a measurable source of decoherence, nuclear coherence decays could provide sensitive probes of the symmetries of electronic wave functions. For the phosphorus donor system, we obtain T-2n values of order 1 second for both the far-bath and equivalent-pair models, confirming the suitability of proximate nuclei in silicon as very-long-lived spin qubits.