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

Abstract

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 $^{29}\mathrm{Si}$ 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 comprising many $(\ensuremath{\gtrsim}{10}^{8})$ 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.