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 ${}^{29}\mathrm{Si}$ 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 $\left(\gtrsim {10}^8\right)$ 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.

Figure 1

Decoherence of electronic spin qubits by a flip-flopping nuclear spin bath in natural silicon. The background plots the spatial electronic wave function; blue denotes the strong-detuning region, where the energy cost of a bath spin flip ${\mathrm{\Delta }}_e^{\pm }\propto \pm (J_1-J_2)$ exceeds the strongest intrabath coupling $C_{12}$; it thus corresponds to the usual definition of the frozen-core region. However, electronic spin decoherence is dominated by an active zone (purple color) of pairs of nuclear spins which are actually within the blue strongly detuned region, with $|{\mathrm{\Delta }}_e^{\pm }/C_{12}|=|(J_1-J_2)/C_{12}|\sim 10$ for Si:P. The reason is that, while for large $|{\mathrm{\Delta }}_e^{\pm }|$ flip-flop amplitudes are strongly damped, qubit-state dependence of the quantum-bath evolution, essential for the entanglement between the electronic spin and bath which produces decoherence, is also proportional to ${\mathrm{\Delta }}_e^{\pm }$. Spin pairs for which $J_1=J_2$ (equivalent pairs) have no effect on electronic decoherence and were not considered in previous studies.

Figure 2

Decoherence of a proximate nuclear spin qubit (labelled “A”) by a quantum bath of nuclear spin pairs (a) outside and (b) inside the frozen core. In contrast to electron spin decoherence (for which the detuning is fully state dependent, see Fig. 1), the detuning is now ${\mathrm{\Delta }}_e+{\mathrm{\Delta }}_n^{\pm }$: there is now potentially a very large state-independent component ${\mathrm{\Delta }}_e\propto (J_1-J_2)$ which simply damps the bath noise, in addition to a state-dependent component ${\mathrm{\Delta }}_n^{\pm }\propto \pm (C_1^A-C_2^A)$ which leads to qubit-bath entanglement and thus decoherence. (a) Far-bath model: For large $R$, the bath spin interaction with both the electron spin and nuclear qubit is dipolar, thus $|{\mathrm{\Delta }}_n^{\pm }/{\mathrm{\Delta }}_e|\sim {10}^{-4}$ so very weak contributions from an extremely large bath of ${10}^8$ pairs for $R\lesssim 350\AA$ must be combined to obtain a converged decay. (b) Equivalent-pairs model: In the frozen core there are comparatively few spin impurities. For equivalent pairs however, $J_1=J_2$ so ${\mathrm{\Delta }}_e\simeq 0$. Their density is determined by the symmetry of the electronic wave function. The requirement for strong state-selective detuning implies also that one member of the pair must be close enough to the qubit to allow appreciable direct dipolar coupling (as opposed to long-range coupling between nuclear spins mediated by the electron). Pairs which also satisfy this requirement (exemplified by the upper, but not the lower, equivalent pair) are rare but even a few dozen suffice to exceed the contribution of the $\sim {10}^8$ far-bath spin pairs.

Figure 3

Convergence of large-bath model with respect to (a) intrabath dipolar coupling and (b) bath size. The figure indicates that decoherence is dominated by spins with $C_{12}\sim 0.01$ to 1 Hz and a bath of spins within $R\lesssim 350\AA$ of the origin, combining the contributions from $5\times {10}^8$ spin pairs. Calculations were performed for the case of Si:P, for $X$ band and magnetic field orientation $B_0=\left[100\right]$, yielding a $T_{2n}$ of 2 s for a single nuclear ${}^{29}\mathrm{Si}$ spin sited at the origin. This represents an estimate for the upper bound for the coherence time if the far bath is the dominant process. Due to the large nuclear spin bath, the coherence decays are insensitive to the choice of random spatial realization of the bath.

Figure 4

Density of equivalent pairs (EPs) as a function of distance. The separate contributions from different types of shells and the total density are shown, assuming a purely isotropic contact interaction (left) or a correction for anisotropic behavior (right). The density of EPs is approximately constant for $R\gtrsim 10\AA$, but the innermost proximate spins typically interact with fewer EPs.

Figure 5

Top panels: calculated Hahn-echo decays for proximate spins in a ^natSi:P system for (a) $J=0.1$ MHz and (b) $J=3.8$ MHz where $J$ is the hyperfine coupling between the donor electron spin and the proximate nucleus; blue: EP model with isotropic hyperfine coupling to the EPs; red: EP model including anisotropy correction; gray: far-bath model. (c) Calculated $T_{2n}$ values. There is a weak trend for $T_{2n}$ to increase as the hyperfine coupling increases (red line is a fit), possibly indicative of the decreasing density of EPs as $R\to 0$. In the far-bath model (gray dots), the slight increase in decoherence with lower $J$ (gray line) reflects the fact that the lower-$J$ proximate spins are slightly closer to the far bath. Coherence times were obtained from decays averaged over 100 spatial realizations of the bath, but typical single realizations gave the same timescale of decoherence.

Figure 6

Simulated coherence decays of a set of proximate spins corresponding to a range of hyperfine couplings J between the electron spin and the proximate qubit: (a) $J=0.1$, (b) $J=0.3$, (c) $J=0.5$, (d) $J=0.7$, (e) $J=1$, and (f) $J=3.8$ MHz. The blue lines correspond to isotropic hyperfine coupling to EPs and yield $T_{2n}\approx 0.2$ to 0.3 s; red lines show the effect of symmetry reduction due to the anisotropy of couplings: we compare the effect of desymmetrization if we constrain EPs to have in addition the same orientation condition [same ${({\widehat{\mathbf{n}}}_B·\mathbf{n})}^2$ as discussed in main text]. The effect is to produce $T_{2n}$ in the seconds timescale.