Nuclear spins as quantum memory in semiconductor nanostructures

W. M. Witzel and S. Das Sarma

Phys. Rev. B 76, 045218 – Published 23 July 2007

Abstract

We theoretically consider the possibility of using solid state nuclear spins in a semiconductor nanostructure environment as long-lived, high-fidelity quantum memory. In particular, we calculate, in the limit of a strong applied magnetic field, the fidelity of P donor nuclear spins as a function of time in random bath environments of Si and GaAs, and the lifetime of excited intrinsic spins in polarized Si and GaAs environments. In the former situation, the nuclear spin dephases due to spectral diffusion induced by the dipolar interaction among nuclei in the bath; in the interest of the high fidelity requirements necessary for fault-tolerant quantum computing, we consider the initial quantum memory decay caused by a non-Markovian bath. We calculate this nuclear spin memory time in the context of Hahn and Carr-Purcell-Meiboom-Gill (CPMG) refocused spin echoes using a formally exact cluster expansion technique which has previously been successful in dealing with electron spin dephasing in a solid state nuclear spin bath. With decoherence dominated by transverse dephasing $(T_{2})$ , we find it feasible to maintain high fidelity (losses of less than $10^{- 6}$ ) quantum memory on nuclear spins for times of the order of $100 μ s$ (GaAs:P) and $1 to 2 ms$ (natural Si:P) using CPMG pulse sequences of just a few $(\sim 2 - 4)$ applied pulses. We also consider the complementary situation of a central flipped intrinsic nuclear spin in a bath of completely polarized nuclear spins where decoherence is caused by the direct flip-flop of the central spin with spins in the bath. Exact numerical calculations that include a sufficiently large neighborhood of surrounding nuclei show lifetimes on the order of $1 - 5 ms$ for both GaAs and natural Si. Our calculated nuclear spin coherence times may have significance for solid state quantum computer architectures using localized electron spins in semiconductors where nuclear spins have been proposed for quantum memory storage.

Received 22 January 2007

DOI:https://doi.org/10.1103/PhysRevB.76.045218

Authors & Affiliations

W. M. Witzel and S. Das Sarma

Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA

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Vol. 76, Iss. 4 — 15 July 2007

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Images

Figure 1
(Color online) Numerical results of nuclear spin quantum memory loss for a ${}^{31}P$ donor nucleus that replaces an As atom in bulk GaAs. We define memory loss as 1 minus the normalized echo and plot this in a log-log scale as a function of the total echo time. The dashed line gives the Hahn echo results and the solid lines give CPMG echo results for two and four pulses. At some point for each type of echo sequence, the cluster expansion fails to converge.Reuse & Permissions
Figure 2
(Color online) Numerical results of nuclear spin quantum memory loss for a ${}^{31}P$ donor nucleus in bulk Si. We define memory loss as 1 minus the normalized echo and plot this in a log-log scale as a function of the total echo time. The dashed line gives the Hahn echo results and the solid lines give CPMG echo results for two and four pulses. Dotted lines give corresponding results, for comparison, obtained from the lowest order expansions provided by Eqs. (13, 17). At some point for each type of echo sequence, the cluster expansion fails to converge.Reuse & Permissions
Figure 3
(Color online) Equivalent to Fig. 2 except that results are shown for Si purified to 1% ${}^{29}{Si}$ . Lowest order results given by Eqs. (13, 17) are shown by the dotted lines. Isotopic purification enhances coherence as predicted in these equations.Reuse & Permissions
Figure 4
(Color online) Relaxation of a central ${}^{29}{Si}$ nuclear spin in a bath of polarized isotopically pure ${}^{29}{Si}$ nuclei in silicon’s diamond lattice structure in the limit of a strong applied magnetic field along two directions with extremal results: $B$ along [001] (or, equivalently, along any of the conventional axes, $x$ , $y$ , or $z$ ) and $B$ along [111]. The dashed curves give the corresponding approximate solutions from Eq. (31).Reuse & Permissions
Figure 5
Relaxation of a central ${}^{29}{Si}$ nuclear spin in a bath of polarized natural Si nuclei in silicon’s diamond lattice structure in the limit of a strong applied magnetic field along one of the conventional axes (e.g., [100]). Error bars indicate the standard deviation as a result of random isotopic configurations (4.67% ${}^{29}{Si}$ and the remaining ${}^{28}{Si}$ and ${}^{30}{Si}$ is spinless giving no contribution to the relaxation). The dashed curve gives the corresponding approximate solution from Eq. (31).Reuse & Permissions
Figure 6
(Color online) Relaxation of a central ${}^{75}{As}$ , ${}^{69}{Ga}$ , or ${}^{71}{Ga}$ nuclear spin in a bath of polarized GaAs nuclei in GaAs’s zinc-blende lattice structure in the limit of a strong applied magnetic field along one of the conventional axes (e.g., [100]). In this limit, the central spin only interacts with like nuclei. The ${}^{75}{As}$ are arranged with certainty on one of the fcc lattices, while the ${}^{69}{Ga}$ and ${}^{71}{Ga}$ randomly occupy 60.4% and 39.6% of the other fcc lattice, respectively. Error bars for the two Ga results indicate the standard deviation as a result of random isotopic configurations. The dashed curves give the corresponding approximate solutions from Eq. (31).Reuse & Permissions

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