Keeping a spin qubit alive in natural silicon: Comparing optimal working points and dynamical decoupling

S. J. Balian, Ren-Bao Liu, and T. S. Monteiro

Phys. Rev. B 91, 245416 – Published 15 June 2015

Abstract

There are two distinct techniques of proven effectiveness for extending the coherence lifetime of spin qubits in environments of other spins. One is dynamical decoupling, whereby the qubit is subjected to a carefully timed sequence of control pulses; the other is tuning the qubit towards “optimal working points” (OWPs), which are sweet spots for reduced decoherence in magnetic fields. By means of quantum many-body calculations, we investigate the effects of dynamical decoupling pulse sequences far from and near OWPs for a central donor qubit subject to decoherence from a nuclear spin bath. Key to understanding the behavior is to analyze the degree of suppression of the usually dominant contribution from independent pairs of flip-flopping spins within the many-body quantum bath. We find that to simulate recently measured Hahn echo decays at OWPs (lowest-order dynamical decoupling), one must consider clusters of three interacting spins since independent pairs do not even give finite- $T_{2}$ decay times. We show that while operating near OWPs, dynamical decoupling sequences require hundreds of pulses for a single order of magnitude enhancement of $T_{2}$ , in contrast to regimes far from OWPs, where only about 10 pulses are required.

Received 3 March 2015
Revised 22 April 2015

DOI:https://doi.org/10.1103/PhysRevB.91.245416

Authors & Affiliations

S. J. Balian^1,*, Ren-Bao Liu², and T. S. Monteiro¹

¹Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom
²Department of Physics, The Chinese University of Hong Kong, Hong Kong, China

^*s.balian@ucl.ac.uk

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Vol. 91, Iss. 24 — 15 June 2015

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Figure 1
(a) The spectra of donor spin systems such as arsenic, antimony, or bismuth (pictured) are affected by strong mixing between the electron and host nuclear spin, at magnetic fields $B$ smaller or comparable to the hyperfine coupling $A$ , allowing a richer behavior than unmixed electron spins. At particular field values termed optimal working points (OWPs), decoherence can be strongly suppressed; the arrows indicate the transitions with four of the most significant OWPs. (b) Coherences of the central electronic spins are dephased primarily by a surrounding quantum bath of clusters of 2, 3, 4 or more nuclear spin impurities (for natural silicon, pictured) or other donors (for isotopically enriched silicon). (c) Effect of dynamical decoupling (CPMG with even pulse numbers $N$ ) as $N \to \infty$ . Plots $T_{2}^{(N)} / T_{2}^{(1)}$ showing enhancement of the electron spin coherence time $T_{2}$ as a function of pulse number $N$ , relative to the $N = 1$ Hahn echo value. We find that while dynamical decoupling far from the OWP enhances $T_{2}$ by an order of magnitude with about 10 pulses, in contrast, behavior close to an OWP is insensitive to dynamical decoupling for low $N$ . For high $N$ , the behaviors near and far from OWPs become comparable. (d) Illustrates coherence enhancement as $B \to B_{OWP}$ (the Hahn spin echo time $T_{2}^{(1)}$ is plotted). The OWP is for a bismuth donor in natural silicon, investigated experimentally in Refs. [12, 13]. Inset of (d): The CPMG dynamical decoupling sequence consists of the initial $π / 2$ pulse, followed by the $- τ - π - τ - echo$ sequence repeated $N$ times. The coherence times in (c) are when the CPMG decays in Fig. 3 and the fits to the decays in Fig. 4 have fallen to $1 / e$ . OWP results are for the $|14〉 \to |7〉$ transition for which $B_{OWP} = 799$ G. For the field value near the OWP ( $B = 795$ G) in (c), $T_{2}^{(1)} ≃ 96$ ms while $T_{2}^{(1)} ≃ 0.79$ ms in the $\neq OWP$ regime ( $B = 3200$ G). The OWP curve in (d) was calculated using the analytical formula (9).
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Figure 2
Shows quantum many-body calculations of the Hahn echo $(N = 1)$ using the cluster correlation expansion (CCE) method. (a) Near OWPs, calculations using a bath of independent spin pairs only (red, CCE2) do not even predict a finite decay time but, surprisingly, calculations with clusters of three spins (blue, CCE3) are already well converged. The dashed lines used Eq. (9), a closed-form equation derived from the short-time behavior, found in Ref. [12] to yield good agreement with experiments; this indicates that three-cluster results too give good agreement with measurements. Higher-order CCE can encounter numerical divergences (which can be attenuated by ensemble averaging); this accounts for the discrepancies with CCE5. (b) Far from the OWP, independent pairs (CCE2) already give results in good agreement with CCE3–5 as well as experiments. The free induction decay (FID) is also shown for comparison. Note that Eq. (9) approximates the decay by a pure Gaussian. CCE calculations were performed for a bismuth donor in natural silicon for $B$ along $[100]$ and the $|14〉 \to |7〉$ transition for which $B_{OWP} = 799$ G. In (a), $B = 795$ G while for (b), $B = 3200$ G.
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Figure 3
Shows dependence of the coherence on the number of dynamical decoupling pulses $N$ , near (a) an optimal working point (OWP) and (b) far from an OWP, for modest numbers of $N$ . (a) For $B$ close to $B_{OWP}$ , the $T_{2}$ times show comparatively little response to dynamical decoupling. Further, even though the initial coherence is extended with increasing $N$ , the decays become ever more oscillatory. For low $N$ , the independent pairs contribution is largely eliminated. Inset of (a): Showing complete suppression of the independent pairs contribution near an OWP, but showing also its gradual revival as $N$ increases. (b) In contrast, far from the OWP, substantial (order of magnitude) enhancement of the $T_{2}$ time by dynamical decoupling is achieved with a moderate (preferably even) number of pulses. Decays for independent pair contributions (dashed lines, CCE2) and the converged quantum many-body numerics (solid lines, CCE4) are also compared, indicating that as $N ≳ 10$ , once again, the independent pair contribution is sufficient. Parameters are as in Fig. 2 and for $CPMG N$ sequences. The converged CCE in (a) corresponds to CCE3.
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Figure 4
Shows behaviors of coherence decays for large numbers $(N)$ of dynamical decoupling pulses (a) near and (b) far from OWPs; as shown in Fig. 3, in this regime, correlations from independent pairs once again dominate the decays in all regimes so CCE2 is converged and plotted. The behavior at OWPs is now sensitive to $N$ but the decays here become increasingly oscillatory as $N$ and $T_{2}$ both become large; we attribute this to large numbers of bath spin-pair frequencies becoming resonant with the pulse spacing. It indicates the behavior one might expect in a single-shot single-spin study. The smooth lines are fits to the decays and indicate the behavior expected after ensemble averaging. Parameters are as in Fig. 2.
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Figure 5
Sharp $B$ -field dependence of $T_{2}$ for various CPMG orders near an OWP. Inhomogeneous broadening from $29^{Si}$ nuclei can be incorporated by convolving the decays with a Gaussian $B$ -field distribution centered about $B$ (here centered about 797 G) and with standard deviation $w ≃ 2$ G (dashed line). For a donor concentration of $3 \times 10^{15} {cm}^{- 3}, T_{2}$ is limited by donor-donor processes at about 300 ms [13]. The $T_{2}$ lines were calculated for bismuth donors using the CCE up to third order and for $B ∥ [\bar{1} 10]$ . The OWP under investigation is shown in red at 799 G.
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