Abstract
Single electron spins coupled to multiple nuclear spins provide promising multiqubit registers for quantum sensing and quantum networks. The obtainable level of control is determined by how well the electron spin can be selectively coupled to, and decoupled from, the surrounding nuclear spins. Here we realize a coherence time exceeding a second for a single nitrogenvacancy electron spin through decoupling sequences tailored to its microscopic nuclearspin environment. First, we use the electron spin to probe the environment, which is accurately described by seven individual and six pairs of coupled carbon13 spins. We develop initialization, control and readout of the carbon13 pairs in order to directly reveal their atomic structure. We then exploit this knowledge to store quantum states in the electron spin for over a second by carefully avoiding unwanted interactions. These results provide a proofofprinciple for quantum sensing of complex multispin systems and an opportunity for multiqubit quantum registers with long coherence times.
Introduction
Coupled systems of individual electron and nuclear spins in solids are a promising platform for quantum information processing^{1,2,3,4,5,6} and quantum sensing^{7,8,9,10,11}. Initial experiments have demonstrated the detection and control of several nuclear spins surrounding individual defect or donor electron spins^{12,13,14,15,16,17}. These nuclear spins provide robust qubits that enable enhanced quantum sensing protocols^{7,8,9,10,11}, quantum error correction^{2,3,18}, and multiqubit nodes for optically connected quantum networks^{19,20,21,22}.
The level of control that can be obtained is determined by the electron spin coherence and therefore by how well the electron can be decoupled from unwanted interactions with its spin environment. Electron coherence times up to 0.56 s for a single electron spin qubit^{5} and ∼3 s for ensembles^{23,24,25,26} have been demonstrated in isotopically purified samples depleted of nuclear spins, but in those cases the individual control of multiple nuclearspin qubits is forgone.
Here we realize a coherence time exceeding 1 s for a single electron spin in diamond that is coupled to a complex environment of multiple nuclearspin qubits. First, we use the electron spin as a quantum sensor to probe the microscopic structure of the surrounding nuclearspin environment, including interactions between the nuclear spins. We find that the spin environment is accurately described by seven isolated single ^{13}C spins and six pairs of coupled ^{13}C spins (Fig. 1a). We then develop pulse sequences to initialize, control and readout the state of the ^{13}C–^{13}C pairs. We use this control to directly characterize the coupling strength between the ^{13}C spins, thus revealing their atomic structure given by the distance between the two ^{13}C atoms and the angle they make with the magnetic field. Finally, we exploit this extensive knowledge of the microscopic environment to realize tailored decoupling sequences that effectively protect arbitrary quantum states stored in the electron spin for well over a second. This combination of a long electron spin coherence time and selective couplings to a system of up to 19 nuclear spins provides a promising path to multiqubit registers for quantum sensing and quantum networks.
Results
System
We use a single nitrogenvacancy (NV) center (Fig. 1a) in a CVDgrown diamond at a temperature of 3.7 K with a natural 1.1% abundance of ^{13}C and a negligible nitrogen concentration (<5 parts per billion). A static magnetic field of B_{z} ≈ 403 G is applied along the NVaxis with a permanent magnet (Methods). The NV electron spin is read out in a single shot with an average fidelity of 95% through spinselective resonant excitation^{27}. The electron spin is controlled using microwave pulses through an onchip stripline (Methods).
Longitudinal relaxation
We first address the longitudinal relaxation (T_{1}) of the NV electron spin, which sets a limit on the maximum coherence time. At 3.7 Kelvin, spinlattice relaxation due to twophonon Raman and Orbachtype processes are negligible^{28,29}. No cross relaxation to P1 or other NV centers is expected due to the low nitrogen concentration. The electron spin can, however, relax due to microwave noise and laser background introduced by the experimental controls (Fig. 1). We ensure a high on/off ratio of the lasers (>100 dB) and use switches to suppress microwave amplifier noise (see Methods). Figure 1b shows the measured electron spin relaxation for all three initial states. We fit the average fidelity F to
The obtained decay time T_{1} is \((3.6 \pm 0.3) \times 10^3\) s. This value sets a lower limit for the spin relaxation time, and is the longest reported for a single electron spin qubit. Remarkably, the observed T_{1} exceeds recent theoretical predictions based on singlephonon processes by more than an order of magnitude^{30,31}. To further investigate the origin of the decay, we prepare m_{ s } = 0 and measure the total spin population summed over all three states. The total population decays on a similar timescale (\(\sim 3.6 \times 10^3\) s), indicating that the decay is caused by a reduction of the measurement contrast, possibly due to drifts in the optical setup (see Methods), rather than by spin relaxation. This suggests that the spinrelaxation time significantly exceeds the measured T_{1} value. Nevertheless, the long T_{1} observed here already indicates that longitudinal relaxation is no longer a limiting factor for NV center coherence.
Quantum sensing of the microscopic spin environment
To study the electron spin coherence, we first use the electron spin as a quantum sensor to probe its nuclearspin environment through dynamical decoupling spectroscopy^{12,13,14}. The electron spin is prepared in a superposition \(x\rangle = (m_s = 0\rangle + m_s =  1\rangle )/\sqrt 2\) and a dynamical decoupling sequence of N πpulses of the form \((\tau  \pi  \tau )^N\) is applied. The remaining electron coherence is then measured as a function of the time between the pulses 2τ. Loss of electron coherence indicates an interaction with the nuclearspin environment.
The results in Fig. 2a for N = 32 pulses reveal a rich structure consisting of both sharp and broader dips in the electron coherence. The sharp dips (Fig. 2b) have been identified previously as resonances due to the electron spin undergoing an entangling operation with individual isolated ^{13}C spins in the environment^{12,13,14}. For this NV center, the observed signal is well explained by seven individual ^{13}C spins and a background bath of randomly generated ^{13}C spins (Fig. 2b). To verify this explanation we perform direct Ramsey spectroscopy on all seven spins (Supplementary Fig. 1)^{3}. For the electron spin in \(m_s = \pm 1\), each spin yields a single unique precession frequency due to the hyperfine coupling, indicating that all seven spins are distinct and do not couple strongly to other ^{13}C spins in the vicinity (Supplementary Fig. 1).
The electron can be efficiently decoupled from the interactions with such isolated ^{13}C spins by setting \(\tau = m \cdot \frac{{2\pi }}{{\omega _{\mathrm{L}}}}\), with m a positive integer and ω_{L} the ^{13}C Larmor frequency for m_{ s } = 0^{33}. In practice, however, this condition might not be exactly and simultaneously met for all spins due to: the limited timing resolution of τ (here 1 ns), measurement uncertainty in the value ω_{L}, and differences between the m_{ s } = 0 frequencies for different ^{13}C spins, for example caused by different effective gtensors under a slightly misaligned magnetic field (here <0.35°, Supplementary Note 3)^{3,33,34,35}. We numerically simulate these deviations from the ideal condition and find that, for our range of parameters, the effect on the electron coherence is small and can be neglected (Supplementary Fig. 2).
We associate the broader dips in Figs. 2a and 2c to pairs of strongly coupled ^{13}C spins. Such ^{13}C–^{13}C pairs were treated theoretically^{32,36} and the signal due to a single pair of nearestneighbor ^{13}C spins with particularly strong couplings to a NV center has been detected^{16}. In this work, we exploit improved coherence times to detect up to six pairs, including previously undetected nonnearestneighbor pairs. We then develop pulse sequences to polarize and coherently control these pairs to be able to directly reveal their atomic structure through spectroscopy.
Direct spectroscopy of nuclearspin pairs
The evolution of ^{13}C–^{13}C pairs can be understood from an approximate pseudospin model in the subspace spanned by \(\uparrow \downarrow \rangle = \Uparrow \rangle\) and \(\downarrow \uparrow \rangle = \Downarrow \rangle\), following Zhao et al.^{32} (Supplementary Notes 1 and 2). The pseudospin Hamiltonian depends on the electron spin state. For m_{ s } = 0 we have:
and for \(m_s =  1\):
where \(\hat S_x\) and \(\hat S_z\) are the spin−\(\frac{1}{2}\) operators. X is the dipolar coupling between the ^{13}C spins and Z is due to the hyperfine field gradient (Supplementary Note 2)^{32}. The evolution of the ^{13}C–^{13}C pair during a decoupling sequence will thus in general depend on the initial electron spin state, causing a loss of electron coherence.
We now show that this conditional evolution enables direct spectroscopy of the ^{13}C–^{13}C dipolar interaction X. Consider two limiting cases: \(X > > Z\) and \(Z > > X\), which cover the pairs observed in this work. In both cases, loss of the electron coherence is expected for the resonance condition \(\tau = \tau _k = (2k  1)\frac{\pi }{{2\omega _{\mathrm{r}}}}\), with k a positive integer and resonance frequency \(\omega _{\mathrm{r}} = \sqrt {X^2 + (Z/2)^2}\)^{13,32,37}. For \(X > > Z\) the net evolution at resonance is a rotation around the zaxis with the rotation direction conditional on the initial electron state (mathematically analogous to the case of a single^{13}C spin in a strong magnetic field^{13,38}). For \(Z > > X\) the net evolution is a conditional rotation around the xaxis (analogous to the nitrogen nuclear spin subjected to a driving field^{37}). These conditional rotations provide the controlled gate operations required to initialize, coherently control and directly probe the pseudospin states.
The measurement sequences for the two cases are shown in Fig. 3a. First, a dynamical decoupling sequence is performed that correlates the electron state with the pseudospin state. Reading out the electron spin in a single shot then performs a projective measurement that prepares the pseudospin into a polarized state. For \(X > > Z\) the pseudospin is measured along its zaxis and thus prepared in \(\Uparrow \rangle\). For \(Z > > X\) the measurement is along the xaxis and the spin is prepared in \(( \Uparrow \rangle + \Downarrow \rangle)/\sqrt 2\). Second, we let the pseudospin evolve freely with the electron spin in one of its eigenstates (m_{ s } = 0 or \(m_s =  1\)) so that we directly probe the precession frequencies \(\omega _0 = X\) (for m_{ s } = 0) or \(\omega _1 = \sqrt {X^2 + Z^2}\) (for \(m_s =  1\)). For \(Z > > X\), an extra complication is that the initial state \(( \Uparrow \rangle + \Downarrow \rangle)/\sqrt 2\) is an eigenstate of \(\hat H_0\). To access \(\omega _0 = X\), we prepare \(( \Uparrow \rangle+ i \Downarrow \rangle)/\sqrt 2\) − a superposition of \(\hat H_0\) eigenstates—by first letting the system evolve under \(\hat H_1\) for a time \(\pi /(2\omega _1)\). Finally, the state of the pseudospin is readout through a second measurement sequence.
We find six distinct sets of frequencies (Fig. 3b), indicating that six different ^{13}C–^{13}C pairs are detected. The measurements for m_{ s } = 0 directly yield the coupling strengths X and therefore the atomic structure of the pairs (Fig. 4a). We observe a variety of coupling strengths corresponding to nearestneighbor pairs (\(X/2\pi = 2082.7(7)\) Hz, theoretical value 2061 Hz), as well as pairs separated by several bond lengths (e.g., \(X/2\pi = 133.8(1)\) Hz, theoretical value 133.4 Hz). The observed number of pairs is consistent with the ^{13}C concentration of the sample (Supplementary Fig. 4). Note that for pair 4, we have \(X > > Z\), so the resonance condition is mainly governed by the coupling strength X. This makes it likely that additional pairs with the same dipolar coupling X—but smaller Zvalues—contribute to the observed signal at \(\tau = 120\) µs. Nevertheless, the environment can be described accurately by the six identified pairs, which we verify by comparing the measured dynamical decoupling curves for different values of N to the calculated signal based on the extracted couplings (Fig. 4b).
Electron spin coherence time
Next, we exploit the obtained microscopic picture of the nuclear spin environment to investigate the electron spin coherence under dynamical decoupling. To extract the loss of coherence due to the remainder of the dynamics of the environment, i.e., excluding the identified signals from the ^{13}C spins and pairs, we fit the results to:
in which M(t) accounts for the signal due to the coupling to the ^{13}C–^{13}C pairs (Fig. 4b, Methods section). A, T, and n are fit parameters that account for the decay of the envelope due to the rest of the dynamics of the environment and pulse errors. As before, effects of interactions with individual ^{13}C spins are avoided by setting \(\tau = m \cdot \frac{{2\pi }}{{\omega _{\mathrm{L}}}}\). An additional challenge is that at high numbers of pulses the electron spin becomes sensitive even to small effects, such as spurious harmonics due to finite MW pulse durations^{39,40} and nonsecular Hamiltonian terms^{41}, which cause loss of coherence over narrow ranges of τ (<10 ns). Here we avoid such effects by scanning a range of ∼20 ns around the target value to determine the optimum value of τ.
Figure 5a shows the electron coherence for sequences from N = 4 to 10,240 pulses. The coherence times T, extracted from the envelopes, reveal that the electron coherence can be greatly extended by increasing the number of pulses N. The maximum coherence time is T = 1.58(7) s for N = 10,240 (Fig. 5b). We determine the scaling of T with N by fitting to \(T_{N = 4} \cdot (N/4)^\eta\), with T_{N=4} the coherence time for N = 4^{23,42,43,44,45} which gives η = 0.799(2). No saturation of the coherence time T is observed yet, so that longer coherence times are expected to be possible. In our experiments, however, pulse errors become significant at larger N, causing a decrease in the amplitude A (Supplementary Fig. 7).
Protecting arbitrary quantum states
Finally, we demonstrate that arbitrary quantum states can be stored in the electron spin for well over a second by using decoupling sequences that are tailored to the specific microscopic spin environment (Fig. 5c). For a given storage time, we select τ and N to maximize the obtained fidelity by avoiding interactions with the characterized ^{13}C spins and ^{13}C–^{13}C pairs. To assess the ability to protect arbitrary quantum states, we average the storage fidelity over the six cardinal states and do not renormalize the results. The results show that quantum states are protected with a fidelity above the 2/3 limit of a classical memory for at least 0.995 seconds (using N = 10,240 pulses) and up to 1.46 s from interpolation of the results. These are the longest coherence times reported for single solidstate electron spin qubits^{5}, despite the presence of a dense nuclear spin environment that provides multiple qubits.
Discussion
These results provide opportunities for quantum sensing and quantum information processing, and are applicable to a wide variety of solidstate spin systems^{4,5,17,46,47,48,49,50,51,52,53,54,55,56}. First, these experiments are a proofofprinciple for resolving the microscopic structure of multispin systems, including the interactions between spins^{32}. The developed methods might be applied to detect and control spin interactions in samples external to the host material^{10,57,58,59}. Second, the combination of long coherence times and selective control in an electronnuclear system containing up to twenty spins enables improved multiqubit quantum registers for quantum networks. The electron spin coherence now exceeds the time needed to entangle remote NV centers through a photonic link, making deterministic entanglement delivery possible^{60}. Moreover, the realized control over multiple ^{13}C–^{13}C pairs provides promising multiqubit quantum memories with long coherence times, as the pseudospin naturally forms a decoherenceprotected subspace^{61}.
Methods
Setup
The experiments are performed at 3.7 K (Montana Cryostation) with a magnetic field of ∼403 G applied along the NVaxis by a permanent magnet. We realize long relaxation (T_{1} >1 h) and coherence times (>1 s) in combination with fast spin operations (Rabi frequency of 14 MHz) and readout/initialization (∼10 μs), by minimizing noise and background from the microwave (MW) and optical controls. Amplifier (AR 25S1G6) noise is suppressed by a fast microwave switch (TriQuint TGS2355SM) with a suppression ratio of 40 dB. Video leakage noise generated by the switch is filtered with a high pass filter. We use Hermite pulse envelopes^{62,63} to obtain effective MW pulses without initialization of the intrinsic ^{14}N nuclear spin. To mitigate pulse errors we alternate the phases of the pulses following the XY8 scheme^{64}. Laser pulses are generated by direct current modulation (515 nm laser, Cobolt MLD  for charge state control) or by acoustic optical modulators (637 nm Toptica DL Pro and New Focus TLB6704P for spin pumping and singleshot readout^{27}). The direct current modulation yields an on/off ratio of >135 dB. By placing two modulators in series (Gooch and Housego Fibre Q) an on/off ratio of >100 dB is obtained for the 637 nm lasers. The laser frequencies are stabilized to within 2 MHz using a wavemeter (HFANGSTROM WS/U10U). Possible explanations for the observed decay in Fig. 1b are frequency drifts of this wavemeter or spatial drifts of the laser focus over 1h timescales.
Sample
We use a naturally occurring NV center in highpurity type IIa homoepitaxially chemicalvapordeposition (CVD) grown diamond with a 1.1% natural abundance of ^{13}C and a 〈111〉 crystal orientation (Element Six). The NV center studied here has been selected for the absence of verycloseby strongly coupled ^{13}C spins (>500 kHz hyperfine coupling), but not on any other properties of the nuclear spin environment. To enhance the collection efficiency a solidimmersion lens was fabricated on top of the NV center^{27,65} and a singlelayer aluminumoxide antireflection coating was deposited^{66,67}.
Data analysis
We describe the total signal for the NV electron spin after a decoupling sequence in Fig. 2 as:
where t is the total time. M_{bath} is the signal due to a randomly generated background bath of noninteracting spins with hyperfine couplings below 10 kHz. \(M_{\mathrm{C}}^i\) are the signals due to the seven individual isolated ^{13}C spins^{13}. \(M_{{\mathrm{pair}}}^j\) are the signals due to the six \(^{13}{\text{C–}} ^{13}{\mathrm{C}}\) pairs and are given by \(1/2 + {\mathrm{Re}}({\mathrm{Tr}}(U_0U_1^\dagger ))/4\), with U_{0} and U_{1} the evolution operators of the pseudospin pair for the decoupling sequence conditional on the initial electron state (m_{ s } = 0 or \(m_s =  1\))^{32}. The coherence time T and exponent n describe the decoherence due to remainder of the dynamics of the spin environment.
Setting \(\tau = m \cdot 2\pi /\omega _{\mathrm{L}}\) avoids the resonances due to individual ^{13}C spins, so that equation (5) reduces to:
The data in Figs. 3 and 4 are fitted to equation (6) and A, T and n are extracted from these fits.
Data availability
The data that support the findings of this study are available from the corresponding author upon request.
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Acknowledgements
We thank V. V. Dobrovitski, J. E. Lang, T. S. Monteiro, H. P. Bartling, C. L. Degen, and R. Hanson for valuable discussions, P. Vinke, R. Vermeulen, R. Schouten, and M. Eschen for help with the experimental apparatus, and A. J. Stolk for characterization measurements. We acknowledge support from the Netherlands Organization for Scientific Research (NWO) through a Vidi grant.
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M.H.A. and T.H.T. devised the experiments. M.H.A., J.C., and T.H.T. constructed the experimental apparatus. M.M. and D.J.T. grew the diamond. M.H.A. performed the experiments with support from M.A.B. and N.K. M.H.A. and T.H.T. analyzed the data with help of all authors. T.H.T. supervised the project.
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Abobeih, M.H., Cramer, J., Bakker, M.A. et al. Onesecond coherence for a single electron spin coupled to a multiqubit nuclearspin environment. Nat Commun 9, 2552 (2018). https://doi.org/10.1038/s4146701804916z
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