Abstract
Quantum state readout is a key component of quantum technologies, including applications in sensing, computation, and secure communication. Readout fidelity can be enhanced by repeating readouts. However, the number of repeated readouts is limited by measurement backaction, which changes the quantum state that is measured. This detrimental effect can be overcome by storing the quantum state in an ancilla qubit, chosen to be robust against measurement backaction and to allow error correction. Here, we protect the electronicspin state of a diamond nitrogenvacancy center from measurement backaction using a robust multilevel ^{14}N nuclearspin memory and perform repetitive readout, as demonstrated in previous work on bulk diamond devices. We achieve additional protection using error correction based on the quantum logic of coherent feedback to reverse measurement backaction. The repetitive spin readout scheme provides a 13fold enhancement of readout fidelity over conventional readout and the error correction a 2fold improvement in the signal. These experiments demonstrate full quantum control of a nitrogenvacancy center electronicspin coupled to its host ^{14}N nuclear spin inside a ~25 nm nanodiamond, creating a sensitive and biologically compatible platform for nanoscale quantum sensing. Our errorcorrected repetitive readout scheme is particularly useful for quadrupolar nuclear magnetic resonance imaging in the low magnetic field regime where conventional repetitive readout suffers from strong measurement backaction. More broadly, methods for correcting longitudinal (bitflip) errors described here could be used to improve quantum algorithms that require nonvolatile local memory, such as correlation spectroscopy measurements for high resolution sensing.
Introduction
Quantum devices rely on the readout of a quantum system after a period of controlled evolution or interaction with other systems. High readout fidelity is critical to such devices, because it enables heralded state initialization,^{1,2,3} quantum error correction,^{4} and improved sensitivity of quantum sensors.^{5,6} Weak signals are typical for quantum systems and limit the information that can be gained about the state of the system in one readout step. In addition, all quantum measurements exhibit backaction, which changes the device’s state during readout. In the absence of backaction, namely for an ideal quantum nondemolition measurement, the problem of a weak signal could be overcome by multiple repeated readout steps. In this article, we describe and demonstrate an error correction protocol that protects spin memories from measurement backaction in the form of depolarization due to spin–spin interactions.
We choose to implement our protocol in a nitrogenvacancy (NV) center in diamond, a versatile system for quantum sensing, due to the ease of controllability of its electronicspin and the inbuilt multilevel local nuclear memory: the threespin states of the nitrogen14 nuclear spin (see illustration in Fig. 1a). The NV center has emerged as a promising candidate for quantum information^{7,8} and sensing^{6,9,10,11} applications because its electronic spin has a long coherence time at room temperature and can be easily initialized, readout and controlled with optical and microwave (MW) fields. However, room temperature optical readout using the NV center’s spindependent fluorescence yields only ≈0.02 detected photons in typical experiments like ours before the spin state is repolarized, leading to a low readout fidelity. Recent experiments on bulk diamond NV centers overcome this poor readout by using nearby hyperfinecoupled nuclear spins as memory qubits to store and repetitively readout the NV center’s electronicspin state.^{3,5,12,13,14,15,16} Other techniques^{17}—predominantly excitation of optical cycling transitions^{18,19}—can also improve readout fidelity, but such techniques are not widely used in room temperature sensing experiments as they require additional lasers, cryogenic temperatures, or sensitive electrical measurements. The fidelity achievable with the nuclearspin repetitive readout technique is limited by measurement backaction in the form of nuclearspin depolarization that occurs when the NV center is optically excited. Here we show that our error correction protocol can partially reverse this depolarization and improve readout fidelity by using the NV electronic spin as an ancilla qubit to control the host ^{14}N nuclear spin. Crucially, our protocol does not require measurement, which has low fidelity for our system, because we use the quantum logic of coherent feedback^{20,21} to perform error correction. We demonstrate that this protocol can enhance the repetitive readout fidelity of a NV center electronic spin by >50% in the regime where backaction is strong.
We apply our protocol to an NV center in a highpurity nanodiamond (Fig. 1a) with typical diameter of 25 nm.^{22} Building upon our recent demonstration of preparation, readout, and coherent control of a ^{13}C nuclear spin in a nanodiamond,^{23} here we demonstrate repetitive readout of a ^{14}N nuclear spin in a nanodiamond. These techniques, which have been widely used in bulk NV center experiments, enable a wide range of nuclearspinassisted measurements and enhanced sensitivity.^{5,13,24} By combining nuclearspinassisted readout with the small size of these nanodiamonds we hope to enhance NV sensors for applications such as scanning magnetometry, interferometry of mechanical motion,^{25} and intracellular nanoscale measurements.^{26,27}
Results
Coherent control and repetitive readout of a host ^{14}N nuclear spin in a nanodiamond
Figure 1b presents the NV electronicspin (S = 1) and the ^{14}N nuclearspin (I = 1) hyperfine levels in the groundstate manifold, including the electronic MW and nuclear radio frequency (RF) transitions, and the optically induced spin flipflop transitions, γ_{±} that occur between the electronic and nuclear spin in the electronic excited state. In order to drive direct coherent rotations that target individual hyperfine energy levels, NV center electronicspin coherence times \(T_{2}^ {\ast}\, >rsim\, 1\,{\mathrm{\mu} \mathrm{s}}\) are required so that the hyperfine transitions are well resolved.^{7,28,29} Here, we are able to reach this regime by using highpurity nanodiamond crystals with a relatively low nitrogen defect concentration of 50 ppm.^{22}
Figure 1c shows three wellresolved dips in the optically detected magnetic resonance (ODMR) signal (black curve) resulting from the nuclearspin selective electronicspin transitions (MWB, MWC, and MWE — see Fig. 1b). The dip amplitudes of these three transitions correspond to the nuclearspin population, providing a direct measure of nuclearspin polarization. The middle green ODMR curve of Fig. 1c, taken with a magnetic field B_{0} = 51 mT aligned with the NV crystal axis, shows the nuclear spin is polarized into the state +1〉_{n} with high fidelity (>95%). This nuclearspin polarization is caused by the combination of fast spin flipflop transitions γ_{−} and the optical pumping of the electronicspin into 0〉_{e}, illustrated by the solid green wavy and straight arrows, respectively, in Fig. 1b. The γ_{±} transitions are allowed due to the strong perpendicular components of the hyperfine interaction in the NV center’s excited state, which induce spin mixing.^{30} In a single optical cycle, the spin flipflop probabilities γ_{±}/γ_{o} for each transition are
where γ_{o} is the optical pumping rate induced by laser excitation, g_{e} ≈ 2 is the electron gfactor, μ_{B} is the Bohr magneton, and A_{es} and D_{es} are the perpendicular component of the hyperfine interaction and the electronicspin zerofield energy splitting of the NV center’s optical excited state, respectively.^{31,32} The γ_{±} transitions are strongest near the excited state level avoided crossing (ESLAC) at \(B_0 = \mp B_{{\mathrm{ESLAC}}} = \mp D_{{\mathrm{es}}}/(g_e\mu _B) = \mp 51\,{\mathrm{mT}}\) due to the strong spin mixing (Fig. 1d inset). In the moderatefield regime where B_{0} ≈ B_{ESLAC}, the γ_{−} transitions are 10^{3} times stronger than the γ_{+} transitions, as shown in Fig. 1d. This imbalance between γ_{+} and γ_{−} together with the optical electronicspin polarization into 0〉_{e} creates the dynamic nuclear polarization seen in the middle green curve of Fig. 1c. In contrast, in the lowfield regime (B_{0} ≪ B_{ESLAC}), γ_{+} and γ_{−} are comparable, and polarization does not occur, as shown in the top black curve of Fig. 1c.
In the highfield regime where B_{0} ≫ B_{ESLAC}, the γ_{±} transition strengths have similar magnitude and are much weaker overall, which reduces the steadystate degree of polarization and the rate of polarization. To polarize the nuclear spin in this regime, we transfer electron spin polarization onto the nuclear spin using repeated qubit swapping (SWAP) operations, which exchange the electronic and nuclearspin states and are created by direct RF rotations (see Supplementary Note 1) of the nuclearspin and hyperfine selective MW pulses.^{31,33} Figure 1c shows 85% polarization into 0〉_{n} reached at 244 mT (bottom blue curve), indicating that a high degree of nuclearspin polarization can be reached in nanodiamondhosted NV centers in the highfield regime.
In this highfield regime, the slow γ_{±} flipflop transitions make it possible to perform repetitive readout of the nuclearspin state with weak measurement backaction,^{3} as described earlier. We use the protocol shown in Fig. 2a to characterize repetitive readout of the NV center’s electronicspin state after it has been stored in the nuclear spin. After initializing the system into the state 0〉_{e}+1〉_{n}, we use the MWB transition (see Fig. 1b) to either prepare the electronicspin state in −1〉_{e} or leave it in 0〉_{e}. A controlledNOT (CNOT) gate created by two RF pulses maps the electronicspin state onto a nuclearspin state so that 0〉_{e}+1〉_{n} → 0〉_{e}+1〉_{n} and −1〉_{e}+1〉_{n} → −1〉_{e}−1〉_{n}. Finally, the nuclearspin state is measured by repetitive readout of the ODMR on the MWE transition. We store the original electronicspin state in the {±1〉_{n}} code space because this mapping provides a slow loss of relative polarization due to the buffer 0〉_{n} state.^{3} We quantify the repetitive readout by the total number of detector counts, C_{0} and C_{1}, which are collected when the electronic spin is initialized into the 0〉_{e}or −1〉_{e} state, respectively. The figure of merit for this readout scheme is the readout fidelity,^{6,34}
This takes into account both shot noise and spin projection noise and quantifies the efficiency of a readout scheme, approaching unity for an ideal projectionlimited readout.
As shown in Fig. 2b, the repetitive readout protocol can improve readout fidelity by more than a factor of 10 from F = 0.03 for a single electronicspin readout to F = 0.4 for N = 2300 repetitive nuclearspin readouts. The cumulative signal C_{0} − C_{1} increases linearly with repeated readout steps initially but saturates exponentially with a characteristic scale N_{1/e} ~ 1700 (Fig. 2b inset). This saturation occurs because information about the nuclearspin state is lost to measurement backaction induced by the γ_{±} transitions. As the cumulative signal saturates and the shot noise increases with \(\sqrt N\), the readout fidelity decreases for large numbers of readouts as shown in the main plot of Fig. 2b.
These results are consistent with a model that includes only the depolarization errors due to the γ_{±} flipflop transitions during optical readout without any additional depolarization mechanisms (Supplementary Note 2). We thus predict that repetitive readout in our nanodiamonds is limited only by this flipflop interaction occurring in the electronic excited state, and that working in an even higherfield regime (B_{0} ≫ B_{ESLAC}, with reduced γ_{±} rates) would lead to additional enhancements in repetitive readout fidelity, including singleshot readout.^{3}
Errorcorrected repetitive readout
We now present an error correction protocol, which can improve repetitive readout fidelity by correcting errors caused by the γ_{±} transitions. We take advantage of the error imbalance between the weak γ_{+} and strong γ_{−} transitions in the moderatefield regime (Fig. 1d) to design a classical error correction protocol using the three nuclearspin states as our logical storage space. The frequent γ_{−} type flipflop errors from −1〉_{n} to 0〉_{n} can be corrected by periodically performing coherent feedback to transfer any population residing in 0〉_{n} back into the −1〉_{n} state (see illustration in Fig. 3a, top panel). The feedback procedure may actually create an additional error if a γ_{+} type flipflop from +1〉_{n} to 0〉_{n} has occurred, but this type of flipflop error is rare because γ_{+} is weak.
We implement this protocol using the pulse sequence shown in Fig. 3a, bottom panel. This protocol follows that of conventional repetitive readout, but after an error correction period of N_{r} nuclearspin readouts, we use a combination of a MW and a RF pulse to create a SWAP gate in the relevant subspace of {0〉_{e}, −1〉_{e}} ⊗ {0〉_{n},−1〉_{n}}. This coherent feedback operation can be understood by considering a density matrix \(\rho _{0} \propto \left 0 \right\rangle _{\mathrm{e}}\left\langle 0 \right_{\mathrm{e}} \otimes \left( {\left {  1} \right\rangle _{\mathrm{n}}\left\langle {  1} \right_{\mathrm{n}} + {\it{\epsilon }}\left 0 \right\rangle _{\mathrm{n}}\left\langle 0 \right_{\mathrm{n}}} \right) + {\cal O}({\it{\epsilon }}^{2})\), an approximate representation of the system state after an error correction period of N_{r} readout steps, where \({\it{\epsilon }}\) is the probability of a memory qubit incrementing error after these readouts. If small, this error probability can be approximated as \({\it{\epsilon }} {\approx} {\gamma} _ {}t_{r}N_{r}\), where t_{r} is the laser excitation time used for electron spin readout. As long as the error correction period is short enough to keep this error probability low, such an error can be corrected. An ideal SWAP gate transforms the density matrix into \(\rho \propto \left {  1} \right\rangle _{\mathrm{n}}\left\langle {  1} \right_{\mathrm{n}} \otimes \left( {\left 0 \right\rangle _{\mathrm{e}}\left\langle 0 \right_{\mathrm{e}} + {\it{\epsilon }}\left {  1} \right\rangle _{\mathrm{e}}\left\langle {  1} \right_{\mathrm{e}}} \right) + {\cal O}({\it{\epsilon }}^{2})\). The effect of the SWAP gate can be understood as a transfer of the entropy of the nuclearspin state onto the electronic spin. After this transfer, the entropy on the electronic spin could be pumped away by repolarizing the electronic spin with a dedicated laser pulse after every SWAP gate. However, such a laser pulse would drive the γ_{±} transitions and thus induce nuclearspin depolarization and reduce performance. To avoid this additional depolarization, our protocol instead relies on subsequent readouts to repolarize the electronic spin after the SWAP gate.
This combination of N_{r} readout steps and one error correction sequence is repeated until N total readouts have been performed. Figure 3c shows a comparison of the readout fidelity with and without error correction when working in the moderatefield regime (B_{0} = 82 mT) and using frequent error correction (N_{r} = 5). Without error correction, the repetitive readout fidelity reaches a maximum of F = 0.08 for N = 120 readouts, much lower than the highfield fidelity presented earlier due to the strong flipflop error rate γ_{−} in this moderatefield regime. The error correction protocol improves repetitive readout fidelity by 57 ± 8% to F = 0.13 for N = 235 readouts. This improvement is equivalent to increasing the NV center brightness by 250 ± 25%. As shown in Fig. 3c, decreasing the error correction period N_{r} improves readout fidelity because lower repumping periods reduce the probability of an uncorrectable doubleerror (−1〉_{n} → 0〉_{n} → +1〉_{n}) occurring between error correction sequences. The improvement in readout fidelity scales as \(N_{\mathrm{r}}^{  1/2}\) for small N_{r}. This scaling is an intrinsic feature of the protocol and can be explained by a simple model that assumes ideal error correction operations (Supplementary Note 2).
The solid curve in Fig. 3c shows the results of a masterequation model of the population dynamics. This model includes several nonidealities of our system, such as the finite probability for pulse errors and driving of offresonant spin transitions (Supplementary Note 3). The only free parameter in the fit to the data in Fig. 3c is the electronicspin polarization fidelity, which we extract as \(P_{\left 0 \right\rangle _{\mathrm{e}}}\, = \,0.81\), consistent with previous studies.^{12,35,36} The model shows that this low electronicspin polarization fidelity reduces the readout fidelity achievable through the error correction protocol by roughly 30% compared with the ideal scenario in which \(P_{\left 0 \right\rangle _{\mathrm{e}}} = 1\). This reduced performance is due to the finite probability of the system occupying the −1〉_{e}−1〉_{n} state before the error correction operation, which causes the SWAP gate to induce an error by transferring the system into the 0〉_{n} state. The subunity charge state fidelity (ratio of the neutrally charged NV^{0} and the negatively charged NV^{−} population after optical excitation), estimated at 0.75 from our nuclearspin Rabi oscillation data,^{12} also reduces the corrected readout fidelity by roughly 10% because the error correction sequence has no effect in the NV^{0} state. The model further predicts an improvement in readout fidelity of around 100% over conventional repetitive readout for N_{r} = 1.
We also perform errorcorrected repetitive readout measurements in the highfield regime at B_{0} = 244 mT and observed a 7% improvement in readout fidelity due to error correction (Supplementary Note 4). The smaller improvement is expected as our scheme targets only γ_{−} type errors and thus operates most effectively in the moderatefield regime where the error imbalance \(\frac{{\gamma _  }}{{\gamma _ + }}\) is large. Similar error correction protocols with a larger logical storage space would enable the correction of more types of errors.
Using the model described above, we can quantitatively predict the performance of the protocol under different magnetic field conditions. To do so, we fit Eq. (1) to the measured depolarization rates γ_{+} and γ_{−} (Supplementary Note 3) to estimate those rates as a function of the applied magnetic field B_{0}. For simplicity, we assume all other parameters in the model do not change with applied magnetic field. Figure 4 shows the calculated and measured improvement in readout fidelity from error correction, for different error correction periods N_{r}. In the low and high magnetic field regimes, the similarity between the depolarization rates γ_{+} and γ_{−} reduces the improvement from error correction as described above. The reduced improvement around B_{0} ≈ B_{ESLAC} is caused by the fast error rates caused by strong spin mixing. Excluding this regime near B_{ESLAC}, at moderate fields B_{0} ≈ 20 to 140 mT the estimated improvement can reach >1.5fold for frequent error correction.
Discussion
The improvements in readout fidelity achieved with repetitive readout (in the highfield regime) and errorcorrected repetitive readout (in the moderatefield regime) come at the cost of a longer readout time. Reaching the optimal readout fidelity in the highfield repetitive readout measurements (Fig. 2b) requires a 3.2 ms long measurement. The errorcorrected protocol requires 30 μs per correction sequence, dominated by the RF pulse ringdown time, 20 μs, which could be reduced with improved engineering of the RFsignal delivery system. For long measurements such as T_{1} or T_{2}sensing schemes, the longer readout length may not be prohibitive. Indeed, nanodiamondhosted NV centers have most widely been applied to relaxometry measurements that detect changes in the electronicspin T_{1}, which can exceed 1 ms in nanodiamonds.^{37,38,39} This suggests that practical nanodiamondNV sensors could benefit from the readout enhancing techniques described here.
We emphasize that our error correction protocol reverses depolarization that is intrinsic to the physics of the NV center and is thus also applicable to NV centers in bulk diamond. The long T_{2} time (>10 ms) accessible in bulk diamond NV centers means that error correction could improve the sensitivity of NVbased magnetometry measurements. In particular, the techniques described here enable high fidelity repetitive spin readout in the moderate magnetic field regime, increasing the sensing range of the NV center. Lower magnetic field nuclear magnetic resonance (NMR) detection with NV centers could be used to study nanoscale chemical structures in the limit where quadrupolar^{5,40,41} coupling dominates over the Zeeman splitting. Quadrupolar coupling has a typical magnitude of order 1 MHz, corresponding to a strong coupling regime in magnetic fields below about 100 mT. In the strong coupling regime, the energy levels contain rich spatial information about the molecular geometry and motion that is inaccessible in traditional ensemble measurements in the Zeeman regime.^{5,42} In addition, lower magnetic fields require less expensive and bulky apparatus for portable measurement technology.
Correction of measurement backaction errors can be extended in several ways: our protocol uses a pulsed error correction sequence derived from quantum logic. A continuous protocol in which error correction transitions are driven at the same time as readout pulses could provide many of the same improvements without the extension in readout time. Additionally, our protocol used only three states as logical resources, which allows us to correct only one type of error. Other nuclearspin memories with larger dimension, such as a register of hyperfinecoupled ^{13}C nuclei or the host nucleus of the germanium vacancy center in diamond (I_{73Ge} = 9/2), could see dramatically improved performance with similar protocols.
Here, we overcome measurement backaction by using a simple error correction code to protect longitudinal information stored in the ^{14}N memory from bitflip errors. Such classical error correction codes may prove useful to other quantum information protocols that rely on quantumaddressable classical memories. As a nearterm example, recent quantum correlation spectroscopy^{24,43} experiments on NV centers achieve highresolution measurement of NMR signals by storing classical information in the ^{14}N nuclearspin state for long correlation waiting periods. However, to prevent the NV electronic spin from dephasing target spins during the waiting period, the NV must be optically pumped, which reduces the ^{14}N classical storage time. The error correction protocol described here could be used to extend the classical storage time, and hence improve spectral resolution of these measurements.
Methods
We used a homebuilt confocal microscope (NA = 0.9 Nikon air objective) with galvanometer mirror scanning to isolate individual nanodiamondhosted NV centers. A 532 nm laser was modulated by an AOM (AcoustoOptical Modulator, AA optoelectronics MT80A1VIS) to create pulses. The laser intensity was calibrated to the saturation power of the NV center, and 850 ns and 350 ns pulses were used to initialize and readout the electronicspin state, respectively. NV center fluorescence during readout was filtered to select the 600–800 nm band, and coupled into a singlemode optical fiber and detected by an APD (Avalanche PhotoDiode, Excelitas SPCMAQRH14FC). The APD output was gated by a MW switch (MiniCircuits ZASWA250DR+).
MW and RF signals were generated by a Tektronix 70002A 10 GHz AWG (Arbitrary Waveform Generator) on separate channels, then amplified (using MiniCircuits ZHL16W43+ or ZX6014012L+ MW amplifiers and LZY22+ RF amplifiers), combined (Microwave Circuits D02G18G1 diplexer) and delivered to the sample through a 100 μm diameter singleloop inductor. This inductor was fabricated by electron beam deposition of 2 nm of titanium and 200 nm of gold through a steel shadow mask, followed by electrodeposition of silver and a protective layer of gold to a total thickness of roughly 3 μm on a 500 μm thick intrinsic silicon substrate mounted on an aluminum heatsink. This system enables us to apply π rotations of the NV electronic spin in <4 ns with minimal heating effects. We note, however, that nuclearspin selective MW transitions are driven with 399 ns πpulses to minimize the probability to drive nearest neighbor transitions.
We note that in our implementation of the repetitive readout protocol, the NV electronic spin is sometimes left in the −1〉_{e} state after the CNOT gate, which inverts the usual relationship between the fluorescence signal and the initial electronicspin state that is being readout. However, this has a small effect on our overall measurement because the electronic spin repolarizes after only a few readout cycles.
The nanodiamonds were purchased from Nabond Technologies Co. and dispersed in a solution of 10 mg of nanodiamond powder to 25 mL of ethanol via sonication. This solution was deposited on the sample using an Omron U22 nebulizer. This method produces a welldispersed distribution of nanodiamond locations and allows individual addressing by the confocal microscope. Previous work has shown these nanodiamonds have a typical diameter of (23 ± 7) nm.^{22} Approximately 1% of the NV centers detected with our confocal microscope show the coherence required to resolve the hyperfine coupling in ODMR measurements \({({T_{2}^{\ast}}\;{>rsim}\;1 {\mathrm{\mu}\mathrm{s}})}\).
The static magnetic field was created by a large axially magnetized \(\left( {\emptyset 5{\mathrm{X}}4} \right)\) cm neodymium permanent magnet mounted on a twoaxis rotation mount, centered on the sample. This setup can produce fields up to 0.25 T limited by the size of the sample heatsink, which prevents the magnet from being placed closer to the sample. Magnetic field alignment was performed by using ODMR measurements to monitor the diagnostic parameter f_{MWA} ± f_{MWB}, which displays an extremum as a function of magnetic field orientation when the field is aligned. The + (−) condition is chosen for magnetic fields below (above) the groundstate anticrossing. Based on our calibration, we estimate that using this method we can reduce the offaxis component of the magnetic field to less than 50 μT.
Data availability
The data sets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
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Acknowledgements
We thank Gavin Morely for stimulating discussions. We gratefully acknowledge financial support by the Leverhulme Trust Research Project Grant 2013337, the European Research Council ERC Consolidator Grant Agreement No. 617985 and the Winton Programme for the Physics of Sustainability. H.S.K. acknowledges financial support by St John’s College through a Research Fellowship. J.H. acknowledges the UK Marshall Aid Commemoration Commission for financial support.
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J.B. designed and built the experimental apparatus. J.H. fabricated the sample. J.H. performed experiments and data analysis, with input of J.B., H.S.K., and D.K. All authors contributed in the writing of the manuscript. M.A., H.S.K., and D.K. advised and coordinated all efforts.
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Correspondence to Mete Atatüre.
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Holzgrafe, J., Beitner, J., Kara, D. et al. Error corrected spinstate readout in a nanodiamond. npj Quantum Inf 5, 13 (2019). https://doi.org/10.1038/s4153401901262
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