Abstract
A promising approach for multiqubit quantum registers is to use optically addressable spins to control multiple dark electronspin defects in the environment. While recent experiments have observed signatures of coherent interactions with such dark spins, it is an open challenge to realize the individual control required for quantum information processing. Here, we demonstrate the heralded initialisation, control and entanglement of individual dark spins associated to multiple P1 centers, which are part of a spin bath surrounding a nitrogenvacancy center in diamond. We realize projective measurements to prepare the multiple degrees of freedom of P1 centers—their JahnTeller axis, nuclear spin and charge state—and exploit these to selectively access multiple P1s in the bath. We develop control and singleshot readout of the nuclear and electron spin, and use this to demonstrate an entangled state of two P1 centers. These results provide a proofofprinciple towards using dark electronnuclear spin defects as qubits for quantum sensing, computation and networks.
Introduction
Optically active defects in solids provide promising qubits for quantum sensing^{1}, quantuminformation processing^{2,3,4}, quantum simulations^{5,6}, and quantum networks^{7,8,9}. These defects, including the nitrogenvacancy (NV) and siliconvacancy (SiV) centers in diamond and various defects in siliconcarbide^{10,11,12}, combine long spin coherence times^{4,13,14,15,16,17,18}, highquality control and readout^{2,3,4,14,19,20,21}, and a coherent optical interface^{7,8,9,15,19,22}.
Largerscale systems can be realized by entangling multiple defects together through longrange optical network links^{7,8,9} and through direct magnetic coupling, as demonstrated for a pair of ionimplanted NV centers^{23,24}. The number of available spins can be further extended by controlling nuclear spins in the vicinity. Multiqubit quantum registers^{4,24,25,26,27}, quantum error correction^{2,3}, enhanced sensing schemes^{28}, and entanglement distillation^{29} have been realized using nuclear spins.
The ability to additionally control dark electron–spin defects that cannot be directly detected optically would open new opportunities. Examples are studying single defect dynamics^{30}, extended quantum registers, enhanced sensing protocols^{28,31,32}, and spin chains for quantum computation architectures^{33,34,35,36}. Two pioneering experiments reported signals consistent with an NV center coupled to a single P1 center (a dark substitutional nitrogen defect)^{37,38}, but the absence of the expected P1 electron–spin resonance signal^{39} and later results revealing identical signals due to NV–^{13}C couplings in combination with an excited state anticrossing^{40}, make these assignments inconclusive. Recent experiments have revealed signatures of coherent interactions between NV centers and individual dark electronspin defects, including P1 centers^{41,42,43}, N2 centers^{44}, and notyetassigned defects^{31,45,46,47,48,49}. Those results have revealed the prospect of using dark spin defects as qubits. However, highquality initialization, measurement, and control of multiqubit quantum states is required to exploit such spins as a quantum resource.
Here, we demonstrate the control and entanglement of individual P1 centers that are part of a bath surrounding an NV center in diamond (Fig. 1a). A key property of the P1 center is that, in addition to its electron spin, it exhibits three extra degrees of freedom: the Jahn–Teller axis, a nuclear spin, and the charge state^{50,51,52}. Underlying our selective control of individual centers is the heralded preparation of specific configurations of these additional degrees of freedom for multiple P1 centers through projective measurements. In contrast, all previous experiments averaged over these additional degrees of freedom^{41,42,53}. We use this capability to develop initialization, singleshot readout, and control of the electron and nuclear spin states of multiple P1s, and investigate their spin relaxation and coherence times. Finally, we demonstrate the potential of these dark spins as a qubit platform by realizing an entangled state between two P1 electron spins through their direct magnetic–dipole coupling.
Results
A spin bath with multiple degrees of freedom
We consider a bath of P1 centers surrounding a single NV center at 3.3 K (Fig. 1a). The diamond is isotopically purified with an estimated ^{13}C concentration of 0.01%. The P1 concentration is estimated to be ~75 ppb (see Supplementary Note 5). Three P1 charge states are known^{51,52}. The experiments in this work detect the neutral charge state and do not generate a signal for the positive and negative charge states. In addition to an electron spin (S = 1/2), the P1 center exhibits a ^{14}N nuclear spin (I = 1, 99.6% natural abundance) and a Jahn–Teller (JT) distortion, which results in four possible symmetry axes due to the elongation of one of the four N–C bonds^{54}. Both the ^{14}N state and the JT axis generally fluctuate over time^{55,56,57}. The Hamiltonian for a single neutrally charged P1 defect in one of the four JT axes i ∈ {A, B, C, D} is^{50}
where γ_{e} (γ_{n}) is the electron (^{14}N) gyromagnetic ratio, B the external magnetic field vector, S and I are the electron spin1/2 and nuclear spin1 operator vectors, and \({\hat{{\bf{A}}}}_{{\bf{i}}}\) (\({\hat{{\bf{P}}}}_{{\bf{i}}}\)) the hyperfine (quadrupole) tensor. We label the ^{14}N (m_{I} ∈ − 1, 0, + 1) and JT states as \(\left{m}_{I},i\right\rangle\), and the electron spin states as \(\left\uparrow \right\rangle\) and \(\left\downarrow \right\rangle\). For convenience, we use the spin eigenstates as labels, while the actual eigenstates are, to some extent, mixtures of the ^{14}N and electron spin states.
We probe the bath surrounding the NV by double electron–electron resonance (DEER) spectroscopy^{41,42,45,47,53}. The DEER sequence consists of a spinecho on the NV electron spin, which decouples it from the environment, plus a simultaneous πpulse that selectively recouples resonant P1 centers. Figure 1b reveals a complex spectrum. The degeneracy of three of the JT axes is lifted by a purposely slightly tilted magnetic field with respect to the NV axis (θ ≈ 4°). In combination with the long P1 dephasing time (\({T}_{2}^{* } \sim \,50\,\) µs, see below) this enables us to resolve all 12 main P1 electron–spin transitions—for four JT axes and three ^{14}N states—and selectively address at least one transition for each JT axis.
Several additional transitions are visible due to the mixing of the electron and nuclear spin in the used magnetic field regime (γ_{e}∣B∣ ~ A_{∥}, A_{⊥}). We select 11 wellisolated transitions to fit the P1 Hamiltonian parameters and obtain {A_{∥}, A_{⊥}, P_{∥}} = {114.0264(9), 81.312(1), − 3.9770(9)} MHz and B = {2.437(2), 1.703(1), 45.5553(5)} G (Supplementary Note 4), closely matching ensemble ESR measurements^{58}. The experimental spectrum is well described by the 60 P1 transitions for these parameters. No signal is observed at the bare electron Larmor frequency (≈128 MHz), confirming that the P1 centers form the dominant electron spin bath.
To probe the coupling strength of the P1 bath to the NV, we sweep the interaction time in the DEER sequences (Fig. 1c). The curves for the different \(\left+1,i\right\rangle\) states show oscillatory features, providing a first indication of an underlying microscopic structure of the P1 bath. However, like all previous experiments^{41,42,53}, these measurements are a complex averaging over ^{14}N, JT, and charge states for all the P1 centers, which obscures the underlying structure and hinders control over individual spins.
Detecting and preparing single P1 centers
To investigate the microscopic structure of the bath we repeatedly apply the DEER sequence and analyze the correlations in the measurement outcomes^{30}. Figure 2a shows a typical time trace for continuous measurements, in which groups of K = 820 measurements are binned together (see Fig. 2b for the sequence). We observe discrete jumps in the signal that indicate individual P1 centers jumping in and out of the \(\left+1,{\rm{D}}\right\rangle\) state. The resulting histogram (Fig. 2a) reveals multiple discrete peaks that indicate several P1 centers with different coupling strengths to the NV center, as schematically illustrated in Fig. 2c. We tentatively assign four P1 centers S1, S2, S3 and S4 to these peaks.
We verify whether these peaks originate from single P1 centers by performing crosscorrelation measurements. We first apply a DEER measurement on \(\left+1,{\rm{D}}\right\rangle\) followed by a measurement on \(\left+1,{\rm{A}}\right\rangle\) (Fig. 2d). For a single P1, observing it in \(\left+1,{\rm{D}}\right\rangle\) would make it unlikely to subsequently also find it in state \(\left+1,{\rm{A}}\right\rangle\). We observe three regions of such anticorrelation (red rectangles in Fig. 2d). We define the correlation
where \({N}_{{\rm{A}}}^{\min }\), \({N}_{{\rm{A}}}^{\max }\), \({N}_{{\rm{D}}}^{\min }\), and \({N}_{{\rm{D}}}^{\max }\) define the region, and where P(X) is the probability that X is satisfied. Assuming that the states of different P1 centers are uncorrelated, a value C < 0.5 indicates that the signal observed in both the DEER sequences on \(\left+1,{\rm{A}}\right\rangle\) and \(\left+1,{\rm{D}}\right\rangle\) is associated to a single P1 center, while C < 2/3 indicates 1 or 2 centers (Supplementary Note 8).
For the three areas, we find C = 0.40(5), 0.22(4), and 0.47(5) for S1, S2 and S3/S4, respectively. These correlations corroborate the assignments of a single P1 to both S1 and S2 and one or two P1s for S3/S4 (the result is within one standard deviation from 0.5). In addition, these results reveal which signals for different \(\left+1,i\right\rangle\) states belong to which P1 centers. This is nontrivial because the NV–P1 dipolar coupling varies with the JT axis, as exemplified in Fig. 2d (see Supplementary Note 3 for a theoretical treatment).
Next, we develop singleshot readout and heralded initialization of the ^{14}N and JT state of individual P1 centers. For this, we represent the time trace data (Fig. 2a) as a correlation plot between subsequent measurements k and k + 1 (Fig. 2e)^{59,60,61}. We bin the outcomes using K = 820 repetitions, where K is chosen as a tradeoff between the ability to distinguish S1 from S2 and the disturbance of the state due to the repeated measurements (1/e value of ~1.5 × 10^{4} repetitions, see Supplementary Note 6). Separated regions are observed for the different P1 centers. Therefore, by setting threshold conditions, one can use the DEER measurement as a projective measurement to initialize or readout the \(\left{m}_{I},i\right\rangle\) state of selected P1 centers, which we illustrate for S1.
First, we set an initialization condition N(k) > N_{S1} (blue dashed line) to herald that S1 is initialized in the \(\left+1,{\rm{D}}\right\rangle\) state and that S2, S3/S4 are not in that state. We use N(k) ≤ N_{notS1} to prepare a mixture of all other other possibilities. The resulting conditional probability distributions of N(k + 1) are shown in Fig. 2f. Second, we set a threshold for state readout N_{RO} to distinguish between the two cases. We then optimize N_{S1} for the tradeoff between the success rate and signal contrast, and find a combined initialization and readout fidelity F = 0.96(1) (see “Methods”). Other states can be prepared and read out by setting different conditions (Supplementary Note 8).
Control of the electron and nuclear spin
To control the electron spin of individual P1 centers, we first determine the effective dipolar NV–P1 coupling. We prepare, for instance, S1 in \(\left+1,{\rm{D}}\right\rangle\) and perform a DEER measurement in which we sweep the interaction time (Fig. 3a). By doing so, we selectively couple the NV to S1, while decoupling it from S2 and S3/S4, as well as from all bath spins that are not in \(\left+1,{\rm{D}}\right\rangle\). By applying this method we find effective dipolar coupling constants ν of 2π ⋅ 1.910(5), 2π ⋅ 1.563(6) and 2π ⋅ 1.012(8) kHz for S1, S2 and S3/S4, respectively. Note that, if the signal for S3/S4 originates from two P1 centers, the initialization sequence prepares either S3 or S4 in each repetition of the experiment.
We initialize and measure the electron spin state of the P1 centers through a sequence with a modified readout axis that we label DEER(y) (Fig. 3b). Unlike the DEER sequence, this sequence is sensitive to the P1 electron spin state. After initializing the charge, nuclear spin and JT axis, and setting the interaction time τ ≈ π/(2 ⋅ ν), the DEER(y) sequence projectively measures the spin state of a selected P1 center (Fig. 3c). We first characterize the P1 electron spin relaxation under repeated application of the measurement and find a 1/e value of ~250 repetitions (Supplementary Note 6). We then optimize the number of repetitions and the initialization and readout thresholds to obtain a combined initialization and singleshot readout fidelity for the S1 electron spin of \({F}_{\left\uparrow \right\rangle /\left\downarrow \right\rangle }\) = 0.95(1) (Fig. 3d).
We now show that we can coherently control the P1 nitrogen nuclear spin (Fig. 4a). To speed up the experiment, we choose a shorter initialization sequence that prepares either S1 or S2 in the \(\left+1,{\rm{D}}\right\rangle\) state (K = 420, “Methods”). We then apply a radiofrequency (RF) pulse that is resonant with the m_{I} = +1 ↔ 0 transition if the electron spin is in the \(\left\uparrow \right\rangle\) state. Varying the RF pulse length reveals a coherent Rabi oscillation. Because the P1 electron spin is not polarized, the RF pulse is on resonance 50% of the time and the amplitude of the Rabi oscillation is half its maximum.
We use the combined control over the electron and nuclear spin to determine the sign of the NV–P1 couplings (Fig. 4b). First, we initialize the ^{14}N, JT axis and electron spin state of a P1 center. Because the DEER(y) sequence is sensitive to the sign of the coupling (Fig. 3c), the sign affects whether the P1 electron spin is prepared in \(\left\uparrow \right\rangle\) or \(\left\downarrow \right\rangle\). Second, we measure the P1 electron spin through the ^{14}N nuclear spin. We apply an RF pulse, which implements an electroncontrolled CNOT gate on the nuclear spin (see Fig. 4a). Subsequently reading out the ^{14}N spin reveals the electron spin state and therefore the sign of the NV–P1 coupling. We plot the normalized difference R (“Methods”) for two different initialization sequences that prepare the electron spin in opposite states. The results show that the NV–P1 coupling is positive for the cases of S1 and S3/S4, but negative for S2 (Fig. 4b). If S3/S4 consists of two P1 centers, then they have the same coupling sign to the NV.
Spin coherence and relaxation
To assess the potential of P1 centers as qubits, we measure their coherence times. First, we investigate the relaxation times. We prepare either S1 or S2 in \(\left+1,{\rm{D}}\right\rangle\), the NV electron spin in m_{s} = 0, and vary the waiting time t before reading out the same state (Fig. 5a). This sequence measures the relaxation of a combination of the nitrogen nuclear spin state, JT axis and charge state, averaged over S1 and S2. An exponential fit gives a relaxation time of \({T}_{\left+1,{\rm{D}}\right\rangle }\) = 40(4) s (Fig. 5b, green).
We measure the longitudinal relaxation of the electron spin by preparing either \(\left\uparrow \right\rangle\) (S1) or \(\left\downarrow \right\rangle\) (S2) (Fig. 5a). We postselect on the \(\left+1,{\rm{D}}\right\rangle\) state at the end of the sequence to exclude effects due to relaxation from \(\left+1,{\rm{D}}\right\rangle\), and find T_{1e} = 21(7) s. The observed electron spin relaxation time is longer than expected from the typical P1–P1 couplings in the bath (order of 1 kHz). A potential explanation is that flipflops are suppressed due to couplings to neighboring P1 centers, which our heralding protocol preferentially prepares in other \(\left{m}_{I},i\right\rangle\) states. Below, we will show that S1 and S2 have a strong mutual coupling, which could shift them offresonance from the rest of the bath.
Second, we investigate the electron and nitrogen nuclear spin coherence via Ramsey and spinecho experiments (Fig. 5c, d). We find \({T}_{2{\rm{e}}}^{* }\) = 50(3) μs and T_{2e} = 1.00(4) ms for the electron spin, and \({T}_{2{\rm{N}}}^{* }\) = 0.201(9) ms and T_{2N} = 4.2(2) ms for the nitrogen nuclear spin. The ratio of dephasing times for the electron and nitrogen nuclear spins is ~4, while the difference in bare gyromagnetic ratios is a factor ~9000. The difference is partially explained by electronnuclear spin mixing due to the large value of A_{⊥}, which changes the effective gyromagnetic ratios of the nitrogen nuclear spin and electron spin. Based on this, a ratio of dephasing times of 12.6 is expected (see Supplementary Note 13). The remaining additional decoherence of the nitrogen nuclear spin is currently not understood.
The electron Ramsey experiment shows a beating frequency of 21.5(1) kHz (Fig. 5d). As the data is an average over S1 and S2, this suggests an interaction between these two P1 centers. Note that, whilst the signal is expected to contain 11 frequencies due to the different Jahn–Teller and nitrogen nuclear spin state combinations, the observation of a single beating frequency indicates that these are not resolved. Next, we will confirm this hypothesis and use the coupling between S1 and S2 to demonstrate an entangled state of two P1 centers.
Entanglement of two dark electron spins
Thus far we have shown selective initialization, control and singleshot readout of individual P1 centers within the bath. We now combine all these results to realize coherent interactions and entanglement between the electron spins of two P1 centers.
We first sequentially initialize both P1 centers (Fig. 6a). To overcome the small probability for both P1 centers to be in the desired state, we use fast logic to identify failed attempts in realtime and actively reset the states (“Methods”). We prepare S1 in the \(\left+1,{\rm{D}}\right\rangle\) state and S2 in the \(\left+1,{\rm{A}}\right\rangle\) state. By initializing the two P1 centers in these different states, we ensure that the spin transitions are strongly detuned, so that mutual flipflops are suppressed and the interaction is effectively of the form S_{z}S_{z}. We then sequentially initialize both electron spins to obtain the initial state \({\left\uparrow \right\rangle }_{{\rm{S}}1}{\left\downarrow \right\rangle }_{S{\rm{2}}}\). As consecutive measurements can disturb the previously prepared degrees of freedom, the number of repetitions in each step is optimized for high total initialization fidelity and success rate (Supplementary Note 15C).
Next, we characterize the dipolar coupling J between S1 and S2 (Fig. 6b). We apply two π/2 pulses to prepare both spins in a superposition. We then apply simultaneous echo pulses on each spin. This double echo sequence decouples the spins from all P1s that are not in \(\left+1,{\rm{D}}\right\rangle\) or \(\left+1,{\rm{A}}\right\rangle\), as well as from the ^{13}C nuclear spin bath and other noise sources. This way, the coherence of both spins is extended from \({T}_{2}^{* }\) to T_{2}, while their mutual interaction is maintained. We determine the coupling J by letting the spins evolve and measuring 〈XZ〉 as a function of the interaction time 2t through a consecutive measurement of both electron spins (Fig. 6b). From this curve we extract a dipolar coupling J = −2π ⋅ 17.8(5) kHz between S1 in \(\left+1,{\rm{D}}\right\rangle\) and S2 in \(\left+1,{\rm{A}}\right\rangle\).
Finally, we create an entangled state of S1 and S2 using the sequence in Fig. 6a. We set the interaction time 2t = π/J so that a 2qubit CPHASE gate is performed. The final state is (see Supplementary Note 14)
with \(\left\pm \right\rangle\) = \(\frac{\left\uparrow \right\rangle \pm \left\downarrow \right\rangle }{\sqrt{2}}\). We then perform full twoqubit state tomography and reconstruct the density matrix as shown in Fig. 6c. The resulting state fidelity with the ideal state is F = (1 + 〈XZ〉 − 〈ZX〉 − 〈YY〉)/4 = 0.81(5). The fact that F > 0.5 is a witness for twoqubit entanglement^{62}. The coherence time during the echo sequence (~700 μs, see “Methods”) is long compared to π/J (~28 μs), and thus the dephasing during the 2qubit gate is estimated to be at most 2%. Therefore we expect the main sources of infidelity to be the final sequential singleshot readout of the twoelectron spin states—no readout correction is made—and the sequential initialization of the twoelectron spins (Supplementary Note 15).
Discussion
In conclusion, we have developed initialization, control, singleshot readout, and entanglement of multiple individual P1 centers that are part of a bath surrounding an NV center. These results establish the P1 center as a promising qubit platform. Our methods to control individual dark spins can enable enhanced sensing schemes based on entanglement^{28,31,32}, as well as electron spin chains for quantum computation architectures^{33,34,35,36}. Larger quantum registers might be formed by using P1 centers to control nearby ^{13}C nuclear spins with recently developed quantum gates^{4}. Such nuclear spin qubits are connected to the optically active defect only indirectly through the P1 electron spin and could provide isolated robust quantum memories for quantum networks^{63}. Finally, these results create new opportunities to investigate the physics of decoherence, spin diffusion, and Jahn–Teller dynamics^{30} in complex spin baths with control over the microscopic singlespin dynamics.
Methods
Sample
We use a single NV center in a homoepitaxially chemicalvapordeposition (CVD) grown diamond with a 〈100〉 crystal orientation (Element Six). The diamond is isotopically purified to an approximate 0.01% abundance of ^{13}C. The nitrogen concentration is ~75 parts per billion, see Supplementary Note 5. To enhance the collection efficiency a solidimmersion lens was fabricated on top of the NV center^{64,65} and a singlelayer aluminumoxide antireflection coating was deposited^{66,67}.
Setup
The experiments are performed at 3.3 K (Montana Cryostation) with the magnetic field B applied using three permanent magnets on motorized linear translation stages (UTS100PP) outside of the cryostat housing. We realize a long relaxation time for the NV electron spin (T_{1} > 30 s) in combination with fast NV spin operations (peak Rabi frequency ~26 MHz) and readout/initialization (~40 μs/100 μs), by minimizing noise and background from the microwave and optical controls^{13}. Amplifier (AR 20S1G4) noise is suppressed by a fast microwave switch (TriQuint TGS2355SM). Video leakage noise generated by the switch is filtered with a high pass filter.
Error analysis
The data presented in this work is either a probability derived from the measurements, the mean of a distribution, or a quantity derived from those. For probabilities, a binomial error analysis is used, where p is the probability and \(\sigma =\sqrt{p\cdot (1p)/Q}\), Q being the number of measured binary values. For the mean μ of a distribution, σ_{μ} is calculated as \(\sigma /\sqrt{Q}\), where σ is the square root average of the squared deviations from the mean and Q is the number of measurements. Uncertainties on all quantities derived from a probability or a mean are calculated using error propagation.
NV spin control and readout
We use Hermite pulse envelopes^{68,69} to obtain effective microwave pulses without initialization of the intrinsic ^{14}N nuclear spin of the NV. We initialize and read out the NV electron spin through spin selective resonant excitation (F = 0.850(5))^{64}. Laser pulses are generated by acoustic optical modulators (637 nm Toptica DL Pro, for spin pumping and New Focus TLB6704P for singleshot spin readout) or by direct current modulation (515 nm laser, Cobolt MLD—for charge state control, and scrambling the P1 center state, see Supplementary Note 7). We place two modulators in series (Gooch and Housego Fiber Q) for an improved on/off ratio for the 637 nm lasers.
Magnetic field stabilization
During several of the experiments, we actively stabilize the magnetic field via a feedback loop to one of the translation stages. The feedback signal is obtained from interleaved measurements of the NV \(\left0\right\rangle \leftrightarrow \left1\right\rangle\) transition frequency. We use the P1 bath as a threeaxis magnetometer to verify the stability of the magnetic field during this protocol (see Supplementary Note 11), and find a magnetic field that is stable to <3 mG along z and <20 mG along the x, y directions.
Heralded initialization protocols
Initialization of the P1 ^{14}N spin, JT axis, charge, and electron spin states is achieved by heralded preparation. Before starting an experimental sequence, we perform a set of measurements that, given certain outcomes, signals that the system is in the desired state.
A challenge is that the probability for the system to be in a given desired state is low, especially in experiments with multiple P1 centers (e.g., Fig. 6). We realize fast initialization by combining the heralded preparation with fast logic (ADwinPro II) to identify unsuccessful attempts in realtime and then actively reset the system to a random state. This way each step is performed only if all previous steps were successful, and one avoids being trapped in an undesired state.
To reset the P1 centers to a random state, we use photoexcitation^{70} of the P1s. We apply a ~5 μs 515 nm laser pulse to scramble the ^{14}N, JT, and charge states of P1 centers. See Supplementary Note 7 for details and the optimization procedure.
The most timeconsuming step is the selective initialization of the JahnTeller and ^{14}N spin states, as K = 820 repetitions are required to distinguish the signals from the P1 centers (S1, S2 and S3/S4). However, cases for which none of these P1 centers are in the desired state can be identified already after a few repetitions (Supplementary Note 7). So after K = 5 repetitions we infer the likelihood for the desired configuration and use fast logic to determine whether to apply a new optical reset pulse or continue with the full sequence (K = 820). This procedure significantly speeds up the experiments (Supplementary Note 7). For creating the entangled state (Fig. 6) we use a more extensive procedure, which is detailed in Supplementary Note 15C.
In the experiments in Figs. 4a and 5, we take an alternative approach to speed up the experiments by using a shorter initialization sequence (K = 420) that does not distinguish between S1 and S2. Such a sequence prepares either S1 or S2, and the resulting data is an average over the two cases. Note that this method cannot be used in experiments where a selective initialization is required (e.g., Fig. 3, Fig. 4b, Fig. 6).
The optimization of the heralded initialization fidelities is discussed in Supplementary Note 15.
Initialization and singleshot readout fidelity
We define the combined initialization and readout fidelity for S1 in \(\left+1,{\rm{D}}\right\rangle\) and S2, S3/S4 not in that state as
whereas for a mixture of all other possibilities we define
In both cases P(X∣Y) is the probability to obtain X given Y. We then take the average fidelity of these two cases:
We initialize and measure the electron spin state of P1 centers through a DEER(y) sequence following the initialization of the \(\left+1,{\rm{D}}\right\rangle\) state. Similarly, we use the correlation of consecutive measurements M(k) and M(k + 1) to determine the combined initialization and readout fidelity \({F}_{\left\uparrow \right\rangle /\left\downarrow \right\rangle }\). First, we define the fidelity for \(\left\uparrow \right\rangle\) as
and the fidelity for \(\left\downarrow \right\rangle\) as
Finally, the average combined initialization and readout fidelity is given as
For a description of the optimization of the singleshot readout fidelities, we refer to Supplementary Note 15.
Data analysis
The DEER measurements in Fig. 1c are fitted to
from which we find T_{2,DEER} of 0.767(6), 0.756(7), 0.802(6), and 0.803(5) ms for \(\left+1,{\rm{A}}\right\rangle\), \(\left+1,{\rm{B}}\right\rangle\), \(\left+1,{\rm{C}}\right\rangle\), and \(\left+1,{\rm{D}}\right\rangle\), respectively. The obtained values for ω are 2π ⋅ 2.12(5), 2π ⋅ 2.14(3), and 2π ⋅ 2.78(6) kHz with corresponding amplitudes B_{0} of 0.105(5), 0.218(7), and 0.073(4) for \(\left+1,{\rm{A}}\right\rangle\), \(\left+1,{\rm{B}}\right\rangle\), and \(\left+1,{\rm{C}}\right\rangle\), respectively. For \(\left+1,{\rm{D}}\right\rangle\) we fix B_{0} = 0.
The DEER measurements with P1 initialization (Fig. 3a) and the P1 nitrogen nuclear spin Ramsey (Fig. 5c) are fitted to
For the dephasing time during the DEER sequence (here t = 2τ) we find T = 0.893(5), 0.763(8), and 0.790(8) ms for S1, S2 and S3/S4, respectively. The obtained respective dipolar coupling constants ν are 2π ⋅ 1.894(3), 2π ⋅ 1.572(6), and 2π ⋅ 1.001(6) kHz. For the P1 nitrogen nuclear spin Ramsey we find a dephasing time of T = \({T}_{2{\rm{N}}}^{* }\) = 0.201(9) ms.
Spinecho experiments (Fig. 1c and Fig. 5) are fitted to
For the NV spinecho (Fig. 1c), T = T_{2} = 0.992(4) ms with n = 3.91(7). For the P1 nitrogen nuclear spin and electron (insets of Fig. 5c, d) T is T_{2N} = 4.2(2) ms or T_{2e} = 1.00(4) ms with the exponents n = 3.9(8) and n = 3.1(5), respectively.
The Ramsey signal for the P1 electron spin in Fig. 5d is fitted to a sum of two frequencies with a Gaussian decay according to
which gives a beating frequency f_{b} = 2π ⋅ 21.5(5) kHz.
The value R (Fig. 4b) is defined as
where P_{(+y)} (P_{(−y)}) is the probability to read out the ^{14}N spin in the m_{I} = +1 state when using a +y (−y) readout basis in the DEER(y) sequence used to initialize the electron spin (Fig. 4b, see Supplementary Note 9).
Twoqubit gate fidelity
We estimate the dephasing during the twoqubit CPHASE gate in Fig. 6 by extrapolation of the measured P1 electron T_{2e} = 1.00(4) ms for a single spinecho pulse (decoupled from all spins except those in \(\left+1,{\rm{D}}\right\rangle\)). We use the scaling \({T}_{2}\propto 1/\sqrt{\langle {n}_{{\rm{spins}}}\rangle }\) with 〈n_{spins}〉 the average number of spins coupled to during the measurement^{53}. The twoqubit gate is implemented by a double echo and the two P1s are thus not decoupled from spins in \(\left+1,{\rm{D}}\right\rangle\) and \(\left+1,{\rm{A}}\right\rangle\), resulting in \({T}_{2} \sim {T}_{2e}/\sqrt{2}\approx\) 700 μs. Assuming the same decay curve as for T_{2e} (n = 3.1) this implies a loss of fidelity due to dephasing of ~0.4%. For a Gaussian decay (n = 2) the infidelity would be ~2%.
Data availability
The data and code underlying the figures of this research article are available online through https://doi.org/10.4121/14376611.
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Acknowledgements
We thank V.V. Dobrovitski, G. de Lange, and R. Hanson for useful discussions. This work was supported by the Netherlands Organization for Scientific Research (NWO/OCW) through a Vidi grant and as part of the Frontiers of Nanoscience (NanoFront) program. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 852410). This project (QIA) has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No. 820445.
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M.J.D., S.J.H.L., and T.H.T. devised the project and the experiments. C.E.B., M.J.D., S.J.H.L., and H.P.B. constructed the experimental apparatus. M.J.D. and S.J.H.L. performed the experiments. M.J.D., S.J.H.L., H.P.B., and T.H.T. analyzed the data. A.L.M. and M.J.D. performed the preliminary experiments. M.M. and D.J.T. grew the diamond sample. M.J.D., S.J.H.L., and T.H.T. wrote the paper with input from all authors. T.H.T. supervised the project.
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Degen, M.J., Loenen, S.J.H., Bartling, H.P. et al. Entanglement of dark electronnuclear spin defects in diamond. Nat Commun 12, 3470 (2021). https://doi.org/10.1038/s41467021234549
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