Protecting solid-state spins from a strongly coupled environment

We consider a system of two spins interacting via an hyperfine coupling A that we describe semi-classically using a spin-fluctuator model [37].

We model the intrinsic ¹⁴N nuclear spin I of the NV center as a random walker, whose phase evolves subject to the state of its neighboring electronic spin S that acts as a strongly coupled fluctuator and generates RTN (figure 1). In such analogy, the velocity v of the random walker is linked to the phase accumulation rate of the qubit, and is set by the hyperfine interaction, v = m_S A. The fluctuator flips at a rate γ between any two m_S states due to spin-lattice relaxation, inducing a change in the value of the hyperfine coupling, m_S A, thus at each such event the walker's velocity v also changes. In between fluctuator jumps, the walker covers a distance φ, representing the accumulated phase between qubit states.

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For a 2-level fluctuator (2LF), the random walker has only two possible directions of motion, left or right. The probability ${\hat{p}}_{r}$(${\hat{p}}_{l}$) of reaching a position φ from the right(left)—that is, from $\phi +(-)vdt$—is therefore the sum of the probabilities of keeping going to the left(right) and of turning left(right) with the 'flipping' probability γ. The random walker's motion can then be described by a system of differential equations [37]:

$$\begin{eqnarray}\left\{\begin{array}{rcl}{\partial }_{t}{\hat{p}}_{l}(\varphi ,t) & = & -\gamma [{\hat{p}}_{l}(\varphi ,t)-{\hat{p}}_{r}(\varphi ,t)]+v{\partial }_{\varphi }{\hat{p}}_{l}(\varphi ,t)\\ {\partial }_{t}{\hat{p}}_{r} & = & -\gamma ({\hat{p}}_{r}-{\hat{p}}_{l})-v{\partial }_{\varphi }{\hat{p}}_{r}.\end{array}\right.\end{eqnarray} \tag{ 1 }$$

The spin coherence is given by the average accumulated phase, $\langle {{\rm{e}}}^{{\rm{i}}\varphi }(t)\rangle =\int \hat{P}(\varphi ,t){{\rm{e}}}^{{\rm{i}}\varphi }{\rm{d}}\varphi$, where $\hat{P}(\varphi ,t)={\hat{p}}_{l}+{\hat{p}}_{r}$ is the total probability of reaching φ at time t. Note that this is the Fourier transform of $\hat{P}$, P(k, t), evaluated at k = −1. We can thus more compactly write the relevant equations of motion as ${\partial }_{t}\vec{P}(k,t)={M}_{k}\vec{P}(k,t)$, that is

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial t}\left(\begin{array}{l}P\\ p\end{array}\right)=\left(\begin{array}{cc}0 & -{\rm{i}}{kv}\\ -{\rm{i}}{kv} & -2\gamma \end{array}\right)\left(\begin{array}{l}P\\ p\end{array}\right),\end{eqnarray} \tag{ 2 }$$

where $p(k,t)={p}_{r}(k,t)-{p}_{l}(k,t)$.

We can similarly describe the spin dynamics in the presence of a three-level fluctuator (3LF), where one of the levels corresponds to a 'rest' state v = 0 (no phase accumulation), with corresponding probability p₀:

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial t}\left(\begin{array}{l}P\\ {p}^{+}\\ p\end{array}\right)=\left(\begin{array}{ccc}0 & 0 & -{\rm{i}}{k}{v}\\ 2\gamma & -3\gamma & -{\rm{i}}{k}{v}\\ 0 & -{\rm{i}}{k}{v} & -3\gamma \end{array}\right)\left(\begin{array}{l}P\\ {p}^{+}\\ p\end{array}\right),\end{eqnarray} \tag{ 3 }$$

where p⁺ = p_l + p_r and here P = p_l + p_r + p₀. The spin decay is then given by $\langle {{\rm{e}}}^{{\rm{i}}\varphi }(t)\rangle ={{\rm{e}}}^{{M}_{-1}t}P(-1,0)$ and is characterized by a typical timescale ${T}_{2}^{* n}$. Note that since we consider the fluctuator to be either a spin-1/2 or a spin-1, we have v = A/2(A) and γ = 1/2 ${T}_{1}^{e}$(1/3 ${T}_{1}^{e}$) for the 2LF(3LF).

In the strong fluctuator regime that we focus on, intuitively a single fluctuator jump is enough to totally decohere the nuclear spin. Then, provided v/γ ≫ 1, the spin decay rates doe not depend anymore on v but only on the jump rate. Indeed, we find ${T}_{2}^{* n}=2{T}_{1}^{e}(1.5{T}_{1}^{e})$ for the 2LF(3LF) (see appendix B.5). This strict limit on ${T}_{2}^{* n}$ for the strong fluctuator is in sharp contrast to the weak fluctuator case, where ${T}_{2}^{* n}$ increases as the hyperfine interaction strength decreases (see appendix B.4).

To experimentally test the spin-fluctuator model, we measure and compare ${T}_{1}^{e}$ of the NV center electronic spin and ${T}_{2}^{* n}$ of its native ¹⁴N nuclear spin. All experiments are performed using a home-built confocal microscope, with single NV centers in an electronic grade diamond sample (Element 6, ¹⁴N concentration ${[}^{14}N]\lt 5$ ppb, natural abundance of ¹³C). We work at a magnetic field of 424G, close to the excited state level anti-crossing, to polarize the ¹⁴N nuclear spin [38]. A 1.5mW laser of 2 μs duration polarizes the hybrid NV-¹⁴N system into $| {m}_{S}=0,{m}_{I}=+1\rangle$ with high fidelity. Microwave (MW) and radiofrequency (RF) pulses are delivered through a 25 μm wide copper wire to have precise control of the NV and ¹⁴N spin states.

For ${T}_{1}^{e}$ measurement, a laser pulse first initializes the system into $| 0,+1\rangle$. Then we apply a strong MW pulse (t_π = 44 ns) to prepare it to the desired state $| -1,+1\rangle$. The NV is free to fluctuate due to ${T}_{1}^{e}$ process before we measure the remaining population in $| -1,+1\rangle$ obtaining the signal ${S}_{-1}^{-1}$ where the sub(super)script refers to the initial (final) electronic spin state. We also measure the population in the state $| 0,+1\rangle$ obtaining the signal ${S}_{-1}^{0}$. ${T}_{1}^{e}$ is extracted to be 4.3 ± 0.3 ms by fitting to the difference of the two measurements ${S}_{-1}^{-1}-{S}_{-1}^{0}$ (here and throughout the paper, uncertainty in all fitted values are 95% confidence interval).

For ${T}_{2}^{* n}$ measurement, we implement a nuclear Ramsey sequence in the electronic m_s = −1 manifold, where the larger nuclear spin energy splitting (due to the hyperfine coupling) allows faster driving. The system is first prepared to $| -1,+1\rangle$, as described above, before being coherently driven to a nuclear superposition state $(| -1,+1\rangle +| -1,0\rangle )/\sqrt{2}$ using on-resonant RF field. After a free evolution period, we convert nuclear spin coherence to populations with a second RF π/2 pulse, $\alpha | -1,+1\rangle +\beta | -1,0\rangle$. Finally, the nuclear spin is read out by mapping its state to the NV electronic spin, using a selective MW pulse (t_π = 1.1 μs). This pulse creates the entangled state $\alpha | 0,+1\rangle +\beta | -1,0\rangle$ between the NV and ¹⁴N, allowing optical readout of the nuclear spin state population $| \alpha {| }^{2},| \beta {| }^{2}$ with high SNR.

Isolating the bare contribution of the nuclear spin dephasing in the fluorescence signal decay is nonetheless not straightforward as the consequences of NV random flips are three-fold. First, they induce the nuclear spin dephasing that we aim at measuring. Second, they modify the NV state, leading the second RF π/2 pulse to be off-resonance and preventing the transfer of the nuclear coherence into population difference. Finally, they also induce errors in the mapping between nuclear and electronic spin states as the NV state is not fully polarized anymore.

Fortunately, in the strong fluctuator regime, one flip of the fluctuator is enough to decohere our qubit, which allows us to neglect the two last errors. The bare contribution of the nuclear spin dephasing can be isolated by recording the signals obtained from nuclear Ramsey sequence with (1) no phase difference between the two RF π/2 pulses and (2) a π phase shift. The last two effects that cause imperfect readout of the nuclear spin have an equivalent contribution in both Ramsey sequences, just creating a common error that is suppressed when subtracting the two Ramsey signals (appendix A.1). We can then measure a dephasing time ${T}_{2}^{* n}$ = 5.6 ± 1.7 ms as shown in figure 2(b). This measurement is consistent with our prediction from spin-fluctuator model with 3 levels, ${T}_{2}^{* n}=1.5{T}_{1}^{e}$.

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To protect the nuclear spin qubit from RTN generated by the NV, we resort to DD. Usual DD schemes are highly effective in protecting qubits from noise provided the pulses are applied at a higher repetition rate than the typical correlation time of the noise. Unfortunately, because of the Markovian nature of RTN, this condition does not apply here. Instead of being set by the fluctuator rate γ, in order for DD to be effective the π-pulse separation time τ must satisfy

$$\begin{eqnarray}&&A\cdot \tau \lesssim 1.\end{eqnarray} \tag{ 4 }$$

When applying DD following the well-known CPMG sequence [39] with time between pulses τ, the spin coherence after N pulses is given by

$$\begin{eqnarray}&&\langle {{\rm{e}}}^{{\rm{i}}\varphi }(N\tau )\rangle =\{{[{{\rm{e}}}^{{M}_{-1}\tau /2}\cdot {U}_{\pi }\cdot {{\rm{e}}}^{{M}_{-1}\tau /2}]}^{N}\vec{P}\}{}_{1},\end{eqnarray} \tag{ 5 }$$

where U_π is the π-pulse operator in the $\vec{P}$ basis. For a 2LF this is diag(1, −1), while it is diag(1, 1, −1) for a 3LF. As discussed below, it might be possible, and even more convenient, to apply π-pulses on the fluctuator instead of the qubit. Indeed, the desired effect is to invert the sign of the coupling between the two systems. For a 3LF, this can be achieved by driving the double quantum (DQ) transition $| +1\rangle \leftrightarrow | -1\rangle$. However, for a 3LF there is some freedom on the type of pulses applied. In addition to driving the DQ transition, one can also drive one of the single quantum (SQ) transitions, $| 0\rangle \leftrightarrow | \pm 1\rangle$, resulting in

$$\begin{eqnarray*}{U}_{\pi }=\left(\begin{array}{ccc}1 & 0 & 0\\ 1 & -1/2 & \mp 1/2\\ 1 & \mp 3/2 & 1/2\end{array}\right).\end{eqnarray*}$$

Since the qubit decay under DD is not necessarily purely exponential [appendix B.3], we define an effective coherence time ${T}_{2}^{n}(\tau )$ through:

$$\begin{eqnarray}&&\langle {{\rm{e}}}^{{\rm{i}}\varphi }(N\tau ={T}_{2}^{n}(\tau ))\rangle =1/e.\end{eqnarray} \tag{ 6 }$$

The dependence of T₂ⁿ on the DD interval τ is shown in figure 4(a). As expected, smaller τ's are, in general, better at decoupling the qubit from RTN and at extending ${T}_{2}^{n}$. Interestingly, the behavior for the 2LF and 3LF is different. For 2LF, DD leaves the decay approximately exponential (see appendix B.2), with a decay rate

$$\begin{eqnarray}&&1/{T}_{2}^{n}=\gamma -\displaystyle \frac{1}{\tau }\mathrm{ln}\left[\displaystyle \frac{\gamma \sin (W\tau )+\sqrt{{v}^{2}-{\gamma }^{2}{\cos }^{2}(W\tau )}}{W}\right],\end{eqnarray} \tag{ 7 }$$

where $W=\sqrt{{v}^{2}-{\gamma }^{2}}$. For a 3LF, however, the coherence decay is not exponential (see appendix B.3). Still, we see that, by using DQ pulses to refocus the fluctuator (or applying pulses directly to the qubit), one could in principle fully decouple the qubit from RTN noise when $\tau \to 0$, until nuclear–nuclear dipolar interactions become the dominant noise source [25]. With SQ drive, however, one only protects the qubit from RTN half of the time, therefore the decay rate can at most be reduced to half of its value without DD (figure 4(a)).

We now apply these ideas to protect the nuclear spin from the NV RTN and extend its ${T}_{2}^{n}$ beyond the limit of ${T}_{1}^{e}$.

Due to the strong hyperfine coupling A = 2.16 MHz between the electronic and nuclear spin of the NV, the condition (4) requires τ ≲ 1 μs, which is not feasible given the slow control of the nuclear spin qubit (with typical π-pulse times 50 μs). However, as the hyperfine interaction is symmetric with respect to the state of both spins, applying π-pulses on either the qubit or the fluctuator modulates the hyperfine interaction sign and will lead to an effectively weaker averaged hyperfine coupling and thus a slower rate at which nuclear states acquire a random phase. It is consequently possible to take benefit from the electronic driving strength that are typically a few tens to a few hundreds of MHz [40–42], yielding π-pulses fast enough to meet the requirement of equation (4).

Another challenge in the experiment is due to the nuclear spin readout, which is indirectly obtained by measuring the NV spin. As mentioned in section 2.2, as the NV center state is unknown at the end of the evolution due to ${T}_{1}^{e}$ processes, the nuclear spin mapping from coherence to population states and its readout via the NV electronic spin might fail. This problem is exacerbated when DD is applied, as we expect some qubit coherence to be stored in all NV manifolds. Thus, the differential measurement scheme applied above no longer provides an accurate picture of the nuclear spin coherence decay. In particular, it is no longer possible to fully measure gains in ${T}_{2}^{n}$ beyond ${T}_{2}^{* n}$ (see appendix A.1).

To remove this undesired effect and more precisely verify the protection of the nuclear spin afforded by DD, we engineer a short-lived 2LF, decoupling its evolution (and final state) from the state needed for the correct readout of the nuclear spin. The engineered noise also allows shorter experiments, further avoiding slow external experimental drifts that could hide the gains in coherence time. In addition, engineering a 2LF instead of 3LF eliminates the need of dual frequency driving. The artificial 2LF is engineered by applying fast, on-resonant MW pulses to flip the NV electronic spin state between $| 0\rangle$ and $| -1\rangle$ (engineered ${T}_{1}^{e}$ flip) at random times following a Poisson distribution. Figure 3(a) displays one of the 200 engineered ${T}_{1}^{e}$ traces that once averaged simulate an exponential ${T}_{1}^{e}$ decay process. We set the flipping constant of the artificial 2LF in order to obtain a relaxation time ${T}_{1}^{e}$ = 10 μs. This time scale is much longer than the π-pulse length t_p = 44 ns, and is two orders smaller than that of the natural ${T}_{1}^{e}$, guaranteeing the third level $| +1\rangle$ of the NV center is not involved in the dynamics and we indeed have an effective 2LF.

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Figure 3(b) shows measurement of the engineered ${T}_{1}^{e}$ decay, matching very well with the simulation of the applied 200 engineered ${T}_{1}^{e}$ traces. An exponential fit gives a decay time of 10.0 ± 0.4 μs, in good agreement with our 10 μs design. We then use the same engineered ${T}_{1}^{e}$ traces to perform a nuclear Ramsey experiment and measure the resulting coherence time by fitting data to an oscillating exponential curve, obtaining an engineered ${T}_{2}^{* n}$ = 22 ± 4 μs = 2 ${T}_{1}^{e}$ (figure 4(b)), as expected from the 2LF theory. In order for the nuclear spin state readout to be accurate, if the NV ends up in $| 0\rangle$ due to the engineered flips we apply an extra π-pulse immediately before the readout process, which brings it back to $| -1\rangle$.

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Finally, in addition to the engineered ${T}_{1}^{e}$ traces, we apply the Knill-dynamical decoupling (KDD) sequence [43] with an interval of τ = 200 ns on the NV to decouple ¹⁴N from the RTN. We choose KDD instead of CPMG because it is robust against pulse errors (in the following experiments, we use DD to refer to KDD pulses). We apply ∼100 π pulses to measure the nuclear spin coherence decay, which according to equation (7), is expected to follow an exponential behavior. In figure 4(b) we compare the experimental (and theoretical) decays with and without DD, clearly showing the improvement achieved by applying a decoupling scheme, proving the successful protection of nuclear spin from its RTN environment. This is confirmed by the measured ${T}_{2}^{n}$ value, extracted from an oscillating exponential fit to be ${T}_{2}^{n}$ = 67 ± 17 μs, (figure 4(b)), clearly exceeding ${T}_{2}^{* n}$, and matching well with theoretical prediction of 71 μs. We repeat this experiment with different DD intervals τ, to compare the trend in ${T}_{2}^{n}$ with our theory. The results, shown in figure 4(a), are in quantitative agreement with the predicted behavior, including the somewhat counterintuitive result for τ = 600 ns, where the applied DD accelerates decoherence, giving ${T}_{2}^{n}$ < ${T}_{2}^{* n}$.

While we demonstrated that DD can be effective in increasing the coherence time affected by a strong random fluctuator only for engineered noise, we remark that our experimental results show that it would be possible to refocus the natural noise as well. Indeed, using a direct readout of the nuclear spin [36, 44] or with a different protocol to map its state onto the electronic spin, it would be possible to avoid seeing direct effects of the NV ${T}_{1}^{e}$ on the measured nuclear spin coherence. We further performed DD experiments to show that it is practically feasible to implement the necessary number of pulses for decoupling (see appendix A.4) without introducing additional noise due to pulse errors.

In our experiments we do not have a direct measurement of the nuclear spin qubit, which would reveal its coherence time. A common strategy to overcome this limitation is to initialize the NV and apply a CNOT gate, flipping the NV state conditional on the nuclear spin state. This effectively maps the state information from the nuclear to the electronic spin. In the nuclear Ramsey experiment in section 2.2, we create the entangled state $\alpha | 0,+1\rangle +\beta | -1,0\rangle$ with this protocol. The NV is then optically read out, giving the same probability distribution $| \alpha {| }^{2},| \beta {| }^{2}$ as if directly measuring the nuclear spin. However, when we measure the nuclear spin coherence, the NV undergoes ${T}_{1}^{e}$ flips, potentially introducing errors in this readout process. First, if the NV final state is different from the nominal one, the second RF pulse operates in the incorrect NV manifold and is thus off-resonance, failing to transfer the nuclear coherence into state populations. Second, the mapping between nuclear and electronic spin states might fail as the NV state is not fully polarized anymore.

To account for these errors, we perform a differential readout to obtain ${T}_{2}^{* n}$: the first measurement is the regular nuclear Ramsey experiment, measured by applying a CNOT on the NV, subject to possible readout failure; for the second measurement, we add a π phase shift on the second RF π/2 pulse in the nuclear Ramsey sequence, and then apply the same readout. The difference of the two measurements yields the expected ${T}_{2}^{* n}$ decay, as shown in figure A1(a). The intuitive explanation is that since one jump of the strong fluctuator totally decoheres the qubit, any nuclear spin coherence is preserved only when the NV stays in the original state. No coherence is left when the final state is different. Therefore, measurements (including unsuccessful ones) already contain all the information about the nuclear spin coherence. The purpose of the differential readout is to remove the two errors mentioned above, which are not related to the nuclear spin coherence decay. Therefore, one could even choose a different measurement as the second data set, as long as it contains the same common mode error. One such choice is to apply a CNOT gate conditional on the nuclear spin being in the $| {m}_{I}=0\rangle$ state (instead of $| {m}_{I}=+1\rangle$). We chose to apply a π phase shift on the nuclear Ramsey because when taking the difference of the two measurements this also doubles the signal amplitude. When the qubit is protected against RTN by the DD control, this differential measurement is no longer effective because some nuclear spin coherence is stored in all NV manifolds (figure A1(b)). We emphasize that this is not a fundamental limit. Single-shot readout, either using on-resonant laser at cryogenic temperature [44] or using intermediate-high magnetic field at room-temperature [36] solves this issue and has been experimentally demonstrated.

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In the engineered ${T}_{1}^{e}$ experiment, we consider N_T = 200 predetermined traces. Each trace is repeated 4 × 10⁴ times in order to build enough statistics to determine the final population state of the NV. In figure A2(a), we plot the average signal at t = 16 μs for each one of the 200 ${T}_{1}^{e}$ traces, clearly showing that the final state of each trace can be reliably determined to be either $| 0\rangle$ or $| -1\rangle$. As another demonstration, in figure A2(b) we plot out the #47 engineered ${T}_{1}^{e}$ trace in black solid line, and the experimental result in red diamonds. The error bars in all engineered ${T}_{1}^{e}$ experiments are then the standard deviation resulting from the 200 × 4 × 10⁴ acquired data for each time point, divided by $\sqrt{{N}_{T}}$, which corresponds to the usual standard error of the mean used for the other experimental results.

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In order to simulate a ${T}_{1}^{e}$ flip, we apply a π pulse, the same pulse used for DD. As the pulse length (44 ns) is comparable to the smallest time interval between pulses of the DD sequence (200 ns), there is a non-negligible probability that a ${T}_{1}^{e}$ flip overlaps with DD pulses for some of the ${T}_{1}^{e}$ traces. We deal with this possibility in two ways: if the overlap of the ${T}_{1}^{e}$ flip and DD π-pulse is larger than half the pulse duration, we do not apply either pulses; also, we discard all traces that contain an overlap of less than half of the pulse duration. We verify that this strategy does not bias the overall engineered noise by measuring ${T}_{1}^{e}$ with and without DD sequence. The results in figure A3(a) show that the fitted ${T}_{1}^{e}$ ${}_{{\rm{DD}}}=10.0\pm 0.4\,\mu {\rm{s}}$  is the same as ${T}_{1}^{e}$ without DD (10.0 ± 0.4 μs) within the error bars. This verifies that our treatment of overlapping π-pulses does not change the underlying physics, and that the protection of nuclear spin coherence derives from DD, rather than from changes in the engineered ${T}_{1}^{e}$ under DD.

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To show that even for the natural ${T}_{1}^{e}$ noise our DD method can efficiently decouple the nuclear spin from its RTN, we measure the natural ${T}_{1}^{e}$ while applying a DD sequence (figure A3(b)). In order to have a sequence robust against flip angle error and off-resonance pulses, we employ the KDD pulse sequence [43],

$$\begin{eqnarray*}&&{{\rm{KDD}}}_{\varphi }=\tau /2-{(\pi )}_{\pi /6+\varphi }-\tau -{(\pi )}_{\varphi }-\tau -{(\pi )}_{\pi /2+\varphi }-\tau -{(\pi )}_{\varphi }-\tau -{(\pi )}_{\pi /6+\varphi },\end{eqnarray*}$$

and concatenate this 5-pulse block following the XY-16 phase cycle [47]: ${{\rm{KDD}}}_{x}-{{\rm{KDD}}}_{y}-{{\rm{KDD}}}_{x}-{{\rm{KDD}}}_{y}-{{\rm{KDD}}}_{y}$ $-\,{{\rm{KDD}}}_{x}-{{\rm{KDD}}}_{y}-{{\rm{KDD}}}_{x}-{{\rm{KDD}}}_{\bar{x}}-{{\rm{KDD}}}_{\bar{y}}$ $-\,{{\rm{KDD}}}_{\bar{x}}-{{\rm{KDD}}}_{\bar{y}}$ $-\,{{\rm{KDD}}}_{\bar{y}}-{{\rm{KDD}}}_{\bar{x}}-{{\rm{KDD}}}_{\bar{y}}-{{\rm{KDD}}}_{\bar{x}}$. We measure up to 8 ms, corresponding to 4 × 10⁴ pulses, longer than ${T}_{2}^{* n}$. We fit the data to an exponential decay and find the 1/e time to be 3.7 ± 1.3 ms, matching ${T}_{1}^{e}$ = 4.3 ± 0.3 ms. At this τ, we expect more than a factor of 3 gain in ${T}_{2}^{n}$, therefore there should be a net improvement in nuclear spin coherence. This suggests that for up to about ∼10⁵ pulses, pulse errors in the DD do not accumulate so significantly to counteract potential gains in ${T}_{2}^{n}$. We expect our method to protect ¹⁴N  beyond the limit set by natural ${T}_{1}^{e}$ of a few ms, as it could be verified with single-shot readout of the nuclear spin coherence.

Given the large uncertainty in the measured natural ${T}_{1}^{e}$ = 4.3(3) ms and ${T}_{2}^{* n}$ = 5.6(1.7) ms (errors are 95% confidence interval from fit), we find a ${T}_{2}^{* n}$/${T}_{1}^{e}$ ratio of 5.6/4.3 = 1.3(2), which is compatible with the 3LF model prediction (${T}_{2}^{* n}$/${T}_{1}^{e}$ = 1.5), but does not exclude other models. It is then worth to examine more in depth whether the data does indeed support the three-level spin-fluctuator model or whether other models could be a better match.

Instead of extracting ${T}_{1}^{e}$ and ${T}_{2}^{* n}$ independently from the fit, we fit the two experimental datasets together to four models: (1) fixing ${T}_{2}^{* n}$/${T}_{1}^{e}$ = 1.5; (2) fixing ${T}_{2}^{* n}$/${T}_{1}^{e}$ = 2; (3) fixing ${T}_{2}^{* n}$/${T}_{1}^{e}$ = 1; (4) leaving ${T}_{2}^{* n}$/${T}_{1}^{e}$ as a free fitting parameter. We compare the mean square error (MSE) of the fit and the uncertainty of the fitted ${T}_{1}^{e}$ for the four models; results are summarized in table 1. This analysis reveals that the model assuming ${T}_{2}^{* n}$/${T}_{1}^{e}$ = 1.5 (model 1) yields the best result both in terms of smallest MSE and smallest uncertainty in ${T}_{1}^{e}$. A more general fit (model 4) converges to ${T}_{2}^{* n}$/${T}_{1}^{e}$ = 1.3(5), similar to the result when the two datasets are fitted independently, but results in larger uncertainty in ${T}_{1}^{e}$. We therefore argue that the experimental data, although with a relatively large uncertainty, are consistent with the spin-fluctuator model predictions.

Table 1.  Comparing different models for ${T}_{2}^{* n}/{T}_{1}^{e}$.

	MSE	T1e(ms)	${\sigma }_{{T}_{1}^{e}}$(ms)	${T}_{2}^{* n}/{T}_{1}^{e}$
model 1	0.008 3	4.1	0.8	3/2
model 2	0.009 1	3.6	0.8	2
model 3	0.008 6	5.0	1.0	1
model 4	0.008 4	4.3	1.2	1.3(5)

We further note that we can ascribe the slightly smaller ${T}_{2}^{* n}$ than expected (${T}_{2}^{* n}$/${T}_{1}^{e}$ < 1.5) to environmental drift. Each data point in figure 2(b) is averaged over 100,000 repetitions. We observe ∼20 Hz drift in the nuclear Larmor frequency even after recalibrating the experiment about every hour by measuring the magnetic field drift with the NV and compensating its effect on the nuclear Larmor frequency by adding a corresponding phase shift to the second π/2 pulse of the nuclear Ramsey sequence. Although small, the frequency drift is non-negligible compared to the detuning in the nuclear Ramsey experiment (∼800 Hz). The resulting off-resonance pulses cause the average data to have a reduced contrast at long time, which is interpreted as a shorter ${T}_{2}^{* n}$.

While the random walker model provides an intuitive semiclassical picture of the decay process, one could also solve the coupled fluctuator-qubit dynamics with a fully quantum mechanical framework. In particular, as the ${T}_{1}^{e}$ process of the NV center is purely Markovian, it is valid to describe the combined electron-nuclear spin dynamics using a master equation. We can then write a Lindblad master equation

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}\rho }{{\rm{d}}{t}}={ \mathcal L }[\rho ]=-{\rm{i}}[{{ \mathcal H }}_{i},\rho ]+\displaystyle \sum _{k=1}^{M}({L}_{k}\rho (t){L}_{k}^{\dagger }-\displaystyle \frac{1}{2}{L}_{k}^{\dagger }{L}_{k}\rho (t)-\displaystyle \frac{1}{2}\rho (t){L}_{k}^{\dagger }{L}_{k}),\end{eqnarray} \tag{ 8 }$$

where the jump operators L_k describe the ${T}_{1}^{e}$ flips of NV, and are therefore ${L}_{k}={\rm{\Gamma }}| {m}_{s}\rangle \langle {m}_{s}^{\prime} |$, where ${m}_{s},{m}_{s}^{\prime} =\{-1,0,+1\}$. ${\rm{\Gamma }}=1/\sqrt{2{T}_{1}^{e}}$ ($1/\sqrt{3{T}_{1}^{e}}$) for 2LF (3LF). We note that we do not need to explicitly write jump operators for the nuclear spin, as its decoherence is mediated by the Hamiltonian ${{ \mathcal H }}_{2}={\omega }_{e}{S}_{z}+{\omega }_{n}{I}_{z}+S\cdot { \mathcal A }\cdot I$ (${{ \mathcal H }}_{3}={{DS}}_{z}^{2}+{\omega }_{e}{S}_{z}+{\omega }_{n}{I}_{z}+S\cdot { \mathcal A }\cdot I$) for the electron-nuclear spin register for an electronic spin-1/2 (spin-1). The evolved density operator can be simply found by vectorizing this equation to obtain $\rho (t)={{\rm{e}}}^{{ \mathcal L }t}\rho (0)$. We find that the numerical results from the spin-fluctuator model and the master equation match, indicating the validity of using the semiclassical spin-fluctuator model to describe a fully quantum process. We note that the quantum mechanical treatment could handle more general cases, such as the initial NV state being a superposition state [48].

In the 2LF case, we can obtain an intuitive picture of the dynamics under DD by diagonalizing the block $[{{\rm{e}}}^{{M}_{-1}\tau /2}\cdot {U}_{\pi }\cdot {{\rm{e}}}^{{M}_{-1}\tau /2}]={V}^{-1}{{\rm{\Lambda }}}_{{\rm{dd}}}V$. The diagonal elements of Λ_dd are real:

$$\begin{eqnarray}&&{\lambda }^{\pm }=\displaystyle \frac{{{\rm{e}}}^{-\gamma \tau }}{W}[\gamma \sin (W\tau )\pm \sqrt{{v}^{2}-{\gamma }^{2}{\cos }^{2}(W\tau )}\,],\end{eqnarray} \tag{ 9 }$$

with $W=\sqrt{{v}^{2}-{\gamma }^{2}}$. They satisfy ${\lambda }^{+}\geqslant 0\geqslant {\lambda }^{-}$, with the equal sign only for v = γ. As we do not expect a net growth of coherence, the negative eigenvalue λ⁻ is not expected to contribute. As we will see soon, its coefficient is almost 0. For an initial state $\vec{P}=[1;p]$, $p\in [-1,1]$, we can express the coherence explicitly:

$$\begin{eqnarray}\langle {{\rm{e}}}^{{\rm{i}}\varphi }(N\tau )\rangle & = & \{V{{\rm{\Lambda }}}_{{\rm{dd}}}^{n}{V}^{-1}\vec{P}\}{}_{1}\\ & = & {{pc}}_{1}{\lambda }_{+}^{n}-{{pc}}_{2}{\lambda }_{-}^{n}+{c}_{3}{\lambda }_{+}^{n}+{c}_{4}{\lambda }_{-}^{n}\end{eqnarray} \tag{ 10 }$$

with c_n as follows

$$\begin{eqnarray}{c}_{1} & = & {c}_{2}={\rm{i}}{v}\gamma {\sin }^{2}(W\tau /2)B,\\ {c}_{3} & = & \displaystyle \frac{1}{2}+\displaystyle \frac{B}{2}[{v}^{2}-{\gamma }^{2}\cos (W\tau )],\\ {c}_{4} & = & \displaystyle \frac{1}{2}-\displaystyle \frac{B}{2}[{v}^{2}-{\gamma }^{2}\cos (W\tau )],\end{eqnarray} \tag{ 11 }$$

where $B=1/[W\sqrt{{v}^{2}-{\gamma }^{2}{\cos }^{2}(W\tau )}]$. In the strong fluctuator regime, $v\gg \gamma$, we have ${c}_{1},{c}_{2}\sim \gamma /v\approx 0$, c₃ ≈ 1, c₄ ≈ 0, yielding an exponential decay $\langle {{\rm{e}}}^{{\rm{i}}\varphi }(N\tau )\rangle \approx {\lambda }_{+}^{n}$. Note that equations (9)–(11) are valid even in the weak fluctuator regime.

The 3-level fluctuator case is more complicated, and we cannot derive an elegant analytical form. When we apply the DQ drive, the coherence approximately follows a simple form

$$\begin{eqnarray}&&\langle {{\rm{e}}}^{{\rm{i}}\varphi }(n\tau )\rangle \approx {c}_{+}{\lambda }_{+}^{n}+{c}_{0}{\lambda }_{0}^{n}+{c}_{-}{\lambda }_{-}^{n},\end{eqnarray} \tag{ 12 }$$

where $\{{\lambda }_{+},{\lambda }_{0},{\lambda }_{-}\}$ are the eigenvalues of ${{\rm{e}}}^{{M}_{-1}\tau /2}\cdot {U}_{\pi }\cdot {{\rm{e}}}^{{M}_{-1}\tau /2}$. Instead of writing down their cumbersome expressions, to obtain some intuition in figure B1 we plot how the eigenvalues and their corresponding coefficients change as a function of τ. Similar to the 2LF case, there is one eigenvalue, λ₀, that is negative but has a vanishing contribution to the dynamics. The coherence behavior also depends on the initial state (which sets c_k). Assuming ${T}_{1}^{e}$ = 4 ms and starting from the subspace spanned by $| {m}_{s}=\pm 1\rangle$, we obtain the solid lines in figure B1; the dashed lines represent the case of starting from $| {m}_{s}=0\rangle$. Interestingly, in the latter case, the coefficients will go beyond 1 and below 0, causing better coherence for some τ value than the best decay rate of the three eigenvalues.

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For completeness, we discuss briefly the 3-level weak fluctuator regime, $v\ll \gamma$ (results for the 2LF can be found in [37]). When evaluating the decay without DD (to obtain the dephasing time ${T}_{2}^{* n}$) only one of the three eigenvalues of M₋₁ contributes significantly to the decay. Therefore the decoherence has an exponential form, with

$$\begin{eqnarray}\displaystyle \frac{1}{{T}_{2}^{* }} & = & 2\gamma -\displaystyle \frac{3{\gamma }^{2}-{v}^{2}}{{3}^{1/3}K}-\displaystyle \frac{K}{{3}^{2/3}}\\ K & = & {(9{\gamma }^{3}+\sqrt{3}v\sqrt{{v}^{4}-9{v}^{2}{\gamma }^{2}+27{\gamma }^{4}})}^{1/3}.\end{eqnarray} \tag{ 13 }$$

The relationship between ${T}_{2}^{* n}$ and v is shown in figure B2, covering the full range from weak fluctuator to strong fluctuator. The black circles are the approximated result from the expression in equation (13), which ignores small contributions from the other two eigenvalues. We note that this approximation captures the average behavior for any v, even if the exact result (red curve obtained from numerical calculations) shows additional features. Unlike in the strong fluctuator regime, where ${T}_{2}^{* n}$ = 1.5 ${T}_{1}^{e}$ is independent of v, ${T}_{2}^{* n}$ for a weaker fluctuator is longer in the regime where v < γ and increases with γ/v. We also see that both in the weak and strong fluctuator regimes, an exponential decay is a good approximation, but the behavior is more complicated in the intermediate regime.

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When we apply DD to protect the nuclear spin qubit in the weak fluctuator regime, we find that only one eigenvalue mainly contributes to the decay, while we had two eigenvalues contributing in the strong coupling regime. The decoherence process is thus exponential. Interestingly, the main contribution comes from the slowest decay term. In figure B3, we show example of the decay component (we did not plot the other two fast decaying components because they are many orders larger) and their contributions to the total decay under DQ drive, similar to figure B1. Here we take ${T}_{1}^{e}$ = 10 μs, v = 0.01γ. A different result due to initial state as in strong fluctuator regime is not seen here (solid lines overlap with dashed lines). The oscillatory behavior seen in figures 4(a) and 8(a) is also missing for the weak fluctuator.

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In the strong fluctuator regime, ${T}_{2}^{* n}$ becomes independent of v (the fluctuator/qubit coupling). This had been observed in [37] for the TLF. For the 3LF, we can find a similar result by taking the limit of v ≫ γ in equation (13),

$$\begin{eqnarray}\mathop{\mathrm{lim}}\limits_{v\gg \gamma }\displaystyle \frac{1}{{T}_{2}^{* }} & = & 2\gamma =\displaystyle \frac{2}{3{T}_{1}^{e}}\\ {T}_{2}^{* n} & = & \displaystyle \frac{3}{2}{T}_{1}^{e}\quad (3\mathrm{LF}).\end{eqnarray} \tag{ 14 }$$

Similarly, for 2LF, the two eigenvalues of M₋₁ are $\gamma \pm \sqrt{{\gamma }^{2}-{v}^{2}}$. In the strong fluctuator regime, the only real contribution to decay is γ,

$$\begin{eqnarray}&&{T}_{2}^{* n}=\displaystyle \frac{1}{\gamma }=2{T}_{1}^{e}\quad (2\mathrm{LF}).\end{eqnarray} \tag{ 15 }$$