Free induction decay of single spins in diamond

We consider the electronic spin S = 1 associated with a single negatively-charged NV defect in diamond (figure 1(a)). This central spin is nestled in a high-purity diamond lattice, where electron spin impurities such as nitrogen donors (P1 centers) are below 1 ppb and thus do not contribute to the decoherence of the central spin. Each lattice position can be occupied either by ¹²C atoms (spinless) or by ¹³C isotopes (nuclear spin $I=\frac {1}{2}$), which form a nuclear-spin bath (figure 1(b)). The natural abundance of ¹³C in diamond is p_nat = 1.1%, while isotopically modified diamond samples exhibit ¹³C concentrations as low as p = 0.01% [43]. In a high-purity diamond sample, the decoherence of the NV defect electron spin is dominated by hyperfine coupling with the ¹³C nuclear spins. The full Hamiltonian of the system is given by

$$\begin{eqnarray} &&H = H_{\rm{NV}} + H_{\mathrm{bath}} + H_{\mathrm{NV\mbox{--}bath}} . \end{eqnarray} \tag{ 1 }$$

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In a magnetic field B, the Hamiltonians describing the NV defect H_NV, the bath H_bath and their interaction H_NV–bath read

$$\begin{equation} H_{\mathrm{NV}} = DS_z^2 + \gamma_{\mathrm{e}}\mathbf{B}\cdot \mathbf{S}, \end{equation} \tag{ 2 }$$

$$\begin{equation} H_{\mathrm{bath}} = \sum_n \gamma_n \mathbf{B}\cdot\mathbf{I}^{(n)} + \sum_{n<m} \mathbf{I}^{(n)}\cdot\mathbbm{C}^{(nm)}\cdot\mathbf{I}^{(m)}, \end{equation} \tag{ 3 }$$

$$\begin{equation} H_{\rm{NV-bath}} = \sum_n \mathbf{S}\cdot\mathbbm{A}^{(n)}\cdot\mathbf{I}^{(n)} , \end{equation} \tag{ 4 }$$

where D/2π = 2.87 GHz is the zero-field splitting of the NV defect electron spin, γ_e/2π = 2.8 MHz G⁻¹ (resp. γ_n/2π = 1.07 kHz G⁻¹) is the gyromagnetic ratio of the electronic (resp. nuclear) spin, $\mathbbm {A}^{(n)}$ is the hyperfine tensor between the central spin and the nuclear spin n and $\mathbbm {C}^{(nm)}$ is the dipole–dipole interaction between nuclei n and m. We note that we consider magnetic field amplitudes large enough to neglect strain-induced splitting terms in equation (2). The decoherence of the NV defect electron spin in the very low magnetic field regime (B < 0.1 G), for which strain-induced splitting protects the central spin against magnetic field fluctuations, has been addressed elsewhere [44].

We focus on the FID of a coherent superposition of the central spin as a function of the amplitude of a magnetic field applied along the NV defect axis, denoted as z (figure 1(a)). In this condition, the electron spin quantization axis is fixed by the NV defect axis and the eigenstates of H_NV are denoted as |m_s〉 with m_s = 0,±1. The FID signal can be obtained by performing a Ramsey sequence $\frac {\pi }{2}-\tau -\frac {\pi }{2}$ on the central spin, where a first $\frac {\pi }{2}$ pulse rotates the central spin from a prepared state |0〉 to a coherent superposition $|{+}\rangle =\frac {1}{\sqrt {2}}\left (|{0}\rangle +|{1}\rangle \right )$, which evolves over a time τ. During such a free precession time, the central spin only interacts with the external magnetic field and the nuclear-spin bath. Finally, a second $\frac {\pi }{2}$ pulse rotates the central spin again, thus converting residual spin coherence into population. The FID signal can be interpreted as the probability to retrieve the initial state |0〉 and can be formally calculated as

$$\begin{equation} s(\tau) = {\mathrm{Tr}}_{\rm bath}\left[\langle{0}|\rho(\tau)|{0}\rangle\right], \end{equation} \tag{ 5 }$$

$$\begin{equation} \rho(\tau) = U_{\mathrm{T}}(\tau)\rho_{\mathrm{e}}\otimes\rho_{\mathrm{bath}} U_{\mathrm{T}}(\tau)^\dagger , \end{equation} \tag{ 6 }$$

where the trace is taken over the Hilbert space of the bath. Here ρ_e = |0〉〈0| is the initial electronic spin density matrix and $\rho _{\mathrm {bath}} = \prod _n \rho _n^\otimes$ is the initial density matrix of the nuclear-spin bath. The total unitary evolution operator of the system is denoted as $U_{\mathrm {T}}(\tau ) = R(\frac {\pi }{2})U(\tau )R(\frac {\pi }{2})$, where $R(\frac {\pi }{2})$ corresponds to a $\frac {\pi }{2}$-rotation operator over the electronic spin and $U(\tau )=\exp (-{\mathrm {i}}H\tau )$.

We neglect the slow dynamics related to dipolar interactions between nuclear spins within the bath, which is at most 2 kHz for nearest-neighbor interaction. Indeed, experiments have shown that the NV defect coherence time T*₂ does not exceed a few μs in diamond samples with natural abundance of ¹³C [31], indicating that the interaction between the central spin and the bath induces decoherence at a much faster rate than the dipolar coupling between ¹³C's within the bath. In addition, the large value of the zero-field splitting allows us to perform the secular approximation for the electronic spin even at small magnetic field values. Therefore, the terms proportional to S_x,S_y are neglected in the Hamiltonian. For a magnetic field pointing along the z-axis, the Hamiltonian can thus be written as

$$\begin{equation} H = DS_z^2 + \gamma_{\mathrm{e}} BS_z + \sum_n \gamma_n BI^{(n)}_{z} + \sum_{n} S_z\mathbf{A}^{(n)}\mathbf{I}^{(n)} , \end{equation} \tag{ 7 }$$

where the last two terms sum over each nuclear spin n and A⁽ⁿ⁾ = (A⁽ⁿ⁾_zx,A⁽ⁿ⁾_zy,A⁽ⁿ⁾_zz) is now a hyperfine vector. For Ramsey experiments, we focus on the rotating frame at which the electron spin transition |0〉 → |1〉 is addressed so that the Hamiltonian reduces to the last two terms in equation (7). We note that this Hamiltonian is diagonal in the electronic spin subspace and therefore it can be considered as a Hamiltonian for the nuclei, H_ms, that is conditional to the state of the electron spin m_s. The full Hamiltonian of the system can thus be written as

$$\begin{equation} H = \sum_{m_{s}=0}^{1} |{m_{s}}\rangle\langle{m_{s}}|\otimes \ H_{m_s} , \end{equation} \tag{ 8 }$$

$$\begin{equation} H_{m_s} = \sum_n H_{m_s}^{(n)}=\sum_n {\boldsymbol{\Omega}}_{m_s}^{(n)}\cdot \mathbf{I}^{(n)}, \end{equation} \tag{ 9 }$$

where Ω⁽ⁿ⁾_ms = γ_nB + m_sA⁽ⁿ⁾ represents the vector around which the nuclear spin n rotates, i.e. the Larmor vector. The evolution of the bath can then be written as the direct product of the evolution operator of each nuclear spin, $U_{m_s}(\tau ) = \prod _n U_{m_s}^{(n)}(\tau )$, where $U_{m_s}^{(n)}(\tau ) = \exp (-{\mathrm {i}}H_{m_s}^{(n)}\tau )$.

It follows from equation (5) that the FID signal is given by

$$\begin{equation} s(\tau) = \frac{1}{2} - \frac{1}{2}{\mathrm{Re}}\prod_n\mathcal{S}_n(\tau) , \end{equation} \tag{ 10 }$$

where the product runs over all nuclear spins and $\mathcal {S}_n = {\mathrm {Tr}}_n\left [\rho _n U_{1}^{(n)\dagger } U_{0}^{(n)}\right ]$ is the contribution of nuclear spin n to the signal. In order to obtain the effect of the nuclear-spin bath on the FID signal envelope, we exclude the contribution of the intrinsic ¹⁴N nuclear spin of the NV defect (I = 1) and of strongly coupled nuclear spins, because they only contribute to coherent oscillations of the signal [29]. For all other nuclear spins, we assume an initial state given by the high temperature limit, $\rho _n= \mathbbm {1}/2$. In this case,

$$\begin{equation} \mathcal{S}_n(\tau) = \cos\frac{\Omega_{0}^{(n)}\tau}{2}\cos\frac{\Omega_{1}^{(n)}\tau}{2} + \cos\beta^{(n)} \sin\frac{\Omega_{0}^{(n)}\tau}{2}\sin\frac{\Omega_{1}^{(n)}\tau}{2} , \end{equation} \tag{ 11 }$$

where β⁽ⁿ⁾ is the angle between Ω⁽ⁿ⁾₀ and Ω⁽ⁿ⁾₁ (see the appendix).

As a figure of merit, we consider the envelope of the FID signal,

$$\begin{equation} E(\tau) = {\mathrm{Re}}\prod_n\mathcal{S}_n(\tau) , \end{equation} \tag{ 12 }$$

where the product runs over lattice positions occupied by a ¹³C atom. For a diamond lattice with a given concentration p of ¹³C not all positions of the lattice are occupied by a ¹³C. We therefore define the random variable $x_n = \left \{1,0\right \}$, with mean p, to denote the presence (x_n = 1) or absence (x_n = 0) of a ¹³C at the lattice position n. For a given configuration of ¹³C $\left \{x_n\right \}$, the FID envelope can be written as

$$\begin{equation} E(\tau, \left\{x_n\right\}) = {\mathrm{Re}}\prod_n\left[x_nS_n(\tau) + 1-x_n\right] . \end{equation} \tag{ 13 }$$

We note that the product now runs over all positions of the lattice. Assuming that random variables x_n are independent, the average envelope is

$$\begin{equation} \bar{E}(\tau) = {\mathrm{Re}}\prod_n\left[pS_n(\tau) + 1-p\right] . \end{equation} \tag{ 14 }$$

We will use the last equation to analyze the average trend of the FID signal in the next sections. We now briefly discuss two simple limits to gain an insight into the FID dynamics.

In the weak magnetic field limit (γ_nB ≪ A⁽ⁿ⁾), the FID envelope is given by

$$\begin{equation} \bar{E}(\tau) \approx \prod_n\left(1 - \frac{p}{8}[A^{(n)}\tau]^2\right) ={\mathrm{e}}^{-(\tau/T_{2,{\tiny{\rm LF}}}^*)^2}, \end{equation} \tag{ 15 }$$

$$\begin{equation} T_{2,{\tiny{\rm LF}}}^* = \left( \frac{p}{8}\sum_n [A^{(n)}]^2\right)^{-1/2} , \end{equation} \noindent \tag{ 16 }$$

where [A⁽ⁿ⁾]² = [A⁽ⁿ⁾_zx]² + [A⁽ⁿ⁾_zy]² + [A⁽ⁿ⁾_zz]². The decay of the FID signal is Gaussian and decoherence is caused because each nuclear spin precesses differently with magnitude A⁽ⁿ⁾.

In the large magnetic field limit, γ_nB ≫ A⁽ⁿ⁾, since cos β⁽ⁿ⁾ ∼ 1, the FID envelope reads

$$\begin{equation} \bar{E}(\tau) \approx \prod_n\left(1 - \frac{p}{8}[A_{zz}^{(n)}\tau]^2\right) = {\mathrm{e}}^{-(\tau/T_{2,{\tiny{\rm HF}}}^*)^2}, \end{equation} \tag{ 17 }$$

$$\begin{equation} T_{2,{\tiny{\rm HF}}}^* = \left( \frac{p}{8}\sum_n [A_{zz}^{(n)}]^2\right)^{-1/2}. \end{equation} \noindent \tag{ 18 }$$

The signal also exhibits a Gaussian decay and decoherence is only due to the difference in Larmor precession along the direction of the external magnetic field because the anisotropic components of the hyperfine interaction are suppressed by the nuclear Zeeman energy. Since A⁽ⁿ⁾_zz ⩽ A⁽ⁿ⁾ is always true, the NV defect coherence time is enhanced at high magnetic field. The enhancement factor η can be calculated as

$$\begin{equation} \eta=\frac{T_{2,{\tiny{\rm HF}}}^*}{T_{2,{\tiny{\rm LF}}}^*} =\sqrt{\frac{\sum_n [A^{(n)}]^2}{\sum_n [A_{zz}^{(n)}]^2}}. \end{equation} \noindent \tag{ 19 }$$

As justified in section 2.2, we neglect the contact interaction and therefore consider a pure dipole–dipole interaction between the central spin and the nuclei. The hyperfine components of nuclear spin n can then be written as

$$\begin{equation} A^{(n)}_{zz}= \alpha_n(1-3\cos^2\theta_n), \end{equation} \tag{ 20 }$$

$$\begin{equation} A^{(n)}_{\perp} = \sqrt{\left[A_{zx}^{(n)}\right]^2 + \left[A_{zy}^{(n)}\right]^2} = 3\alpha_n\sin\theta_n\cos\theta_n, \end{equation} \tag{ 21 }$$

$$\begin{equation} A^{(n)} = \sqrt{\left[A^{(n)}_{\perp}\right]^2 + \left[A_{zz}^{(n)}\right]^2} = \alpha_n\sqrt{1+3\cos^2\theta_n}, \end{equation} \noindent \tag{ 22 }$$

where α_n is the magnitude of the interaction and θ_n is the angle between the NV symmetry axis and the imaginary line that joins the central spin and nucleus n. Considering a homogeneous distribution of nuclei and integrating over θ and ϕ, the average enhancement factor is finally given by

$$\begin{equation} \eta=\frac{T_{2,{\tiny{\rm HF}}}^*}{T_{2,{\tiny{\rm LF}}}^*} =\sqrt{\frac{5}{2}} = 1.58\ldots, \end{equation} \noindent \tag{ 23 }$$

independent of the concentration of nuclear-spin impurities.

The FID signal is numerically simulated using two different approaches. The first method (N1) consists in a statistical analysis over a large number of randomly sorted ¹³C distributions $\left \{x_n\right \}$ for which equation (13) is used to calculate the envelope of the FID signal. For each ¹³C distribution, the result of the simulation is fitted to the function ${\mathrm {exp}}\left [-(\tau /T_2^*)^{\epsilon }\right ]$ in order to extract the exponent of the FID signal decay  and the coherence time T*₂. The second approach (N2) consists in calculating the average envelope of the FID signal using equation (14), thus considering that each lattice site has a probability p to host a ¹³C. This approach takes less computational time to infer the mean behavior of the single electronic central spin which otherwise must be obtained using the first approach by averaging the signal over a large number of different ¹³C distributions. For both methods, a lattice of 100 000 atoms is considered, corresponding to approximatively 1000 nuclear spins for a diamond sample with natural abundance of ¹³C (p_nat = 1.1%). In addition, the nuclear-spin bath is divided into shells and cones in order to study the contribution to the FID dynamics of each ¹³C with respect of its position relative to the central spin.

We only consider the dipolar part of the hyperfine interaction in our model. Although a strong contact interaction has been measured for several ¹³C's close to the central spin [45, 46] and calculated theoretically [47], we restrict the study to low ¹³C concentrations p ⩽ 4%, where decoherence is dominated by the dipolar interaction with the nuclear-spin bath [31]. In this framework, strongly coupled nuclear spins only contribute to coherent oscillations of the FID signal but do not change its envelope. Therefore, we exclude from our simulations the lattice sites associated with nuclei that strongly interact with the central spin such as the well-known 130-MHz-splitting linked to the nearest-neighbor sites of the vacancy [48] and those with a contact interaction larger than a few MHz. The neglected lattice positions correspond to about 100 lattice points, i.e. on average one nuclear spin for a natural abundance of ¹³C.

From the experimental side, individual NV defects hosted in a high-purity diamond crystal with a natural abundance of ¹³C are optically isolated at room temperature using a scanning confocal microscope. A laser operating at 532 nm wavelength is focused onto the sample through a high numerical aperture oil-immersion microscope objective. The NV defect photoluminescence (PL) is collected by the same objective and directed to a photon counting detection system. Under optical illumination, the NV defect is efficiently polarized into the m_s = 0 spin sublevel [49]. This process provides an efficient state preparation for the Ramsey sequence. In addition, the PL intensity is significantly higher (∼30%) when the m_s = 0 state is populated. Such a spin-dependent PL response enables detection of electron spin resonances (ESR) on a single NV defect by optical means [50]. As discussed in the previous section, the FID signal is measured by applying a Ramsey sequence $\frac {\pi }{2}-\tau -\frac {\pi }{2}$ to the NV defect electron spin (figure 2(a)). For that purpose, the ESR transition $|0\rangle \leftrightarrow |1\rangle$ is driven with a microwave field applied through a copper microwire directly spanned on the diamond surface. In addition, a permanent magnet placed on a three-axis translation stage is used to apply a static magnetic field with controlled orientation and amplitude. The magnetic field is aligned along the NV axis with a precision better than 1°.

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We first study the dynamics of the FID signal as a function of the amplitude of a magnetic field applied along the NV defect axis for a natural abundance of ¹³C. Typical FID signals recorded from a single NV defect at different magnetic field magnitudes are shown in figure 2(b). Since the intrinsic nitrogen atom of the NV defect is a ¹⁴N isotope (99.6% abundance), corresponding to a nuclear spin I = 1, each electron spin state is split into three sublevels by hyperfine interaction. The ESR spectrum thus exhibits three hyperfine lines, split by A_N = −2.16 MHz [48] and corresponding to the three nuclear-spin projections. As a result, beating frequencies can be observed in the FID signal at low field (figure 2(b), upper and middle traces). By increasing the magnetic field, the NV defect gets closer to the excited-state level anti-crossing (LAC) which occurs at B_LAC ≈ 510 G [51, 52]. In this case, electron–nuclear-spin flip-flops mediated by hyperfine interaction in the excited state lead to an efficient polarization of the ¹⁴N nuclear spin [53]. Consequently, the beating frequencies gradually disappear in the FID signal while increasing the magnetic field magnitude and the oscillations are then only due to the detuning of the microwave excitation (figure 2(b), lower trace).

From these measurements, the exponent of the FID decay and the coherence time T*₂ are extracted through data fitting with the function $\exp [-(\tau /T_{2}^{*})^{\epsilon }]\sum _{m=-1}^{m=1}\beta _{m}\cos [2\pi (\delta +mA_{\mathrm {N}}) \tau ]$, where δ is the microwave detuning and β_m is the population of each ¹⁴N nuclear-spin state. As expected from the model developed in the previous section, we observe a significant enhancement of the coherence time with the amplitude of the magnetic field (figure 2(b)). In the high field limit, since the nuclear Zeeman energy is much larger than the strength of the hyperfine coupling, the quantization axis is fixed by the external magnetic field for all ¹³C nuclear spins and the anisotropic components of the hyperfine interaction A⁽ⁿ⁾_⊥ are suppressed (see equation (18)). In the weak field limit, the quantization axis of each nuclear spin is rather given by their hyperfine fields and all the components of the hyperfine interaction contribute to decoherence of the central spin (see equation (16)). For a specific distribution of ¹³C, the numerical simulation reproduces well the behavior of the experimental results, as shown in figure 2(c).

To gain more insights into the FID dynamics, the coherence time T*₂ and the exponent of the FID decay are plotted in figures 3(a) and (b) as a function of the magnetic field amplitude for two different NV defects. For both, the coherence time increases with the magnetic field strength. In addition, we also observe that the exponent of the FID decay deviates from  = 2 in the magnetic field range where the coherence time is rising, as recently reported in  [41]. In this intermediate magnetic field regime, the nuclear Zeeman energy and the hyperfine interaction are of the same order of magnitude. Dephasing of the central spin is due to comparable nuclear rotations along non-parallel precession axis, Ω₀ and Ω₁. A more detailed expression for the contribution of a single nucleus to the signal is given in the appendix for the intermediate regime. The simulation of the FID signal using the numerical method N1 agrees fairly well with our experimental measurements, as shown in figures 3(c) and (d). The average coherence time over 1000 different configurations of ¹³C is shown in figure 3(c) as a function of the magnetic field. The coherence time increases around B = 150 G, and at the same time the exponent of the exponential fit deviates from a Gaussian decay ( = 2) (figure 3(d)).

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In order to extract quantitative information on the enhancement factor η, the coherence time was measured at low field (B ≈ 20 G) and at high field (B ≈ 500 G) for a set of 20 single NV defects. Figure 4(a) shows a histogram of the coherence time in both regimes. A large variation of dephasing times is observed owing to the random distribution of ¹³C around the central spin. The mean decoherence time at low field is 〈T*₂ (20 G)〉_exp = 2.3 ± 1 μs, while in the high field limit 〈T*₂ (500 G)〉_exp = 3.7 ± 2 μs (errors are standard deviations). The histogram of the enhancement factor η is shown in figure 4(b), corresponding to 〈η〉 = 1.7 ± 0.5, in good quantitative agreement with the model developed in the previous section (see equation (23)). The histogram of the coherence time extracted from the numerical simulation over 1000 different nuclear-spin bath configurations is shown in figure 4(c), leading to 〈T*₂ (20 G)〉_N1 = 2.4 ± 0.7 μs and 〈T*₂ (500 G)〉_N1 = 3.9 ± 1.4 μs; meanwhile, the histogram of the enhancement factor is shown in figure 4(d), leading to 〈η〉_N1 = 1.7 ± 0.4, in agreement with the experimental data.

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We note that the spread of coherence times increases at high magnetic field, indicating that for some particular distributions of the nuclear-spin bath, there is a sizable enhancement; meanwhile, for others there is little enhancement, as shown in figure 3(a) for two single NV defects having similar coherence times at low magnetic field. This feature is further illustrated in figures 4(c) and (d), which show the histograms of the enhancement factor inferred from the experiments and numerical simulations, respectively. The contribution to the coherence time enhancement of each ¹³C with respect to its position relative to the central spin is addressed in section 3.3 where the bath is divided into shells and cones.

We now study the dynamics of the FID signal as a function of the magnetic field amplitude for different concentrations of ¹³C nuclear spins p. For this purpose, we use the numerical method N2 which directly provides the average trend of the FID signal within a short computational time. Figure 5(a) shows the coherence time extracted from the envelope of the FID signal given by equation (14), as a function of the magnetic field strength for four different concentrations of impurities. By decreasing the content of ¹³C, the coherence time of the central spin gets longer because ¹³C atoms are placed on average farther away from the central spin and thus exhibit weaker hyperfine interaction [17]. In addition, the magnetic field amplitude required to freeze the evolution due to the anisotropic component of the hyperfine interaction gets larger as the concentration of ¹³C nuclear spins increases. As a result, the magnetic field strength linked to the transition between the coherence time in the low field limit ($T_{2,{\tiny {\rm LF}}}^*$) and in the high field limit $(T_{2,{\tiny {\rm HF}}}^*$) decreases with the ¹³C content. We note that for a given ¹³C content, the inner distribution of nearby nuclei might resemble a bath configuration corresponding to a smaller concentration. For this particular case, we expect to have a larger T*₂ and a smaller magnetic field amplitude to increase T*₂, as it might be the case for NV12 in figure 3(a).

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The coherence time normalized to its value in the high field limit is shown in figure 5(b). The enhancement of the coherence time matches very well the value given by equation (23), $T_{2,{\tiny {\rm HF}}}^* / T_{2,{\tiny {\rm LF}}}^* = \sqrt {5/2} = 1/0.632$, and is independent of the impurity concentration. Finally, the coherence time is plotted as a function of p in figure 5(c). In the limit of either a weak or a high magnetic field, the coherence time scales as $1/\sqrt {p}$, as expected from equations (16) and (18). In intermediate regimes, the scaling deviates from this simple behavior owing to the increment of the coherence time starting when γ_nB ∼ A⁽ⁿ⁾ (see figure 5(a)).

In this section, we analyze the contribution to decoherence of each nuclear-spin impurity with respect to its position relative to the central spin, using the numerical method N2 for a natural abundance of ¹³C.

First, we group ¹³C according to their radial distance to the central spin. The nuclear-spin bath is thus divided into shells S_i with 5 Å width. The number of lattice sites in each shell is roughly proportional to the area of the shell times the width, as shown in figure 6(a). To conduct the analysis, we infer the coherence time T*_2,Si from the FID envelope due to the nuclei belonging to the shell S_i.

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For low and high magnetic fields, the total coherence time can be calculated as

$$\begin{equation} \left(\frac{1}{T_2^*}\right)^2 = \sum_i\left(\frac{1}{T_{2,S_{i}}^*}\right)^2. \end{equation} \tag{ 24 }$$

The coherence time for each shell at low and high magnetic fields is shown in figure 6(b). As expected, the closest shell to the central spin contributes the most to decoherence. We note that we have not considered strongly interacting nuclei, which results in neglecting the contribution of all lattice points of the first shell S₁, as explained in section 2.2. We estimate the contribution to decoherence of each shell as $\left ( T_2^*/T_{2,S_{i}}^* \right )^2$. Figure 6(c) indicates that the second shell with ≈650 lattice points, which corresponds to ≈7 nuclear spins at p_nat = 1.1%, contributes 85% to decoherence; meanwhile, the third shell contributes only 10%. In addition, we note that, on average, the coherence time enhancement factor is not modified while changing the radial distance of the nuclear spins with respect to the central spin (see the inset of figure 6(b)).

We now analyze the contribution of each nuclear spin as a function of its angular orientation θ with respect to the symmetry axis of the NV defect. For this purpose, the nuclear-spin bath is divided into 20 angular cones, as shown in figure 7(a). Each cone C_i contains on average an equal number of lattice sites, so that their relative contributions to decoherence can be better compared. The coherence time T*_2,Ci of the central spin is inferred by calculating the FID envelope due to the nuclei belonging to a given cone C_i of the nuclear-spin bath.

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Figure 7(b) shows the FID envelope at weak and high magnetic fields for three different cones C₁, C₅ and C₁₀, containing nuclei close to the north pole, to the magic angle θ_M ≈ 54° and to the equator, respectively. For cones C₁ and C₁₀, the FID signals are almost identical at weak and high magnetic fields, whereas for cone C₅ there is a large enhancement of the coherence time at high magnetic field. The coherence time enhancement factor η is plotted as a function of the cone mid-angle in figure 7(c). This behavior can be understood as follows. Within our model, the hyperfine interaction is assumed to be purely dipolar between the central spin and each nuclear spin of the bath. In this framework, equations (20) and (21) indicate that the anisotropic component of the hyperfine vector A_⊥ is zero at the poles ($\theta = \left \{0,\pi \right \}$) and at the equator (θ = π/2); meanwhile, the A_zz component is zero at the polar magic angle ($\theta _{\mathrm {M}} = \cos ^{-1}(1/\sqrt {3})\approx 54^\circ$). Therefore, since the anisotropic component is suppressed for large magnetic fields, the main contribution to the coherence time enhancement comes from the nuclei with axial component A_zz ≈ 0, i.e. placed at the magic angle θ_M ≈ 54°. This effect is an analogous phenomenon to magic angle spinning in solid-state nuclear magnetic resonance spectroscopy [54]. Following equation (23), the enhancement factor as a function of the polar angle θ can be written as

$$\begin{equation} \eta(\theta) = \frac{T_{2,{\tiny{\rm HF}}}^*(\theta)}{T_{2,{\tiny{\rm LF}}}^*(\theta)} = \left| \frac{\sqrt{1+3\cos^2\theta}}{1-3\cos^2\theta} \right| . \end{equation} \tag{ 25 }$$

This formula accurately matches the enhancement factor inferred from the numerical simulations, as shown in figure 7(c).

From the partition of the bath into shells and cones, we conclude that nuclear-spin baths with a large number of nuclei close to the magic angle and close to the central spin exhibit a large enhancement of their coherence time when the magnetic field is increased. This feature also explains the large variation of the enhancement factor shown in figures 4(c) and (d), which correlates with the presence of nuclei close to the magic cone.