There are a number of possible explanations for the differing reductions of the orbital *g*-factor and spin-orbit parameter from NV^{−} to NV^{0}. Both parameters are affected by the same Ham reduction factor [22], [23] arising from the JT interactions of the NV^{0 2}*E* or NV^{− 3}*E* levels. If the JT interaction of NV^{0} is larger than of NV^{−} (as indicated by features observed in piezospectroscopy of NV^{0} [5]), then this would explain the reduced size of these parameters in NV^{0}. However, this explanation would imply that both parameters should be reduced by the same factor.

An alternate or complimentary explanation can be found in how the parameters depend on the defect’s molecular orbital structure and local electrostatic potentials. The orbital *g*-factor is proportional to the reduced matrix element $\u3008e\mathrm{(}\overrightarrow{r}\mathrm{;}\overrightarrow{R}\mathrm{)}\left|\right|{l}_{z}\left|\right|e\mathrm{(}\overrightarrow{r}\mathrm{;}\overrightarrow{R}\mathrm{)}\u3009,$ whereas the spin-orbit parameter is $\propto \u3008e\mathrm{(}\overrightarrow{r}\mathrm{;}\overrightarrow{R}\mathrm{)}\left|\right|[\overrightarrow{\nabla}{V}_{Ne}\mathrm{(}{\overrightarrow{r}}_{i}\mathrm{,}\overrightarrow{R}\mathrm{)}\times \overrightarrow{p}{\mathrm{]}}_{z}||e\mathrm{(}\overrightarrow{r}\mathrm{;}\overrightarrow{R}\mathrm{)}\u3009,$ where *l*_{z} is the orbital angular momentum operator along the axis of the NV center, ${\mathrm{[}\overrightarrow{\nabla}{V}_{Ne}\mathrm{(}{\overrightarrow{r}}_{i}\mathrm{,}\overrightarrow{R}\mathrm{)}\times \overrightarrow{p}\mathrm{]}}_{z}$ is similarly the axial component of the orbital operator of the spin-orbit interaction, *V*_{Ne} is the electrostatic interaction between the defect electrons and the nuclei, $\overrightarrow{r}$ are the electron coordinates, $\overrightarrow{R}$ are the nuclear coordinates, and $e\mathrm{(}\overrightarrow{r}\mathrm{;}\overrightarrow{R}\mathrm{)}$ are *e* molecular orbitals in the Born-Oppenheimer approximation [14].

Owing to the different charge states of NV^{0} and NV^{−}, their nuclear coordinates and molecular orbitals differ [24], [25]. Thus, the reduced matrix elements demonstrate that both the orbital *g*-factor and spin-orbit parameter will differ between the two charge states due to the molecular orbital differences, and the spin-orbit parameter will additionally differ due to differences in the electrostatic potential *V*_{Ne}. This additional dependence of the spin-orbit parameter over the orbital *g*-factor is the likely reason why the parameter reduction differ when comparing NV^{0} and NV^{−}.

Our observations do not immediately explain why fine structure has not been observed in EPR or PLE measurements of the NV^{0} ground-state. Based on the spin-orbit parameter determined here and assuming the use of a 9.6 GHz X-band EPR spectrometer, the NV^{0 2}*E* EPR features are expected to be at *B*=±(±*f*−*λ*)/μ_{B}g=±0.16T, ±0.50T which are within the available field range of a typical X-band EPR spectrometer. Additionally, it has been shown that strong illumination with a green laser can photo-convert NV^{−} to NV^{0}. Thus, samples could be conveniently pumped to contain more NV^{0} centers [12], [13]. Additionally, a tunable high-resolution laser should also be able to see these features in PLE.

We believe that there are three major reasons why the EPR signals of the ^{2}*E* were previously observed: (1) A reduction in angular momentum could also be influenced by fast averaging over the orbital states by a weak JT coupling to a bath of acoustic *E* modes. This is seen in the NV^{− 3}*E* state at room temperatures [26], [27] as a removal of spin-orbit splittings. The remaining EPR signal would then be obscured by other spin-1/2 paramagnetic spins in the diamond sample, primarily the substitutional nitrogen or P1 center. (2) Another reason would be if there is a large-strain broadening of the resonances. For the limit of a strain distribution width (Γ) which is larger than spin-orbit splitting (Γ≫*λ*), only a central spin-1/2 resonance remains. This will be obscured for the same reason mentioned above. (3) The final reason is that measurements have simply not been performed in the correct spectral range or the signal has been overlooked.

We have modeled the electron-phonon interactions and have found them to produce a negligible effect on the NV^{0} spin resonances (see Supplementary information), ruling out the first reason. For explanation 3, it seems unlikely that the signal was overlooked, given targeted efforts to look for it [1], leaving explanation 2. We have modeled the second explanation (see Supplementary information) and we find that a wide strain distribution significantly reduces the contribution of the strained centers to the total EPR spectrum. We model this by introducing a strain shift to the spin resonances *ℰ*, resulting in the spin-resonances (neglecting *l*=0.0186) at $\mathcal{B}$ and $\mathcal{B}\pm \Delta $ where $\mathcal{B}=g{\mu}_{B}{B}_{z}$ and $\Delta =\sqrt{{\lambda}^{2}+{\mathcal{E}}^{2}}.$ These spin resonances have the associated transition amplitudes cos^{2}φϕ and sin^{2}φ, where φ=tan (*ℰ*/*λ*). By constraining the oscillator strength to be conserved over the integrated spectral band, the intensity of the above resonances are ${I}_{\mathcal{B}}=\frac{\Gamma}{\Gamma +\lambda}$ and ${I}_{\mathcal{E}}=\frac{\lambda}{\Gamma +\lambda},$ where Γ is the width of the strain distribution, here assumed to be Lorentzian. In the limit that Γ≫*λ*, then *I*_{ℰ}→0 and ${I}_{\mathcal{B}}\to 1,$ resulting in spectra with a single resonance only at the free spin−1/2 resonance frequency, this behavior is demonstrated in Figure 4. The limit of Γ≫*λ* is reasonable, as the stress susceptibility and resulting width of the NV^{0} orbital electronic states are both of order 100 s of GHz [5]. Note that the ensemble ZPL width in absorption of this particular sample (~50 GHz) is about 20 times narrower than the ensemble emission ZPL width presented by Davies [5] and Manson et al. [12].

Figure 4: Behavior of ^{2}*E* spin-levels for increasing strain (*ℰ*) in a nonzero magnetic field.

In the high strain limit (*ℰ*≫*λ*) the system is better represented by the {|*X*〉, |*Y*〉} strain basis than the {|+〉, |−〉} orbital basis. The colored arrows represent the observable electron paramagnetic resonance (EPR). For example, EPR spectra for increasing strain are shown at the bottom of the figure, with increasing strain from left to right. The spectra changes from exhibiting spin-orbit, orbital Zeeman, and spin Zeeman behavior to simple spin Zeeman behavior.

The above arguments for EPR also apply to PLE measurements on a single NV^{0}. PLE should still observe some structure even in the presence of large strain. However, similar to EPR, in the large strain limit the PLE spectra will be modified to just show two transitions separated by a splitting ℰ, as such the fine-structure parameters cannot be extracted. This problem is general to measuring the PLE of any centre that has a Kramers doublet ground-state. For a Kramers doublet the degeneracy of the spin-levels will not be lifted by the strain and the eigenenergies will only add in quadrature with the applied strain splitting. This is not the same for NV^{− 3}*E* as it is a not a Kramers doublet and PLE is routinely observed [20] (see figure 4 in the Supplementary information). Our fine structure parameters will put a lower bound on the expected PLE splitting, allowing for low-strain NV^{0} single sites to be found via PLE.

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