Protecting a diamond quantum memory by charge state control

TL;DR

This paper demonstrates how controlling the charge state of NV centers in diamond can significantly extend nuclear spin coherence times and enable individual optical addressability, advancing solid-state quantum memory technology.

Contribution

It identifies and characterizes the positively charged NV center as a new charge state for improved quantum memory performance and control.

Findings

Nuclear spin coherence times increased by a factor of 20.

Charge state control enables individual optical addressability.

Positively charged NV center is electron-spin-less and optically inactive.

Abstract

In recent years, solid-state spin systems have emerged as promising candidates for quantum information processing (QIP). Prominent examples are the Nitrogen-Vacancy (NV) center in diamond, phosphorous dopants in silicon (Si:P), rare-earth ions in solids and V$_{\text{Si}}$-centers in Silicon-carbide (SiC). The Si:P system has demonstrated, that by eliminating the electron spin of the dopant, its nuclear spins can yield exceedingly long spin coherence times. For NV centers, however, a proper charge state for storage of nuclear spin qubit coherence has not been identified yet. Here, we identify and characterize the positively charged NV center as an electron-spin-less and optically inactive state by utilizing the nuclear spin qubit as a probe. We control the electronic charge and spin utilizing nanometer scale gate electrodes. We achieve a lengthening of the nuclear spin coherence times by a factor of 20. Surprisingly, the new charge state allows switching the optical response of single nodes facilitating full individual addressability.

Tables

Table 1. Table 1: Experimental (exp.) and ab initio calculated (calc.) values for nuclear spin quadrupole splittings of 14 N for various NV charge states. In the calculations, the C q subscript 𝐶 𝑞 C_{q} quadrupole splitting is 3 ​ e ​ Q N ​ V z ​ z / ( 4 ​ h ) 3 𝑒 subscript 𝑄 N subscript 𝑉 𝑧 𝑧 4 ℎ 3eQ_{\text{N}}V_{zz}/(4h) where h ℎ h is the Planck-constant and V z ​ z subscript 𝑉 𝑧 𝑧 V_{zz} is the corresponding component of the ab initio electric field gradient of the potential. Q N subscript 𝑄 N Q_{\text{N}} is the nuclear quadrupole moment of the 14 N isotope, which scatters between 0.0193 and 0.0208 barn in the literature stone_table_2005 . This uncertainty is reflected in the calculated values.

	NV- / MHz	NV0 / MHz	NV+ / MHz
calc.	-5.02$\pm$0.19	-4.92$\pm$0.19	-4.82$\pm$0.19
exp.	-4.945	-4.655	-4.619

Equations

$$H=$$

$$+\underset{H^{\mathrm{hf}}}{\underbrace{A_{\parallel }{S}_z{I}_z+A_{\perp }/2\left(S_{+}{I}_{-}+S_{-}{I}_{+}\right)}}$$

Full text

Spin defects are excellent quantum systems. Particularly, defects that possess an electron spin together with a set of well-defined nuclear spins make up excellent, small quantum registers pla_coherent_2014; waldherr_quantum_2014. They have been used for demonstrations in quantum information processing dolde_room-temperature_2013; waldherr_quantum_2014, long distance entanglement bernien_heralded_2013 and sensing zaiser_enhancing_2016. In such systems the electron spin is used for efficient readout (sensing or interaction with photons) whereas the nuclear spins are used as local quantum bits. Owing to their different magnetic and orbital angular momentum, electron and nuclear spins exhibit orders of magnitude different spin relaxation times. As an example, NV electron spins typically relax on a timescale of ms under ambient conditions jarmola_temperature-_2012, while nuclear spins do have at least minutes-long spin relaxation times pfender_nonvolatile_2016. However, the hyperfine coupling of nuclear spins to the fast relaxing electron spins in most cases significantly deteriorates the nuclear spin coherence, and eventually its relaxation time, down to time scales similar to the electron spin. For NV centers at room-temperature, this limits coherence times to about $10\,$ms waldherr_high-dynamic-range_2011; zaiser_enhancing_2016; pfender_nonvolatile_2016. For other hybrid spin ensembles e.g. in Si:P, this strict limit was overcome by ionizing the electron spin donors and thereby removing the electron spin. The resulting $T_{2}$ times were on the order of minutes for Si:P ensembles saeedi_room-temperature_2013 and less than a second for single spins morello_single-shot_2010; pla_high-fidelity_2013.

It is known that the NV center in diamond exists in various charge states. Besides the widely employed negative charge state ($\text{NV}^{-}$), it is known to have a stable $\text{NV}^{0}$ and eventually $\text{NV}^{+}$ configurations. $\text{NV}^{-}$ has a spin triplet ground state with total spin angular momentum $S=1$. Calculations as well as spectroscopic data suggest that the $\text{NV}^{0}$ ground state is $S=1/2$ while $\text{NV}^{+}$ is believed to be $S=0$, i.e. diamagnetic. Several experiments have demonstrated the optical ionization from $\text{NV}^{-}$ to the neutral $\text{NV}^{0}$ charge-state han_metastable_2010; waldherr_dark_2011; han_dark_2012; doi_deterministic_2014; aslam_photo-induced_2013; beha_optimum_2012 and electroluminescence from single $\text{NV}^{0}$ centers mizuochi_electrically_2012. These experiments characterize the $\text{NV}^{-}$ and $\text{NV}^{0}$ charge states via their photophysics and different fluorescence spectral fingerprints (see Fig. 1c).

Apart from optical ionization, deterministic electronic charge state control is feasible using a hydrogen-terminated (H) diamond surface with in-plane or electrolyte gates for Fermi-level manipulation hauf_chemical_2011; hauf_addressing_2014; grotz_charge_2012 (cf. Fig. 1a). Hydrogen termination of diamond creates a two-dimensional profile of free holes on the surface. Recently, it was shown that this surface conductive layer completely quenches the photoluminescence of NV centers in its close proximity. This optically inactive NV state was tentatively attributed to the positively charged $\text{NV}^{+}$ hauf_addressing_2014 (see Fig. 1c). Proper bias applied to lateral gate electrodes on H-terminated surface depletes the charge carrier density around the gate contact area. At full depletion, the Fermi level at the surface is shifted away from the valence band edge. This eventually leads to switching of the charge state ($\mathrm{\text{NV}^{+}\rightarrow\text{NV}^{0}\rightarrow\text{NV}^{-}}$) of such NV centers.

Here we show that such nano-scale gate structures can be used to stabilize a third charge state with an $S=0$ ground state under dark conditions, which we attribute to $\text{NV}^{+}$ (see Fig. 1). To this end, we apply the intrinsic nitrogen nuclear spin qubit of the NV center as a probe to reveal information about the charge and spin state of the NV center waldherr_dark_2011 (cf. Fig. 1d). We utilize this $\text{NV}^{+}$ charge state to prolong the nuclear spin $T_{2}$ lifetime. Furthermore, the absence of fluorescence allows tailoring the optical response from multiple NV centers within a confocal spot and therefore increases individual addressability.

To switch the charge state of NV centers we prepare H-terminated, conductive gate electrodes onto an otherwise oxygen-terminated (O) and therefore non-conductive diamond surface hauf_addressing_2014. NV centers have been created approximately 10 nm below the described surface (see Fig. 1b and Methods section). As H-terminated diamond surfaces quench the fluorescence of NV centers, the gate electrode structure can be seen in the confocal microscopy image (see fig. 1b).

In Figure 2a,b the electrical current and the NV fluorescence response to a varying gate voltage is shown. For up to $\pm 10\,$V the resistance of the capacitor is on the order of $100\,\mathrm{M}\Omega$. For properly located NV centers the voltage change results in a fluorescence response (see Fig. 2b). The fluorescence of those NV centers located in the center of the H- or O-terminated regions are expected to be stable under voltage changes, whereas NV centers in the border regions are likely to switch charge states because the Fermi-level change is most pronounced in these regions hauf_addressing_2014 (cf. Fig. 1a).

Next, we concentrate on NV centers that change into a non-fluorescing state upon a suitable voltage change. For different charge states, the quadrupole splitting $Q$ as well as the hyperfine coupling $A$ of the nitrogen nuclear spin to the NV electron spin vary significantly waldherr_dark_2011. Hence, we can perform charge and electron spin selective control gates on our nuclear spin qubit. To this end, we initialize the nuclear spin into one of its $m_{I}$ states ($\left|1_{\mathrm{n}}\right\rangle$ in Fig. 1d,e) utilizing single shot readout waldherr_quantum_2014. We set a certain gate voltage resulting in a related charge state. In this charge state we perform radiofrequency (RF) pulse sequences in resonance with an NMR transition of the nitrogen nuclear spin. To ensure a steady charge state a waiting time of 2 ms is used after the voltage change (see Fig. 2d). Finally, $\text{NV}^{-}$ is reset by the proper gate voltage and read out the 15N spin state.

For identification of $\text{NV}^{+}$, we start out with 15NV centers because they possess a nuclear spin $I=1/2$ and therefore lack quadrupole splitting. We check for an $m_{S}=0$ state in different charge states, by identifying the corresponding 15N nuclear spin transition $\left|m_{S}=0,\,m_{I}=\downarrow\right\rangle\leftrightarrow\left|0,\uparrow\right\rangle$. For this purpose, the nuclear spin is initialized into $m_{I}=\uparrow$ (denoted by $\left|1_{\mathrm{n}}\right\rangle$ in Fig. 2c), set a gate voltage and perform an RF $\pi$-pulse in resonance with the mentioned transition to flip the nuclear spin. As a control experiment, we also track the amplitude of the $\left|S=1/2,\,m_{I}=\downarrow\right\rangle\leftrightarrow\left|1/2,\uparrow\right\rangle$ 15N spin transition in the $\text{NV}^{0}$ state. Both nuclear spin transitions are displayed on the left side of Figure 2c. The normalized amplitudes of the latter transitions equal the probability of being in the corresponding charge state $W_{\pm}$ and $W_{0}$, where we can not discriminate $W_{+}$ and $W_{-}$. We start out in the $\text{NV}^{-}$ state at a gate voltage of $-8\,$V. $W_{\pm}$ and $W_{0}$ reveal a switch from $\text{NV}^{-}$ to $\text{NV}^{0}$ at around $-2\,$V for the NV center investigated (see Fig. 2c). The latter state is stable until around $+8\,$V, where $W_{0}$ decreases and $W_{\pm}$ increases again. Two reasons for this change are conceivable. Either $\text{NV}^{-}$ reappears at higher voltages, or we detect the presence of $\text{NV}^{+}$ with an electron spin state $m_{S}=0$, which is more likely and will turn out to be indeed the case. Next, we compare nuclear spin Rabi oscillations in the $\text{NV}^{-}$ and the $\text{NV}^{+}$ case (see Fig. 2d). The smaller Rabi frequency for the $\text{NV}^{+}$ case at the same radiofrequency (RF) and the same RF amplitude is explained by the absence of an electron spin and the corresponding impact on the nuclear spin states. The Rabi frequency ratio is calculated to be $\frac{\Omega_{\text{NV}^{-}}}{\Omega_{\text{NV}^{+}}}=1+\frac{\tilde{\gamma}_{\mathrm{e}}}{\tilde{\gamma}_{\mathrm{n}}}\frac{2A_{\perp}D}{\tilde{\gamma}_{\mathrm{e}}^{2}B_{z}^{2}-D^{2}}=1.832$ with the electron and nitrogen nuclear spins’ reduced gyromagnetic ratios $\tilde{\gamma}_{\mathrm{e}}$ and $\tilde{\gamma}_{\mathrm{n}}$, the perpendicular hyperfine interaction $A_{\perp}$, the Zero-field splitting parameter $D$ and the axial magnetic field $B_{z}$. The theoretically derived value agrees well with the experimentally obtained one of $1.81\pm 0.04$. The larger amplitude of the $\text{NV}^{+}$ compared to the $\text{NV}^{-}$ oscillation amplitude directly relates to the higher probability of being in the newly identified charge state. The limited spin readout fidelity let’s the oscillation in figure 2d start with a value of 0.3 on the vertical axis and a spin flip with unity probability would yield a maximum value of $1-0.3=0.7$. Indeed, the displayed oscillations reach 0.7 and hence the chance to be in $\text{NV}^{+}$ at $+8\,$V is around one. However, around $-8\,$V the NV center is only about 70 % of time in $\text{NV}^{-}$ waldherr_dark_2011. As a result, we have observed around unity combined probability to switch into $\text{NV}^{+}$ and preserve the 15N nuclear spin state.

Having identified the electron spin-less $\text{NV}^{+}$ charge state by the 15N probe spin we investigate the accompanying changes of local electric field gradients. The 14N spin’s quadrupole splitting is susceptible to the latter and therefore also is a measure for the change of the charge state (see Fig. 2e). First estimates of the $\text{NV}^{+}$ related quadrupole splitting hint to smaller values than for the $\text{NV}^{0}$ case. We therefore employ ab initio simulations in order to get insight about the 14N quadrupole moment of NV defect in diamond as a function of its charge state. The calculated electric field gradient around the 14N nuclei in different charge states of NV defect is calculated within Kohn-Sham density functional theory (DFT) that results in qualitatively good agreement with the interpretation of experimental data (see Table 1, Fig. 2e and Methods section for technical details). In the negative, neutral and positive charge state the NV center is in the ${}^{3}A_{2}$, ${}^{2}E$ and ${}^{1}A_{1}$ many-body ground state, respectively, exhibiting $C_{3v}$ symmetry. Table 1 summarizes our results with NV center in $C_{3v}$ symmetry. We find that DFT simulations imply a linear decrease in the quadrupole splitting of 14N when removing electrons from the system. Our analysis reveals that this effect is caused by the successive depletion of the charge density on the nitrogen atom. However, experimental data indicate that the 14N quadrupole moment in NV0 is in magnitude closer to the that in NV+ than to that in NV- that breaks the linear trend. We found that this effect is caused by the fact that the ${}^{2}E$ ground state is a so-called correlated many-body state that cannot be accurately described by the applied Kohn-Sham DFT method. We proved by a DFT based configuration interaction method (see Methods for technical details) that accurate calculation of the ${}^{2}E$ ground state results in depletion of the charge density on the nitrogen atom with respect to the Kohn-Sham DFT result. This explains the complex physics and the observed trend on the 14N quadrupole moment of NV defect with various charge states in diamond.

The electron spin-less $\text{NV}^{+}$ state is expected to be a formidable storage state for quantum information residing on proximal nuclear spins. Similar observations were made in ensembles of Si:P saeedi_room-temperature_2013. For the negative charge state $\text{NV}^{-}$, in contrast, the limiting factor for the nuclear spins’ $T_{1}$ and $T_{2}$ lifetimes is the hyperfine coupling to the central electron spin. Whereas the longitudinal relaxation time still reach minutes in moderate magnetic fields ($\sim 1\,$T) pfender_nonvolatile_2016 and therefore set the ultimate limit for $T_{2}$, the transverse relaxation of the nuclear spins is limited to $T_{2}\approx 10\,$ms by the longitudinal relaxation of the electron spin. In a spinless environment like diamond, the $T_{1}$ and $T_{2}$ values are supposed to increase dramatically by removing the central electron spin.

We characterize the nuclear spin lifetimes in $\text{NV}^{+}$ by preparing the 14N spin in $m_{I}=1$ ($\left|1_{\mathrm{n}}\right\rangle$) and subsequently setting the appropriate gate voltage (see Fig. 3b). The 14N dephasing time is deduced from a spin echo measurement in $\text{NV}^{+}$ shown in Figure 3a. In addition, a long Rabi oscillation and measurement of the longitudinal relaxation are depicted. The coherence lifetime is clearly increased. However, the increase is less than anticipated. This can be attributed to two effects: First, the utilized NV centers were created by nitrogen implantation close to the diamond surface, in order for the band bending effect of the gate structure to play a significant role. Near-surface NV centers commonly suffer from short coherence times on the order of $\sim 10\,\mu$s due to electron spins on the surface and paramagnetic defects created during implantation favaro_de_oliveira_toward_2016; ofori-okai_spin_2012; romach_spectroscopy_2015; de_oliveira_tailoring_2017 (see Methods section). Only recently, novel methods have experimentally demonstrated how to overcome these effects (e.g. via plasma treatment of the surface or nano-scale nitrogen doped layers close to the surface) kim_effect_2014; de_oliveira_effect_2015; ohno_engineering_2012; ohno_three-dimensional_2014; mclellan_patterned_2016. If the decoherence is caused by paramagnetic defects on or near the surface, the same effect limits the coherence of the nuclear spin, weakened by the small $g$-factor of the nuclear spin, to the order of $\sim 100\,$ms for a basic spin echo. Another effect is indicated by the comparably short nuclear spin $T_{1}$ time of $0.3\,$s. If the charge state is not stable, other states might induce faster decay. Indeed, we have observed a very slow and inert response to voltage changes, which we attribute to the large capacitor that enabled a characterization of many NV centers in the first place. Further experiments with much smaller capacitors will shed light on this behavior.

Summarizing we have identified and characterized the positive charge state of NV centers in diamond. We demonstrated deterministic and reversible electric switching of single NV centers into the newly detected charge state. It was found that nuclear spin energy eigenstates are resilient under this switching operation and thus serve as probe for $\text{NV}^{+}$. They confirmed the absence of an electronic spin, which enabled electron-spin-unlimited nuclear spin coherence storage in $\text{NV}^{+}$. Furthermore, the $\text{NV}^{+}$ state does not fluoresce under $532\,$nm illumination commonly used for $\text{NV}^{-}$ excitation. Hence, $\text{NV}^{+}$ does not contribute to photon shot noise when other proximal NV centers within the same confocal spot are optically interrogated.

These results pave the way for a fascinating and feasible implementation of the Kane proposal for a scalable, solid-state, spin-based quantum processor kane_silicon-based_1998; pezzagna_3.13_2014. As sketched in Figure 4a an H-terminated diamond surface would mute all NV based electron spins below as already discussed above (see Fig. 1). By adding insulated, nanoscopic top gates selected NV centers might be switched back into $\text{NV}^{-}$ upon request. It is interesting to note, that further increase of the gate potential $V_{\mathrm{G}}$ would lead to a lateral depletion of the surface conductive channel. The scale of the lateral depletion effect can be expressed as $r_{l}\propto V_{\mathrm{G}}^{a}/n_{s}^{b}$ denisenko_lateral_2010, where $n_{s}$ is the areal charge density in the surface channel and the exponents $a$ and $b$ are roughly in the range of 0.5 and 1. For H-terminated diamond the $n_{s}$ value is typically in the $10^{12}-10^{13}\,\mathrm{cm^{-2}}$ range maier_origin_2000. This allows a precise control of the depletion edge $r_{l}$ of about few nanometers per Volt of the gate bias denisenko_lateral_2010. This effect would allow individual addressing of closely located NV centers. Several such NV centers might then coherently interact, for example via magnetic dipole-dipole interaction dolde_high-fidelity_2014, and the resulting quantum state can then be stored unharmed on the nuclear storage qubits. For readout of quantum information, individual NV centers are switched into $\text{NV}^{-}$, their nuclear spin state is transfered to the electron spin and finally readout optically without touching other nuclear spin qubits of NV centers which reside in $\text{NV}^{+}$.

Apart from the potential realization of a scalable quantum processor, the storage of quantum information has proven to be a vital resource for nanoscale quantum metrology. In Ref. zaiser_enhancing_2016 a non-volatile quantum memory enhanced the performance of the NV electron spin sensor. This enables, on the one hand, coherent interactions of the sensor qubit to spectrally highly selective 13C spin qubits. On the other hand, high spectral resolution correlation spectroscopy was demonstrated. In both cases spectral resolution is inversely proportional to the storage time of the quantum memory. Our results would yield an increase of spectral resolution by a factor of five. Furthermore, muting the electron spin sensor also increases the coherence lifetime of sample spins and therefore allow for high resolution spectroscopy in the first place pfender_nonvolatile_2016.

I Methods

**The experimental setup consists a home-built confocal microscope with an oil immersion objective at room-temperature. Furthermore, a permanent magnet creates a field of $B_{z}\approx 470\,$mT along the optical axis, which is collinear with the NV center’s symmetry axis. Microwave (MW) and Radiofrequency (RF) fields for electron and nuclear spin gate realizations are provided through a copper wire close to the NV center’s position.

Diamond sample preparation. (111)-oriented diamond plates were obtained by laser-slicing of a low-strain type-IIa HPHT crystal. One of them was polished and used as a substrate. High-purity homoepitaxial diamond (111) film was deposited using a microwave plasma-assisted chemical-vapor-deposition (MPCVD) apparatus teraji_homoepitaxial_2015. High-purity H2, CH4 specified to 99.999% 12C isotopic enrichment (Cambridge Isotope Laboratories CLM-392), and high-purity O2 was used as source gases. The total gas pressure, microwave power, methane concentration ratio (CH4/total gas), oxygen concentration ratio (O2/total gas), growth duration and substrate temperature employed were $120\,$Torr, $1.4\,$kW, 1%, 0.2%, $11\,$h, and $950-1000^{\circ}$C, respectively. Growth rate under this condition is $3.3\,\mu$m/h, thus the homoepitaxial film thickness is deduced to be $\approx 36\,\mu$m. The pristine MPCVD layer contained a negligible amount of NV centers. We implanted ${}^{15}\mathrm{N}^{+}$ and $\mathrm{He}_{2}^{+}$ ions with kinetic energies of $10\,$keV and $6\,$keV and doses of $8\cdot 10^{9}\,\mathrm{cm}^{-2}$ and $10^{10}\,\mathrm{cm}^{-2}$ for creation of sub-surface NV centers pezzagna_creation_2010; favaro_de_oliveira_toward_2016. While the 15N implantation mainly leads to 15NV centers, helium creates vacancies that form 14NV centers from ingrown 14N impurities. The expected average depths of the created NV centers are $25\,$nm de_oliveira_effect_2015 and $10-30\,$nm favaro_de_oliveira_toward_2016 for the nitrogen and the helium implantations, respectively. For final NV center formation the sample was annealed at $950^{\circ}$C for two hours. The sample was then boiled in acid to remove graphite.

In-plane gate electrodes were created as described in hauf_addressing_2014. Hydrogen plasma treatment of the surface creates a conductive surface layer, which switches NV centers into the $\text{NV}^{+}$ state. Interdigitated capacitors are created by electron-beam lithography and substituting hydrogen by oxygen surface terminations in the selected regions (see Fig. 1c). The new termination in the gaps of the capacitor is nominally not conductive and has a lateral width of $\approx 500\,$nm. The large capacitor with an insulating gap length of more than $1\,$mm (see fig. 1) allows for investigation of many NV centers. Figure 2a reveals a sheet resistance on the order of $10^{11}\,\Omega/\text{m}^{2}$, which leads to a finite current for a few volts of bias field. The capacitor electrodes were fabricated on top of those regions containing NV centers from ion implantation. The electrodes of the capacitor are connected to large H-terminated areas, where gold pads are evaporated and connected by gold wire bonding to the periphery.

The Spin Hamiltonian restricts to the NV center electron spin (S=1) and 15N, 14N nuclear spin ($I=1/2,\,I=1$) for the $\text{NV}^{-}$ case (eq. 1). For the $\text{NV}^{+}$ case only the pure nuclear spin parts ($H^{\mathrm{n}}$) remain including the corresponding quandrupole splitting $Q_{-}\rightarrow Q_{+}$. The electron ($H^{\mathrm{e}}$) as well as hyperfine coupling ($H^{\mathrm{hf}}$), however, can be neglected.**

[TABLE]

In eq. (1) $D$ is the electron spin triplet zero-field splitting, $\tilde{\gamma}_{\mathrm{e}}$ and $\tilde{\gamma}_{\mathrm{n}}$ are the electron and nuclear spin gyromagnetic ratios over $2\pi$, and $A_{\parallel}$ and $A_{\perp}$ are the longitudinal and transverse hyperfine coupling constants to the nitrogen nuclear spin. When an electron spin is present, nuclear spin states get slightly dressed and thus deviate from pure product states due to hyperfine interaction. This effect is observable in nuclear spin Rabi oscillations, which get faster or slower than for a pure nuclear spin. A first order perturbation of the product states is sufficient for an estimation. For a 15NV center the $m_{S}=0$ spin levels change according to $\left|0\uparrow\right\rangle\mapsto\left|0\uparrow\right\rangle-\frac{A_{\perp}}{\sqrt{2}\left(\tilde{\gamma}_{\mathrm{e}}B_{z}+D\right)}\left|+1\downarrow\right\rangle$ and $\left|0\downarrow\right\rangle\mapsto\left|0\downarrow\right\rangle+\frac{A_{\perp}}{\sqrt{2}\left(\tilde{\gamma}_{\mathrm{e}}B_{z}-D\right)}\left|-1\uparrow\right\rangle$, where $\uparrow,\downarrow$ denotes the state of the 15N spin. We derive a 15N nuclear spin Rabi frequency enhancement factor of $1+\frac{\tilde{\gamma}_{\mathrm{e}}}{\tilde{\gamma}_{\mathrm{n}}}\frac{2A_{\perp}D}{\tilde{\gamma}_{\mathrm{e}}^{2}B_{z}^{2}-D^{2}}=1.832$ at $470\,$mT and for $A_{\perp}=3.689\,$MHz. The latter value is derived from the 14N value of $A_{\perp}=-2.630(2)\,$MHz by taking into account the nuclei’s different $g$-factors.

Density functional theory based calculations The density functional theory (DFT) simulations were carried out within plane wave supercell formalism together with the all-electron projector augmented wave (PAW) method blochl_projector_1994 as implemented in vasp scientific code kresse_textitab_1993; kresse_textitab_1994; kresse_efficiency_1996; kresse_efficient_1996. The electron-nuclei system is treated within Born-Oppenheimer approximation where the quantum mechanical problem of electrons and the nuclei systems is separated, and the nuclei are approximated as classical particles. We show the results in the main text as obtained by Perdew-Burke-Ernzerhof (PBE) DFT functional PBE_1996. We found that other DFT functionals including hybrid density functionals provided the same trend for the quadrupole moment of 14N in various charge states of the NV defect. The NV defect was modelled in a 512-atom simple cubic diamond supercell with PBE optimized lattice constant. The Brillouin zone is sampled in the $\Gamma$-point which is convergent within 1%. A high plane-wave cutoff of 1200 eV is applied. The geometry of the NV center was optimized in each charge state until the forces on the atoms fell below 0.0001 eV/Å. In the calculation of the electric field gradient on nitrogen nuclei we applied a hard PAW-potential for nitrogen as provided by vasp package. Since NV0 is a dynamic Jahn-Teller system we calculated its potential energy surface around the $C_{3v}$ configuration. We found that the $C_{1h}$ configuration has the lowest energy within Born-Oppenheimer basis. The absolute value of the calculated nuclear quadrupole moment of 14N is reduced by about 3.7% with respect to that of $C_{3v}$ configuration. By analyzing the DFT orbitals in the reduced $C_{3v}\rightarrow C_{1h}$ symmetry we found that the most important effect on the calculated nuclear quadrupole moment of 14N is the correlation between the $a_{1}\rightarrow a^{\prime}$ and $e\rightarrow a^{\prime},a^{\prime\prime}$ Kohn-Sham DFT orbitals in the band gap. The highly correlated localized orbitals cannot be accurately captured by Kohn-Sham DFT functionals. Indeed, the ${}^{2}E$ ground state within $C_{3v}$ symmetry is a multiplet state of $a_{1}^{(2)}e^{(1)}$ and $a_{1}^{(1)}e^{(2)}$ electron configurations (where the numbers in parentheses label the occupation of orbitals). In order to calculate these many-body ${}^{2}E$ states, we applied an approximate configuration interaction based on DFT wavefunctions wang_linear_1999; franceschetti_many-body_1999; bester_electronic_2009. We found a 5% contribution of $a_{1}^{(1)}e^{(2)}$ configuration to the ${}^{2}E$ ground state. Since only the $a_{1}$ orbital is localized on the nitrogen atom gali_ab_2008 this causes some charge density delocalization around the nitrogen atom which consequently decreases the absolute value of the calculated nuclear quadrupole moment of 14N with respect to that of DFT values.