Photoluminescence at the ground state level anticrossing of the nitrogen-vacancy center in diamond

We employ two different theoretical approaches to study the GSLAC photoluminescence signal of NV ensembles interacting with external fields and environmental spins. For external fields, the density matrix $\varrho$ of a single NV center is propagated over a finite time interval according to the master equation of the closed system,

where $H$ is the ground-state spin Hamiltonian specified in section II. The starting density matrix $\varrho_{0}$ is set to describe 99.99% polarization in the $\left|m_{\text{S}},m_{{}^{14}\text{N}}\right\rangle=\left|0,+1\right\rangle$ state of the electron and the ¹⁴N nuclear spins of the NV center. The PL intensity $\mathcal{I}$ is approximated from the time averaged density matrix according to the formula of

where the $\left\langle p_{0}\right\rangle$ and $\left\langle p_{\pm 1}\right\rangle$ are the time averaged probabilities of finding the electron spin in $m_{\text{S}}=0$ and $m_{\text{S}}=\pm 1$, respectively, and $C=0.3$ is a reasonably experimentally attainable ODMR contrast.

To study the effects of environmental spins on the GSLAC photoluminescence signal, we apply a recently developed extended Lindblad formalism(Ivády, 2020). In this approach spin-relaxation of a selected point defect surrounded by a bath of environmental spins can be simulated over either a fixed simulation time or cycles of ground state time evolution and optical excitation steps. The modeled systems consist of a central NV center, either ¹⁴NV or ¹⁵NV, and a bath of coupled environmental spins of the same kind. Different bath spins considered in our study are spin-1/2 ¹³C, spin-1/2 P1 center with a spin-1 ¹⁴N nuclear spin, and spin-1 ¹⁴NV and ¹⁵NV centers. To create a realistic spin bath, spin defects are distributed randomly in the diamond lattice around the central NV center in a sphere. To obtain ensemble averaged PL spectra, in all cases, we consider an ensemble of configurations, i.e.  a set of random distributions of the spin defects. While each configuration describes a different local environment of the NV center, the ensembles describe a certain spin bath concentration on average. As a main approximation of the method, the many-spin system is divided into a cluster of subsystems. The number of spins included in each cluster determine the order of the cluster approximation. In the first-order cluster approximation no entanglement between the bath spins is taken into account. Higher order modeling allows inclusion of intra-spin bath entanglement. For further details on the methodology see Ref. [Ivády, 2020]. For simplicity, the mean field of the spin bath(Ivády, 2020) is neglected in this study.

In the case of a ¹³C spin bath, the nuclear spin-relaxation time is long compared to the inverse of the optical pump rate, which enables nuclear spin polarization to play considerable role in the GSLAC PL signal of NV centers. Therefore, to simulate the PL signal we simulated a sequence of optical excitation cycles. Each of them included two steps, 1) coherent time evolution in the ground state with a dwell time $t_{\text{GS}}$ set to 3 $\mu$s, and 2) spin selective optical excited process taken into account by a projection operator defined as

where $I$ is the identity operator, $P_{i\rightarrow f}$ is a projector operator from $\left|m_{S}=i\right\rangle$ state to $\left|m_{S}=f\right\rangle$ state of the NV spin, and $p_{s}$ is the probability of finding the system in state $\left|m_{S}=s\right\rangle$ $(s=0,\pm 1)$. Hyperfine coupling tensors between the central NV center and the nuclear spin are determined from first principles density function theory (DFT) calculations as specified in Refs. (Ivády, 2020; Szász et al., 2013). Spin-relaxation in the excited state is neglected in this study. Typically 32 cycles are considered, which corresponds to $\approx 0.1$ ms overall simulations time. We note that simulation of longer pumping is possible, however, beyond $0.1$ ms we experience considerable finite-size effects in our model consisting of 127 nuclear spins, find more details in the appendix. Based on the convergence tests summarized in the appendix, we set the order parameter to 2, meaning that ¹³C-¹³C coupling is included in the model between pairs of close nuclear spins, and considered an ensemble of 100 random spin configurations in all cases when ¹³C nuclear spin bath is considered.

For point-defect spin environments we make an assumption that the spin-relaxation time of the spin defects is shorter than the inverse of the coupling strength and the pump rate, thus dynamical polarization of the spin defects due to interaction with the central NV center may be neglected. Omitting optical polarization cycles, we simulate a ground state time evolution of 0.1 ms dwell time to model such systems. For P1 center and NV center spin environments we assume non-polarized and nearly completely polarized states for the spin bath, respectively. Coupling tensors between the central and environmental spin are calculated from the dipole-dipole interaction Hamiltonian. Our ensembles induce 100 random spin defect configurations, each of them consisting of 127 spin defects. Electron spin defects usually possess shorter coherence time than the inverse of the NV coupling strength, therefore, the bath may be considered uncorrelated and the first-order cluster approximation is appropriate in these cases.(Ivády, 2020)

In our experiments we study different diamond samples with different defect concentration and ¹³C abundance. Table 1 summarizes the most relevant properties of all the studied samples.

We carry out photoluminescence measurements on our samples. The experimental apparatus includes a custom-built electromagnet which provides magnetic field of 0 to about 110 mT. The electromagnet can be moved with a computer-controlled 3-D translation stage and a rotation stage. The NV-diamond sensor is placed in the center of the magnetic bore. The diamond can be rotated around the $z$-axis (along the direction of the magnetic field).

At the GSLAC region the parallel magnetic field is set so that $g_{\text{e}}\beta S_{z}B_{z}\approx D$. Due to the large splitting, $2D\approx 5.8$ GHz , between $m_{S}=+1$ level and the other electron spin levels and also the dominant spin polarization in the $m_{S}=0$ spin state, the $m_{S}=+1$ level can be neglected. The relevant energy levels in the vicinity of the GSLAC are depicted in Fig. 1 (a). The corresponding wavefunctions, expressed in the $\left|S_{z},I_{z}\right\rangle$ basis, are provided in Table 2. Besides the hyperfine interaction induced avoided crossings between levels $\gamma$ and $\varepsilon$ and $\beta$ and $\zeta$, one can identify seven crossings.

In the absence of external field and other spin defects in the environment, the GSLAC PL signal is a straight line with no fine structures at the GSLAC, as the ¹⁴N hypefine interaction does not allow further mixing of the highly polarized $\alpha$ state. External fields, however, give rise to additional spin flip-flop processes that open gaps at the crossings, mix the bright $m_{S}=0$ and the dark $m_{S}=-1$ spin state and thus imply fine structures in the GSLAC PL spectrum. In Table 3 we list spin flip-flop processes that may take place at crossing A-F, also labeled by two Greek letters that refer to the crossing states. We note that except for the $\alpha\delta$ crossing, all crossings allow additional spin mixing. Precession of the electron and ¹⁴N nuclear spin may be induced by external transverse field. Cross relaxation between the electron and nuclear spin can happen when the initial, near unity polarization of the $\alpha$ state is reduced by external perturbations.

As $2\times 2$ Pauli matrices, $\sigma_{x}$, $\sigma_{y}$, and $\sigma_{z}$, and the $2\times 2$ identity matrix $\sigma_{0}$ span the space of $2\times 2$ matrices, the spin Hamiltonian of any external field acting on the reduced two-dimensional basis of the NV electron spin can be expressed by the linear combination of these matrices at the GSLAC, as

This means that any time independent external perturbation acting on the electron spin of the NV center at the GSLAC can be described as an effective magnetic field. Therefore, in the following, we restrict our study to transverse magnetic field perturbations that induce spin mixing. This is sufficient to understand the GSLAC PL signal due to external fields.

We study the effects of transverse magnetic field theoretically, in a spin defect-free NV center model, and experimentally, in our 99.97% ¹²C IS diamond sample. In the simulations we evolve the density matrix according to the master equation of the closed system, Eq. (11), over 0.1 ms and calculate the average PL intensity. This procedure allows us to obtain minuscule PL features caused by weak transverse magnetic fields. In Fig. 1 (b) and (c) the theoretical and experimental PL signals are depicted at different transverse magnetic fields. On top of the wide central dip at $B_{z}=102.4$ mT, that corresponds to crossing E and to the precession of the electron spin due to the transverse field, altogether four (five) pronounced side dips can be seen on the theoretical (experimental) curves. The right most dip M in the experiment is related to cross relaxation with the nitrogen spin of other NV centers, for details see section IV.4. Side dips A, B, C, and F are well resolvable in both theory and experiment and we assign them to spin flip-flop processes induced by the transverse magnetic field. With increasing transverse field, these features shift, broaden, and change amplitude. For example, dips B and C merge with the central dip, while dip A moves away from the central dip. These features are characteristic fingerprints of external transverse fields. We note, that the side dip positions are somewhat different in experiment and theory. We attribute these differences to other, unavoidable couplings in the experiment, e.g. parasitic longitudinal and transverse magnetic field, electric fields, and other spin defect.

Transverse-field dependence of the dip position, width, and amplitude can be understood through the energy level structure altered by the transverse magnetic field and the variation of the population of the states induced by additional spin flip-flop processes. As an example we discuss the case of dip C that appears at the crossing $\alpha\delta$ at 102.305 mT. In Table 3 we marked this crossing as not allowed, which is valid in the limit of $B_{x}\rightarrow 0$. Indeed, by reducing the strength of the transverse magnetic field the dip vanishes rapidly. At finite transverse field the mixing of the states at the $\beta\delta$ crossing changes the character of level $\delta$ that allows new spin flip-flop processes at $\alpha\delta$. This happens only when the transverse magnetic field is strong enough to induce overlap between the anti-crossing at $\beta\delta$ and crossing at $\alpha\delta$.

Next, we discuss the magnetic field dependence of the linewidth of the central dip E at $B_{z}=102.4$ mT, which is relevant for both longitudinal and transverse magnetic-field-sensing applications. We study the full-width at half maximum (FWHM) for the central peak for different transverse magnetic field values both experimentally and theoretically, see Fig. 1 (d). Except the region where dip C merges with the central dip, the theoretical FWHM depends linearly on the transverse field with a gradient of 3.36 mTmT^-1. The experimental FWHM depends also approximately linearly on the transverse magnetic field, however, at vanishing transverse magnetic field it exhibits an offset from zero. This is an indication of parasitic transverse fields and other couplings in the experiment. The slope of the experimental curve is measured to be $2.3\pm 0.1$ mTmT^-1, which is smaller than the theoretical one. Here, we note that inhomogeneous longitudinal fields can also broaden the peaks at the GSLAC. In this case, the broadening is determined by the variance $\Delta B_{\parallel}$ of the longitudinal field.

Next, we theoretically investigate the PL signal of ¹⁵NV center that is subject to transverse magnetic field of varying strength. The energy level structure of the two spin system is depicted in Fig. 2 (a), and Table 4 provides the energy eigenstates as expressed in the $\left|S_{z},{{}^{15}I_{z}}\right\rangle$ basis. Besides the hyperfine interaction induced wide avoided crossing of states $\nu$ and $\pi$, three crossing can be seen in Fig. 2 (a) that may give rise to PL features in the presence of transverse magnetic field.

Figure 2 (b) depicts the simulated PL signal of ¹⁵NV center exhibiting two dips on the PL curves. Based on the position of the dips, the pronounced dip at 102.4 mT can be assigned to the crossing marked by $\Phi$, while the shallow dip at 102.85 mT, observable only at low transverse magnetic fields, is assigned to crossing $\Gamma$. In order to describe the processes activated by the transverse magnetic field at these dips, in Table 5 we detail the crossings observed at the GSLAC of ¹⁵NV center. As can be seen spin precession is only possible at crossing $\Delta$ and $\Gamma$ and forbidden in first order at crossing $\Phi$. Due to the high degree of polarization in state $\mu$ and the weak spin state mixing at $\Gamma$, spin precession is suppressed to a large degree at crossing $\Delta$ and $\Gamma$. The prominent PL signature at $\approx 102.4$ mT is enabled by the interplay of the spin state mixing at crossing $\Delta$ and $\Phi$. For large enough transverse magnetic fields the avoided crossing appears at $\Delta$ overlaps with the crossing at $\Phi$ that enables additional mixing with the highly polarized $\mu$ state. The role of this second order process greatly enhances as the transverse magnetic field increases and eventually gives rise to a prominent PL dip at the GSLAC.

In Fig. 2 (c) we depict the transverse magnetic field dependence of the FWHM of the central dip of the GSLAC PL signal of ¹⁵NV center. Due the second-order process involved in the spin mixing, the FWHM curve is hyperbolic like. The derivative of the curve is approaching zero (3.3 mTmT^-1) for vanishing (large) transverse magnetic field. Due to additional perturbation and field inhomogeneities, we expect that the linewidth of the central dip saturates at a finite minimal value in experiment, similarly as we have seen for the ¹⁴NV center.

The case of ¹⁵NV center demonstrates that second-order processes enabled by the perturbation of the energy level structure can also play a major role at the GSLAC. Eventually, such processes make ¹⁵NV centers interesting for magnetometry applications.

We study the interaction of ¹⁴NV-¹³C spin bath system at the GSLAC. We record the experimental PL spectrum in our W4 sample of natural ¹³C abundance, in which hyperfine interaction with the surrounding nuclear spin bath is the dominant environmental interaction expectedly. A fine structure is observed that exhibits a pair of side dips at $\pm$48 $\mu$T distances from the central dip at 102.4 mT, see Fig. 3 (a). Similar effects have been recently reported in single-NV-center measurement in Ref. [Broadway et al., 2018].

The phenomenon can be qualitatively understood by looking at the energy-level structure of a ¹⁴NV center interacting with a ¹³C nuclear spin at the GSLAC, see Fig. 3 (b), where the level labeling introduced in Fig. 1 is supplemented with either an up or down arrow depending on the spin state of the ¹³C nucleus and the hyperfine interaction is neglected for simplicity. The central dip appears at the place of crossing E_↓ and E_↑, where the $\alpha$ and $\gamma$ electron spin states of down and up ¹³C nuclear spin projection cross, respectively. Electron spin depolarization and consequent drop of the PL intensity occur at this magnetic field due to the precession of the NV electron spin induced by the effective transverse magnetic field of the nuclear spin. The phenomenon is similar to what we have seen for the case of external transfers fields. The transverse field of the nuclear spin arises from the dipolar hyperfine coupling interaction. According to Eqs. (7), the transverse field of individual nuclear spins is proportional to $\cos\theta\sin\theta$, therefore, it vanishes for $\theta=0^{\circ}$ and $90^{\circ}$, while it is maximal for $\theta=45^{\circ}$ and $135^{\circ}$. An important difference between external transverse field and transverse hyperfine field is that the latter varies center-to-center due to distinct local nuclear spin arrangement of individual centers. The varying traverse field induces LAC of varying width at the crossing of $\alpha$ and $\gamma$ levels. Consequently, the central dip observed in an ensemble measurement is a superposition of numerous Lorentzian curves of varying width resulting in a typical line-shape distinguishable from the line-shape observed for homogeneous external fields.

The left (right) satellite dip corresponds to the crossing G (H) of states $\gamma\uparrow$ and $\alpha\downarrow$ ($\gamma\downarrow$ and $\alpha\uparrow$), where hyperfine term $S_{\pm}I_{\mp}$ ($S_{\pm}I_{\pm}$) may open a gap. According to Eq. (8), the strengths of these coupling terms are given by $A_{\parallel}\sin^{2}\theta+A_{\perp}\left(\cos^{2}\theta+1\right)$ and $A_{\parallel}\sin^{2}\theta+A_{\perp}\left(\cos^{2}\theta-1\right)$, respectively. Note that the terms exhibit distinct dependence on the parameters of the hyperfine tensor. Consequently, the left side dip is dominantly due to nuclear spins that are placed on the symmetry axis of the NV center, while the right side dip is dominantly due to nuclear spins that are placed next to the NV center in a plane perpendicular to the NV axis. The PL side dips are caused by mutual spin flip-flops of the electron and nuclear spins that depolarize the electron spin. In turn the nuclear spins can be polarized at the magnetic field values corresponding to the side dips. Due to the different electron and nuclear spin coupling terms efficient at the different side dips, opposite nuclear spin polarization is expected. Indeed, our simulations reveal that the average nuclear polarization $P=\left\langle p_{+1/2}-p_{-1/2}\right\rangle$, where $p_{\chi}$ is the probability of finding individual nuclear spins in state $\left|\chi\right\rangle$, where $\chi=+1/2$ or $-1/2$, and $\left\langle...\right\rangle$ represent ensemble and bath averaging, switches as the magnetic field sweeps through the GSLAC, see Fig. 3 (c). These results are in agreement with previous results(Wang et al., 2013; Broadway et al., 2018).

Dynamic nuclear polarization is demonstrated in Fig. 3 (c), where we depict the average nuclear spin polarization obtained after simulating continuous optical pumping of varying duration. The pumping rate is set to 333 kHz in the simulations. It is apparent from the figure that the average nuclear polarization continuously increases as the pumping period extends. The positive and the largest negative polarization dip correspond to the crossing G and H, respectively. The complicated pattern is however the result of the interplay of different processes that take place at other, not labeled crossings. It is also apparent from the figures that DNP is considerably stronger at the magnetic field corresponding to the right dip. We also note that in the simulations considerable finite-size effects are observed due to the limited number of spins included in the model, see appendix. Therefore, quantitative results reported in Fig. 3 (c) are not representative to the bulk but rather to nano-diamond samples of $\approx 5$ nm size embedding a single, magnetic field aligned NV center. In such small nano-particles nuclear spin diffusion may be negligible, as it is in the simulations.

As the NV center has an effect on the nuclear polarization, the nuclear polarization has also an effect on the NV center, especially on the PL signature at the GSLAC. Similar effects were also seen in single NV center measurements.(Broadway et al., 2018) Polarization of the nuclear spins populates and depopulates certain levels that makes the effects of certain level crossing more or less pronounced. In Fig. 3 (d) we model the GSLAC PL spectrum of NV centers interacting with polarized and non-polarized spin baths. Note that the simulation time is set only to 32 $\mu$s in order not to alter the initial polarization significantly. Polarization in nuclear spin up (down) state completely reduces the left (right) dip but in turn enhances the right (left) dip amplitude. Furthermore, additional shallower satellite dips appears. In contrast, the central dip amplitude is affected only marginally by the degree and sign of the nuclear spin polarization. When the spin bath is not polarized initially, i.e. it only polarizes due to optical pumping according to Fig. 3 (c), we observe two side dips of similar amplitudes in the simulations.

The theoretical PL curve of non-polarized ¹³C spin bath in Fig. 3 (d) resembles the experimental curve Fig. 3 (a), however, the amplitude of the side dips is overestimated. As we have seen, this amplitude depends considerably on the polarization of the bath. The relatively small side dip amplitudes in the experiment indicate considerable polarization. We note that the numerical simulation cannot reproduce these curves completely due to finite-size effects observed in the simulations. As mentioned above, DNP at the higher-magnetic-field side dip, that polarizes in the perpendicular plane, is more efficient. Therefore, polarization reaches the side of the simulation box quickly in the simulation, after which the nuclear polarization increases rapidly and reduces the right side dip, that makes the PL side dips asymmetric in amplitude, see appendix. To circumvent this issue, one may utilize a model including ¹³C nuclear spins in a larger, disk shaped volume centered at the NV center.

Next, we theoretically investigate the PL spectrum of the ¹⁵NV-¹³C system. The simulated GSLAC PL spectrum, depicted in Fig. 4 (a), reveals a multi-dip fine structure with four distinct dips labeled by capital Greek letters. While the central dip $\Phi$ and its satellite dip $\Theta$ are less prominent compared to the central dip of the ¹⁴NV GSLAC PL signal, we observe two major side dips at larger distances. These dips are similar in amplitude and nearly symmetrical to the central dip, however, their origin is completely different. Figure 4 (b) depicts the energy-level structure of ¹⁵NV-¹³C system and identify the level crossings that are responsible for the observed dips. The central dip and the satellite dip $\Theta$ correspond to the precession of the electron spin driven by the effective transverse field of the ¹³C nuclear spin bath. Side dip $\Lambda$ appears at the crossing of states of different ¹³C magnetic quantum number, thus hyperfine flip-flop operators may induce mixing between the nuclear and electron spin states. At the place of the dip $\Lambda$, DNP may be realized. Finally, side dip $\Omega$ appears at the crossing of levels of identical ¹³C nuclear spin quantum numbers suggesting that here electron spin precession plays a major role.

Figure 4 (c) and (d) depict the polarization of the ¹⁵N nuclear spin and the site and ensemble averaged polarization of the ¹³C nuclear spins, respectively. Polarization of ¹⁵N nuclear spin closely follows polarization of the electron spin due to their strong coupling. In absolute terms, the depolarization of the ¹⁵N nuclear spin is twice as large as the electron spin’s depolarization indicating that ¹⁵N nuclear spin plays a role in forming the PL dips. The average polarization of the ¹³C spin bath shows distinct signatures, see Fig. 4 (d). Efficient polarization transfer is only possible at the magnetic field corresponding to dip $\Lambda$, where electron spin-nuclear spin mixing is possible. At the central dip $\Phi$ spin coupling gives rise to a sharp alternating polarization pattern. At dip $\Theta$ and $\Omega$ we observe only shallow dips in the ¹³C polarization. In order to compare ¹⁴NV and ¹⁵NV DNP processes we depicted in Fig. 4 (d) the averaged nuclear spin polarization obtained for ¹⁴NV center as well. After a fixed 0.3 ms optical pumping, we see that nuclear spin polarization achieved in the two cases is comparable.

As can be seen in Fig. 4 (d) polarization transfer can be as efficient as for the ¹⁴NV center. The fact that the crossing states at dip $\Lambda$ contain only slight contribution from the $\left|m_{\text{S}}=-1\right\rangle$ electron spin state suggests that the polarization transfer is suppressed. In contrast, we obtain considerable polarization that we attribute to the absence of competing flip-flop processes at dip $\Lambda$. Processes that could hinder polarization transfer, such as electron spin precession at dip $\Theta$ and $\Phi$, are well separated, in contrast to the case of ¹⁴NV center. These results indicate that besides the most often considered ¹⁴NV center, the ¹⁵NV center system may also be utilized in MW free DNP applications.

Diamond often hosts paramagnetic point defects that can interact with the NV centers at the GSLAC. The spin-1/2 P1 center is a dominant defect in diamond. This defect does not exhibit level crossing at the magnetic field corresponds to the GSLAC, see section II. Due to the large energy gap between the electron spin states, the central NV center couples non-resonantly to P1 center. This limits the range of interactions to some extent, however, as we show below, efficient coupling is still possible.

We study the PL signature of two different samples, E6 and F11, that contain P1 centers in 13.8 ppm and 100-200 ppm concentrations, respectively, see Fig. 5 (a). Depending on the P1 concentration, we observe either three or five dips in the PL intensity curve. Similar signal has been reported recently in Ref. [Armstrong et al., 2010; Anishchik and Ivanov, 2017, 2019]. In sample F11 of higher P1 center concentration, two pairs of side dips, I and L and J and K, can be seen around the central dip E at 102.4 mT. In sample E6 of lower P1 center concentration only side dips J and K can be resolved beside the central dip. Note that the distance of the side dips from the central dip is an order of magnitude larger than in the case of ¹³C spin environment, thus these signatures are not related to the nuclear spin bath around the NV center.

In Fig. 5 (b) we depict the energy-level structure of the magnetic field aligned and 109^∘ aligned P1 centers for $m_{\text{P1}}=-1/2$, where one can see three groups of lines for both $m_{\text{S}}=0$ and $m_{\text{S}}=-1$. These separate groups of lines can be assigned to the different quantum numbers of the ¹⁴N nuclear spin of the P1 center. The corresponding states split due to the strong hyperfine interaction. Note that similar energy level structure can be seen for $m_{\text{P1}}=+1/2$ at 2.8 GHz higher energy. From the comparison of the place of the crossings and the dips in the PL signature we can assign each of the dips to separate crossing regions. The central crossing, labeled E, where the crossing states possess identical P1 center electron and ¹⁴N nuclear spin projection quantum numbers, is responsible for the central dip. Similarly to the case of the ¹³C nuclear spin, the NV electron spin precesses in the effective transverse field of the P1 center caused by the non-secular $S_{\pm}S_{z}^{\text{P1}}$ term of the dipole-dipole interaction. Side dips J and K (I and L) correspond to crossings where the quantum number of the nuclear spin of the P1 center changes by $\pm$1 ($\pm$2). Understanding the mechanisms that give rise to the side dips requires further considerations.

We theoretically study the PL signal of NV centers interacting with P1 centers ensembles of different concentration, see Fig. 5 (c). As one can seen, the theoretical curves resemble the experimental ones, however, there are important differences. Even in large P1 concentrations we only see side dips J and K besides dip E in the simulations. The amplitude of the side dips is also underestimated. Furthermore, the shape of the central peak is different in the simulations and in the experiment, especially in sample E6. The latter can be described by a Lorentzian curve, similarly as we have seen for external fields. This may indicate considerable transverse magnetic field or strain in sample E6.

To understand the mechanism responsible for the side dips, we study the magnetic field dependence of the polarization of the electron spin and the ¹⁴N nuclear spin of the P1 center. The latter can be characterized by $\varrho_{0}^{0}$ monopole, $\varrho_{0}^{1}$ dipole, and $\varrho_{0}^{2}$ quadrupole moments that correspond to population, orientation, and alignment, respectively. (Auzinsh et al., 2019) Orientation and alignment can be obtained from quantities $p_{m}$ defining the probability of finding the nuclear spin in state $\left|m\right\rangle$ as

and

respectively. The polarization curves as a function of the external magnetic field are depicted in Fig. 5 (d). Note that the electron spin does not exhibit any polarization. This is due to the fact that the large, 2.8 GHz splitting of the P1 center electron spin states at the GSLAC suppresses flip-flop processes that could polarize the P1 center. The nuclear spin polarization observed in Fig. 5 (d) might be unexpected, as the electron spin of the P1 center cannot polarize the nuclear spin. Instead, the NV center directly polarizes the nuclear spin of the P1 center. This direct interaction is made possible by the hyperfine coupling that mixes the electron and nuclear spin of the P1 center. Considering only the nuclear spin states, the hyperfine mixing gives rise to an effective $g$-factor that may be significantly enhanced due to the contribution of the electron spin. It is apparent from Fig. 5 (d) that the nuclear polarization exhibits a fine structure at the magnetic fields that correspond to side dips J and K. This fine structure cannot be resolved in the experimental PL spectrum.

As side dips I and L do not appear in the theoretical simulation we can only provide tentative explanation of these dips. The positions of the dips correspond to magnetic fields where the crossings are related to P1 center nuclear spin state $\left|+1\right\rangle$ and $\left|-1\right\rangle$. Therefore, to flip the NV electron spin, the quantum number of the P1 center nuclear spin must change by 2. This may be allowed by the interplay of other spins. For example, ¹³C nuclear spin around the NV center or P1 center-P1 center interation may contribute to this process. As side dips I and L are pronounced only at higher P1 center concentrations, we anticipate that the second process is more relevant.

Next, we investigate the linewidth of the central dip E. The varying local environment of the NV centers can induce magnetic field inhomogeneity in an ensemble that broadens the central GSLAC PL dip. This effect may limit the sensitivity of magnetic field sensors. The FWHM for varying P1 concentration $c$, ranging from 10 ppm to 200 ppm, is considered. By fitting a linear curve to the theoretical points we obtain a slope of $\approx 20$ $\mu$T/ppm.

Finally, we note that the example of P1 center can be easily generalized to the case of other spin-1/2 point defects. Through the effective transverse magnetic field of the defects, one may expect contribution to the central dip at 102.4 mT. Furthermore, depending on the hyperfine interactions at the paramagnetic defect site, one may observe side dips placed symmetrically beside the central dip. When the point defect includes a paramagnetic isotope of high natural abundance that couples strongly to the electron spin, a pronounced PL signature may be observed at the GSLAC.

The example of P1 center demonstrates that nuclear spins around spin-1/2 defects can be polarized by the NV center. This phenomena may enable novel DNP applications at the GSLAC. Furthermore, we demonstrated both experimentally and theoretically that GSLAC PL signature depends on the concentration of the spin defect in the vicinity of the NV centers. This realization may motivate the use of GSLAC PL signal in spin defect concentration measurements.

Next, we investigate the case of ¹⁴NV center coupled to a bath of magnetic-field-aligned ¹⁴NV center spins. Note that in this case both the central spin and the bath spins exhibit crossings and anticrossings at the GSLAC. The simulated PL spectrum is depicted in Fig. 6 (a) for various spin-bath concentrations. As can be seen, two dominant dips, E and M, are observed with additional shoulders appearing at higher concentrations. To track down the origin of the most visible dips, we depict energy curves of allowed spin-state transitions of an individual NV center in Fig. 6b. Transition-energy curves are differences of energy levels that we label by pairs of Greek letters $\mu\nu$, where $\mu$ and $\nu$ represent levels of individual NV centers, see Fig. 1 and Table 2. Crossings in the energy level structure of an individual NV center are represented in the transition energy plot by curves approaching zero at the places of the level crossings. Due to the identical level structure of the central and coupled spins, crossings of transition energy curves represent places of efficient cross relaxation between the two NV centers. Cross relaxation can induce depolarization of the central spin that may give rise to dips in the PL spectrum. There are numerous cross relaxation places that can be identified in Fig. 6 (b). Most of them, however, are not active due to the high polarization of the coupled NV centers. Relevant transition-energy curves associated to the highest populated energy level $\alpha$ in Fig. 1 are highlighted in Fig. 6 (b) by wide solid lines. Dip E appears at the magnetic field where $\alpha\gamma$ transition energy vanishes, ie.  $\alpha$ and $\gamma$ states cross, enabling precession of the central NV center in the transverse field of other NV centers, similarly as we have seen for ¹³C and P1-center spin baths. We assign the source of dip M in the PL spectrum to a crossing between $\alpha\gamma$ and $\alpha\varepsilon$ transition energy curves. At the magnetic field of the transition-energy crossing $\alpha\gamma$ transition corresponds dominantly to $\left|0,+1\right\rangle\leftrightarrow\left|-1,+1\right\rangle$ transition, while $\alpha\varepsilon$ transition corresponds dominantly to $\left|0,+1\right\rangle\leftrightarrow\left|0,0\right\rangle$ transition. Therefore, at dip M efficient cross relaxation between the electron and the nuclear spins of the two centers takes place.

To demonstrate ¹⁴NV-¹⁴NV couplings at the GSLAC experimentally we carry out PL measurement in our IS sample. In Fig. 6 (c) one can see the experimental PL spectrum obtained at near perfect alignment of the external magnetic field and the NV axis. The residual transverse field is estimated to be $\approx 3.6$  $\mu$T. The experimental curve exhibits several dips. To identify each of them we carried out theoretical simulations including additional external transverse magnetic fields of 15 $\mu$T strength. External field is applied solely to the central spin to mimic local transverse inhomogeneities. It is apparent from the comparison that the experimental curve exhibits the signatures of both local field inhomogeneities and interaction with field aligned NV centers in our ¹³C depleted sample.

Finally, in Fig. 6 (d), we theoretically investigate the effect of 109^∘ aligned NV center of 10 ppm concentration. These spin defects act like a source of local inhomogeneous transverse field that gives rise to PL signature similar to the central dip of P1 center induced spin bath. The FWHM of the curve is however twice larger than the FWHM of the P1 center induced PL signature at the same concentration. This is due to the larger magnetic moment of the NV center.