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

The nitrogen-vacancy center (NV center) in diamond at magnetic fields corresponding to the ground state level anticrossing (GSLAC) region gives rise to rich photoluminescence (PL) signals due to the vanishing energy gap between the electron spin states, which enables to have an effect on the NV center's luminescence for a broad variety of environmental couplings. In this article we report on the GSLAC photoluminescence signature of NV ensembles in different spin environments at various external fields. We investigate the effects of transverse electric and magnetic fields, P1 centers, NV centers, and the $^{13}$C nuclear spins, each of which gives rise to a unique PL signature at the GSLAC. The comprehensive analysis of the couplings and related optical signal at the GSLAC provides a solid ground for advancing various microwave-free applications at the GSLAC, including but not limited to magnetometry, spectroscopy, dynamic nuclear polarization (DNP), and nuclear magnetic resonance (NMR) detection. We demonstrate that not only the most abundant $^{14}$NV center but the $^{15}$NV can also be utilized in such applications and that nuclear spins coupled to P1 centers can be polarized directly by the NV center at the GSLAC, through a giant effective nuclear $g$-factor arising from the NV center-P1 center-nuclear spin coupling. We report on new alternative for measuring defect concentration in the vicinity of NV centers and on the optical signatures of interacting, mutually aligned NV centers.

The physics of isolated NV centers at the GSLAC is well-understood, 10,17 however, the effects of environmental perturbations are not comprehensively described. Due to the presence of 14 N ( 15 N) nuclear spin, six (four) mixed electron-nuclear spin states either cross or exhibit an avoided crossing. External perturbations and interaction with the local nuclear and electron spin environment may give rise to additional spin-relaxation mechanisms at specific magnetic fields corresponding to the crossings of the spin states. Through the spin-dependent PL of the NV center, these processes may give rise to various PL signals at the GSLAC 12,[19][20][21] . Besides the optical signal, optically detected magnetic resonance (ODMR) signal of a NV ensemble has been recently recorded. 22 Recently, the ground state level anticrossing at zero magnetic field and related phenomena have attracted considerable attention. 23,24 As increasing number of applications rely on LAC signals of single or ensemble NV systems, quantitative description of the most relevant environmental couplings is essential for further development and engineering of these applications.
Furthermore, interaction between NV centers and 13 C nuclear spins at the GSLAC can potentially be utilized in dynamic nuclear polarization (DNP) [25][26][27] applications. DNP can give rise to a hyperpolarized diamond sample with a potential to transfer spin polarization to adjacent nuclear spins for the improvement of traditional nuclear magnetic resonance methods 16,18,[28][29][30][31][32] . It is therefore of fundamental importance to gain detailed insight into the NV-13 C spin dynamics at the GSLAC.
In this article we aim at establishing a guideline for developing and advancing applications at the GSLAC of the NV center in diamond by collecting and describing the most relevant interactions that may either limit existing applications or give rise to new ones. Indeed, by identifying the PL signals of different environmental couplings we reveal important interactions that enables new spectroscopy, magnetometry and dynamic nuclear polarization applications The rest of the paper is organized as follows: in section II we provide a brief overview of the established physics of the NV center. In section III, we describe our experimental setup and samples and the details of the theoretical simulations. Section IV describes our results in four sections considering interactions of NV centers with external fields, 13 C nuclear spins, P1 centers, and other NV centers at the GSALC. In section V, we discuss implications of our results. Finally, we summarize the findings in section VI.

II. BACKGROUND
The NV center in diamond gives rise to a coupled hybrid register that consists of a spin-1 electron spin and either a spin-1 14 N or spin-1/2 15 N. Hereinafter, we refer to the former as 14 NV center and to the latter, less abundant configuration, as 15 NV center.
The spin Hamiltonian of the 14 NV center can be written as where terms on the right-hand-side describe zero-field splitting, Zeeman, hyperfine, nuclear quadrupole, and nuclear Zeeman interaction, respectively, S and I14 N are the electron and nu- is the hyperfine tensor of the 14 N nuclear spin that can be expressed by its diagonal elements A xx = A yy = A ⊥ = −2.70 MHz and A zz = A = −2.14 MHz 33 .
The spin Hamiltonian of the 15 NV center can be written as where g15 N is the nuclear g-factor of 15 N nucleus, and A15 N is the hyperfine tensor of the 15 N nuclear spin that can be expressed by its non-zero diagonal elements A ⊥ = +3.65 MHz and A = +3.03 MHz 33 .
Diamond contains 1.07% spin-1/2 13 C isotope in natural abundance that can effectively interact with the NV electron spin at the GSLAC through the hyperfine interaction. The Hamiltonian of a 13 C nuclear spin coupled to a NV center can be written aŝ whereÎ13 C is the nuclear spin operator vector, g13 C is the nuclear g-factor of 13 C nucleus, and A13 C is the hyperfine tensor that consists of two terms, the isotropic Fermi contact term and the anisotropic dipolar interaction term, Due to the typically low symmetry of the NV-13 C coupling, all the six independent elements of the hyperfine tensor can be non-zero in the coordinate system of the NV center. These components can be expressed by the diagonal hyperfine tensor elements, A xx ≈ A yy = A ⊥ and A zz = A , as well as angle θ of the principal hyperfine axis e z and the symmetry axis of the NV center. The hyperfine Hamiltonian, expressed in the basis of |m S , m13 where A P1 is the hyperfine interaction tensor that can be expressed by A P1 ⊥ = 81 MHz and A P1 = 114 MHz diagonal elements. For simplicity the quadrupole interaction strength is set to the value of the NV centers quadrupole splitting in this article, i.e. Q P1 = −5.01 MHz, which is comparable with the measured quadrupole splitting of −3.974 MHz of the P1 center 34 . Both the P1 center and other NV centers may exhibit a distinct local quantization axis depending on the C 3v reconstruction and the N-V axis, respectively. We denote the symmetry and quantization axis of the P1 center in Eq. (9) by z . The angle between z and the quantization axis of the central NV center can be either 0 • or 109.5 • . The spin Hamiltonian of NV center that has z orientation can be obtained from Eq. (1) by a proper transformation of the coordinate system.
The interaction Hamiltonian between paramagnetic defects and the central NV center can be written as where S def is the spin operator vector of the spin defect and J is the coupling tensor. Assuming point like electron spin densities, J can be approximated by the dipole-dipole coupling tensor.

Theoretical approaches
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 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 0 is set to describe 99.99% polarization in the |m S , m14 N = |0, +1 state of the electron and the 14 N nuclear spins of the NV center. The PL intensity I is approximated from the time averaged density matrix according to the formula of where the p 0 and p ±1 are the time averaged probabilities of finding the electron spin in m S = 0 and m S = ±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 35  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. [35]. For simplicity, the mean field of the spin bath 35 is neglected in this study.
In the case of a 13 C 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 GS set to 3 µs, and 2) spin selective optical excited process taken into account by a projection operator defined as where I is the identity operator, P i→f is a projector operator from |m S = i state to |m S = f state of the NV spin, and p s is the probability of finding the system in state |m S = s (s = 0, ±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. 35,36 . Spinrelaxation in the excited state is neglected in this study. Typically 32 cycles are considered, which corresponds to ≈ 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 13 C-13 C 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 13 C 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 nonpolarized 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. 35

Samples and experimental methods
In our experiments we study different diamond samples with different defect concentration and 13 C abundance. Table I 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).

IV. RESULTS
In the following four sections we investigate experimentally and theoretically the most relevant interactions that can have significant effects on the GSLAC PL spectrum.

A. External fields
At the GSLAC region the parallel magnetic field is set so that g e βS z B z ≈ D. Due to the large splitting, 2D ≈ 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 |S z , I z basis, are provided in Table II. Besides the hyperfine interaction induced avoided crossings between levels γ and ε and β and ζ, 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 14 N hypefine interaction does not allow further mixing of the highly polarized α 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 III 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 αδ crossing, all crossings allow additional spin mixing. Precession of the electron and 14 N 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 α state is reduced by external perturbations.
As 2 × 2 Pauli matrices, σ x , σ y , and σ z , and the 2 × 2 identity matrix σ 0 span the space of 2 × 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   We study the effects of transverse magnetic field theoretically, in a spin defect-free NV center model, and experimentally, in our 99.97% 12 C 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  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 αδ at 102.305 mT. In Table III  in Fig. 2 (a), and Table IV provides the energy eigenstates as expressed in the |S z , 15 I z basis.
Besides the hyperfine interaction induced wide avoided crossing of states ν and π, three crossing can be seen in Fig. 2 (a) that may give rise to PL features in the presence of transverse magnetic field.    The case of 15 NV 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 15 NV centers interesting for magnetometry applications.

B. Interaction with 13 C spin bath
We study the interaction of 14 NV-13 C spin bath system at the GSLAC. We record the experimental PL spectrum in our W4 sample of natural 13 C 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 ±48 µT 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. [18].
The phenomenon can be qualitatively understood by looking at the energy-level structure of a 14 NV center interacting with a 13 C nuclear spin at the GSLAC, see Fig. 3  of these coupling terms are given by A sin 2 θ + A ⊥ (cos 2 θ + 1) and A sin 2 θ + A ⊥ (cos 2 θ − 1), 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 = p +1/2 − p −1/2 , where p χ is the probability of finding individual nuclear spins in state |χ , where χ = +1/2 or −1/2, and ... 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 16,18 .
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 ≈ 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. 18 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 µs 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 13 C 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 13 C nuclear spins in a larger, disk shaped volume centered at the NV center.
Next, we theoretically investigate the PL spectrum of the 15 NV- 13   At dip Θ and Ω we observe only shallow dips in the 13 C polarization. In order to compare 14 NV and 15 NV DNP processes we depicted in Fig. 4 (d) the averaged nuclear spin polarization obtained for 14 NV 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 14  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. [19][20][21]. 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 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 14 N nuclear spin of the P1 center. The latter can be characterized by 0 0 monopole, 1 0 dipole, and 2 0 quadrupole moments that correspond to population, orientation, and alignment, respectively. 22 Orientation and alignment can be obtained from quantities p m defining the probability of finding the nuclear spin in state |m 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 |+1 and |−1 . 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, 13 C 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. signatures of both local field inhomogeneities and interaction with field aligned NV centers in our 13 C 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.

V. DISCUSSION
The ground-state avoided crossing of the NV center spin states gives rise to a variety of couplings that imply different behavior of the NV center. We considered the most relevant couplings and demonstrated that each of them gives rise to a unique PL signature that enables identification of the dominant environmental couplings in a given sample. This may be informative for optimizing defect concentration in samples and experimental setups.
Due to the strong coupling of the NV center to its environment at the GSLAC, an optical signal is produced that makes single and ensemble NV centers interesting for microwave-free sensing and spectroscopy applications. The results collected in this article provide the necessary information for advancing such applications. In spectroscopy, the fine structure and dip positions of the GSLAC PL signal are analyzed. To avoid misinterpretation of the PL signal, it is indispensable to know the signatures of all parasitic interactions that may interfere with the signal to be measured.
Applications at the GSLAC have focused on the 14 NV center so far. We demonstrate that 15 NV centers can also be utilized in optical applications at the GSLAC with comparable or even superior performance. 15 NV center based magnetometry, spectroscopy, and hyperpolarization application may be of particular interest. In spectroscopy applications target nuclear spins induced PL dips appear with larger spacing than for 14 NV center, due to the low-angle crossing of the energy levels.
This may give rise to better resolution and low sensitivity to other perturbations.
We demonstrated that, despite the suppressed coupling of the P1 center and NV center electron spins, nuclear spins coupled to a P1 center can be polarized by the NV center at the GSLAC through an effective hyperfine interaction greatly enhanced by the electron spin-electron spin coupling and the hyperfine interaction at the P1 center site. This coupling opens new directions for DNP applications through P1 centers and other spin-1/2 defects at the GSLAC. For example, far-ther nuclear spin ensembles can be polarized by the NV center without relying on nuclear spin diffusion. This possibility may be particularly important for near surface NV centers that may polarize nuclear spins at the surface though paramagnetic surface defects. In addition, we demonstrated that the GSLAC PL signal depends considerably on the concentration of paramagnetic point defects, therefore it may serve as a novel tool for measuring spin defect concentration in the vicinity of NV centers.
Finally, we demonstrated that mutually aligned NV centers can also couple at the GSLAC opening new alternatives for gate operations. While the energy-level structure of coupled NV centers is quite involved at the GSLAC, different spin flip-flop processes resonantly enhance at certain magnetic fields. Depending on the states and the magnetic field, all sorts of operations are possible. We note that 15 NV centers are of great potential in this respect as well. Due to the larger hyperfine splitting and the reduced number of states crossing at the GSLAC, the 15 NV centers may be better controllable.

VI. SUMMARY
In summary, we examined, in a joint experimental and theoretical study, most of the relevant interactions at the GSLAC to reveal fine details of the PL signal of NV ensembles. We showed that external fields, 13 C nuclear spin, P1 centers, and other NV centers give rise to unique signatures.
These results make identification of the most relevant environmental interactions possible through the GSLAC PL signal. In addition, we provide comprehensive description of all the relevant factors that are needed to be taken into consideration in microwave-free sensing, spectroscopy, and dynamic nuclear polarization applications at the GSLAC. In order to determine the optimal simulation settings, we carry out initial convergence tests.
We consider 13 C spin bath, that may couple coherently to the central NV center, due to the long coherence time of the nuclear spins and the relatively strong coupling strength for the closest nuclear spins.
Neglect of spin-bath correlation effects is the main approximation of the utilized theoretical approach. Spin-bath correlation can be included systematically in the simulations, however, by increasing the order of cluster approximation, i.e. the number of spins included in each subsystem.
In Fig. 7(a), we depict the PL signal obtained for different order parameters, where one can see a significant difference between the case of non-correlated spin bath, 1× 13 C, and partially correlated cases, N × 13 C, where N > 1. Beyond N = 2, the PL curves change only slightly, thus we use N = 2 in the simulations of 13 C spin bath. Note that other spin defects considered in the main text include electron spins that usually possess much shorter coherence time, therefore the bath may be considered uncorrelated and the first-order cluster approximation is appropriate in those cases.
In Figs. 7(b)-(c), we study ensemble-and bath-size dependence of the GSLAC PL signal. As can be seen 100 and 128 are convergent settings for the ensemble and bath sizes, respectively.
Finally, in Fig. 7(d), we investigate ground-state dwell-time dependence of the PL curves. For increasing dwell time we observe additional fine structures appearing. In the simulations we use 3 µs dwell time that is a reasonable choice knowing the optical laser power usually used in the experiments. each of which includes 127 13 C nuclear spins in an arrangement corresponds to natural abundance.
With increasing time the right PL side dip reduces rapidly, while the left side dip reduces only moderately in the simulations. The corresponding rightmost and leftmost nuclear polarization peaks in Fig. 8(b) grows rapidly and modestly, respectively. This shows that the polarization transfer is most efficient at the magnetic field corresponds to the right PL side dip. As the efficiency of the polarization transfer is varying at the left and right dips, finite-size effects influence the side dips differently. The different pumping duration dependence of the PL side-dip amplitudes observed in Fig. 8(a) is attributed to this effect.