Introduction

The atomic nuclear spins are central elements for NMR and magnetic resonance imaging1 and promising candidates for storing and manipulating long-lived quantum information2 due to their long coherence time. However, the tiny magnetic moment of the nuclear spins makes them completely random in thermal equilibrium, even in a strong magnetic field and at low temperature. This poses severe limitations on their applications. The dynamic nuclear polarization (DNP) technique can bypass this limitation by transferring the electron spin polarization to the nuclear spins via the hyperfine interaction (HFI), but efficient DNP is usually prohibited at room temperature.

An exception is the nitrogen-vacancy (NV) center3 in diamond, which has an optically polarizable spin-1 electronic ground state with a long coherence time4, allowing DNP at room temperature5,6. This prospect has attracted widespread interest due to its potential applications in room-temperature NMR, magnetic resonance imaging and magnetometry7,8, electron-nuclear hybrid quantum register9,10,11 and electron spin coherence protection by suppressing the nuclear spin noise12. In addition to the remarkable success in coherently driving spectrally resolved transitions to initialize, manipulate and readout up to three strongly coupled nuclear spins10,13,14,15,16, there are intense activities aiming to enhance the polarization of many nuclear spins via dissipative spin transfer from the NV to the nuclear spins. To overcome the large energy mismatch for resonant spin transfer, various strategies have been explored, e.g., tuning the NV spin near the excited state level anticrossing6,17,18,19,20,21,22 or ground state level anticrossing (GSLAC)5,23,24, driving the NV-nuclear spins into Hartman-Hahn resonance25,26 or selectively driving certain spectrally resolved transitions between hyperfine-mixed states under optical illumination27,28. Successful polarization of bulk nuclear spins in diamond have dramatically enhanced the NMR signal by up to five orders of magnitudes17,28 and significantly prolonged the NV spin coherence time25,26. In particular, near NV excited state level anticrossing, almost complete polarization has been achieved for the on-site15N (or 14N) and the strongly coupled13C nuclei in the first shell of the vacancy6,20,21,22,29.

Recently, Wang et al.24 exploited the GSLAC to achieve near complete polarization of the strongly couled, first-shell 13C nuclei and observed multiple reversals of the polarization direction over a narrow range (a few mT) of magnetic field. This interesting observation allows sensitive control of the polarization of strongly coupled13C nuclei by tuning the magnetic field, but a clear physical picture remains absent. Furthermore, in most of the existing works, only a few strongly coupled nuclear spins (HFI  kHz) are significantly polarized via direct spin transfer from the NV center, while many weakly coupled nuclear spins are only slightly polarized via nuclear spin diffusion. Enhancing the polarization of these weakly coupled nuclear spins could further improve NMR and magnetic resonance imaging17,27,28 and prolong the NV spin coherence time25,26.

Motivated by the experimental observation of Wang et al.24, in this paper we present an analytical formula for the DNP induced by an optically pumped NV center near the GSLAC at ambient temperature. It not only provides a clear physical picture for the experimentally observed polarization reversal of the strongly coupled13C nuclei (with HFI ~ 100 MHz) over a narrow range (a few mT) of magnetic field24, but also reveals a simple strategy to (i) strongly polarize weakly coupled13C nuclei (with HFI down to ~kHz) and (ii) control the direction of their polarization by tuning the magnetic field over a much narrower range (~0.1 mT). First, we introduce an intuitive physical picture for our strategy. Then we present an analytical formula that substantiates this physical picture. Finally, we perform numerical simulations that demonstrate our strategy for a few hundred weakly coupled 13C nuclei.

Results

Model and intuitive physical picture

Our model consists of a negatively charged NV center coupled to many surrounding nuclear spins at ambient temperature. The NV center has a ground state triplet and separated by zero-field splitting  GHz and an excited state triplet and separated by zero-field splitting 29. In a magnetic field B along the N-V axis (z axis), the electron Zeeman splitting cancels the ground state zero-field splitting at a critical magnetic field , leading to GSLAC between and . The GSLAC reduces the energy mismatch for the NV-nuclei flip flop and enables NV-induced DNP through their HFI , where and is the NV ground state spin.

Now we provide an intuitive physical picture for using an optically pumped NV center near the GSLAC to strongly polarize the nuclear spins and control their polarization direction by the magnetic field. For brevity we focus on one 13C nuclear spin-1/2 and drop the nuclear spin index i. Since is nearly degenerate with the NV steady state under optical pumping, the NV-nucleus flip flop is dominated by the nuclear spin raising transition and the nuclear spin lowering transition , where are nuclear spin Zeeman eigenstates. The energy mismatch of the raising (lowering) transition is , where is the energy separation and comes from the nuclear Zeeman term and the longitudinal HFI term . Since we always work near the GSLAC, where  mT, we can regard ΔΒ as a constant independent of the magnetic field B. The different energy mismatches make it possible to selectively drive one transition into resonance while keeping the other transition off resonance: set the magnetic field to , so the raising (lowering) transition has a vanishing energy mismatch, i.e., on resonance, while the lowering (raising) transition has a finite energy mismatch , i.e., off resonance as long as the linewidth of the transition is smaller than . This highlights the linewidth (hereafter denoted by R) of the NV ground state as a crucial ingredient for achieving strong nuclear polarization: must be smaller than , so that the raising and the lowering transitions can be spectrally resolved. Below we show that the optical pumping is the dominant level-broadening mechanism, so that optical pumping strength. Therefore, strong negative (positive) nuclear polarization can be achieved under sufficiently weak optical pumping by tuning the magnetic field to B− (B+). The direction of the polarization can be reversed by switching the magnetic field between B− and B+. For first-shell 13C nuclei,  mT. This gives a simple explanation to the experimentally observed reversal of the polarization direction over a few mT24. For weakly coupled 13C nuclei,  mT, so the direction of the polarization can be reversed by sweeping the magnetic field over a much smaller range. Below we substantiate this physical picture with an analytical formula.

DNP theory of single nuclear spin

Under optical pumping, seven energy levels of the NV center are relevant (Fig. 1). The NV-nucleus coupled system obeys the Lindblad master equation

Figure 1
figure 1

NV states at ambient temperature responsible for DNP under optical pumping.

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where is the Liouville superoperator governing the NV evolution (including various dissipation processes as shown in Fig. 1) in the absence of the nuclear spin, is the nuclear spin Zeeman Hamiltonian and is the NV-nucleus HFI. Equation (1) can be solved exactly by numerical simulation. However, this approach does not provide a clear physical picture and it quickly becomes infeasible for multiple nuclear spins (to be discussed shortly), because the dimension of the Liouville space grows exponentially with the number of nuclei.

Our work is based on a recently developed microscopic theory30,31,32. For the optically pumped NV center, this theory is applicable as long as the optical initialization time τc of the NV center is much shorter than the timescale of the DNP process, because in this case the NV center can be regarded as a non-equilibrium Markovian bath. Applying this theory to a 13C nuclear spin-1/2, we obtain the rate equation for the nuclear spin populations and , where W+(W−) is the rate of the nuclear spin raising (lowering) transition, as discussed in the previous subsection. The real-time evolution of the nuclear spin polarization is given by , where is the steady-state polarization and is the rate of DNP.

For weak optical pumping far from saturation, we can derive (see Methods) the following Fermi golden rule for the nuclear spin transition rates:

where with , is the steady-state population of the NV center on , is the Lorentzian shape function and the optically induced NV ground state level broadening R is equal to the optical pumping rate, i.e., the number of optical transitions per unit time from the NV ground orbital to the excited orbital (see Methods). This optically induced level broadening is typically much larger than the intrinsic NV spin decoherence rate (~1 kHz) and provides a microscopic explanation for the previously observed NV level broadening under laser illumination23,33. Most importantly, Eq. (2) quantifies the physical picture discussed in the previous subsection: to achive strong negative (positive) nuclear polarization, we can use weak optical pumping and set the magnetic field to B− (B+), so that the rate W− (W+) of the nuclear spin lowering (raising) transition is resonantly enhanced, while the rate W+ (W−) of the nuclear spin raising (lowering) transition is suppressed. The relation also suggests that the polarization pss depends strongly on the HFI tensor A, e.g., for the 15N nucleus with and , we have and and hence .

Equation (2) is accurate only when the DNP time 1/W calculated from Eq. (2) NV optical initialization time , so that the NV center is a Markovian bath. When the optical pumping rate R is so small or the HFI is so strong that the DNP time calculated from Eq. (2) drops below τc, the NV center becomes a non-Markovian bath and Eq. (2) becomes inaccurate, e.g., instead of dropping below τc, the true DNP time would be lower bounded by ~τc. This can be easily understood: since the nuclear spin dissipation originates from the dissipation of the optically pumped NV center, the timescale of the nuclear spin dissipation should be longer than the time sclae τc of the NV center dissipation.

Up to now, we have neglected other nuclear spin relaxation mechanisms, such as spin-lattice relaxation and HFI with NV excited states. The former occurs on the time scale ranging from a few seconds to tens of minutes17,34. The latter can be estimated from the Fermi golden rule, e.g., for a 13C located at 3Å away from the NV center, the dipolar HFI gives a nuclear spin relaxation time >1 s. These effects are described by nuclear depolarization , which increases W by and decreases by a factor , i.e., nuclear depolarization is negligible when .

Below we first specialize to strongly coupled 13C nuclei, where is always satisfied, so the nuclear depolarization is negligible. Then we consider weakly coupled 13C nuclei, where becomes a crucial condition for achieving strong nuclear polarization. In all our calculations, we use the experimentally measured parameters at room temperature (cf. Fig. 1): excited state orbital pure dephasing 35, radiative decay rate γ = 13 MHz36 and intersystem crossing rates 37 and 38,39,40,41. Unless explicitly mentioned, the very small leakage from to is neglected, consistent with the experimentally reported14,42 high optical initialization probability ~96% into . The effect of imperfect NV optical initialization will be discussed shortly.

DNP of strongly coupled 13C nucleus

To begin with, we consider the DNP of a strongly coupled13C nucleus in the first shell of the NV center under optical pumping near the – GSLAC. This configuration has been studied experimentally in ensembles of NV centers24, which shows a reversal of the direction of the 13C nuclear polarization over a narrow range (~a few mT) of magnetic fields. This interesting observation allows sensitive control of the polarization of the strongly coupled 13C nuclei by tuning the magnetic field, but a clear physical picture remains absent. Here the very strong HFI makes the DNP time calculated from Eq. (2) much shorter than , so our analytical formula are not accurate. In this case, we numerically solve Eq. (1) using the experimentally measured ground state HFI tensor19,24,43,44 with nonzero components , MHz, MHz and MHz. To focus on the intrinsic behavior of the DNP, we set . We have verified that due to the strong HFI induced NV-13C mixing, the 13C nuclear polarization depends weakly on the optical pumping up to  MHz. In our numerical calculation, we take  MHz.

In Fig. 2(a), the numerically calculated nuclear polarization (black solid line) correctly reproduces the sign reversal of the experimentally deduced nuclear polarization24 (circles and squares). Actually, although our analytical formula in Eq. (2) is not accurate, it still provides a clear physical picture for the polarization reversal: the negative polarization dip in Fig. 2(a) originates from the resonance of the nuclear spin lowering transition at  mT, while the positive polarization away from B− originates from the dominance of the raising transition rate over the lowering transition rate since  MHz is significantly larger than  MHz. We notice that there is significant difference between our numerical results and the experimentally deduced nuclear polarization from the ODMR spectrum, especially in the magnitude of the negative dip. We tentatively attribute this discrepancy to two factors. First, all the parameters used in our calculation are taken from previous experimental measurements and/or first-principle calculations, which may differ from the particular experiment of Wang et al.24. Second, the estimate of the nuclear polarization from the ODMR spectrum involves a series of assumptions such as the lack of quantum coherence between different NV-13C mixed levels, so the uncertainty of this estimate is relatively large, e.g., the estimates based on different transitions give different results, denoted by the circles and squares in Fig. 2(a).

Figure 2
figure 2

(a) pss of a first-shell 13C nucleus: exact numerical solution to Eq. (1) (orange solid line) compared with the experimentally deduced nuclear polarization24 (filled squares and empty circles) from two different analysis methods. (b) Real-time evolution of nuclear polarization at B+ or far from B+: exact solution to Eq. (1) vs. our analytical formula Eq. (7).

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For the first-shell 13C nucleus24, the NV center is not a Markovian bath and the numerically calculated nuclear polarization [solid lines in Fig. 2(b)] exhibit strong non-Markovian oscillation. In this regime, our analytical results [dashed lines in 2(b)] are not valid. However, as long as the DNP time calculated from Eq. (2) is much longer than , e.g., for weakly coupled 13C nuclei, our analytical formula becomes accurate. To demonstrate this point, we keep invariant, set and reduce and by a factor (i.e., and , so that the nuclear spin transition rates are reduced by a factor . In Fig. 3(a), the agreement between the analytical results and the exact numerical results improves successively with increasing , e.g., excellent agreement is reached for . In Fig. 3(b), when the magnetic field approaches B± (i.e., resonance of the raising or lowering transition), the DNP rate calculated from Eq. (2) sharply peaks and significantly exceeds , while the exact numerical result saturates to ~(1.6 μs)−1. Away from B±, our analytical DNP rate and hence agrees well with the exact result.

Figure 3
figure 3

(a) Steady-state polarization pss of a first-shell 13C nucleus: exact numerical solution to Eq. (1) (solid lines) vs. analytical formula Eq. (2) (dotted line). To illustrate the convergence of our theory, we set and reduce and by a factor (black line), 5 (red line) and 25 (gray line). (b) DNP rate for : exact solution to Eq. (1) (gray solid line) vs. analytical formula Eq. (2) (blue dotted line).

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DNP of weakly coupled 13C nucleus

An important feature in Fig. 2(a) is that when the HFI decreases, the positive (negative) polarization peak (dip) at B+ (B−) approaches +100% (−100%) due to the resonance of the nuclear spin raising (lowering) transition. This indicates the possibility of achieving strong positive (negative) nuclear polarization for weakly coupled nuclear spins by tuning the magnetic field to B+ (B−). In the above calculations, we have assumed perfect optical initialization of the NV center into by neglecting the small intersystem crossing from to . Actually, strong nuclear polarization can be achieved even when the NV initialization is not perfect: as shown in the inset of Fig. 3(a), the maximal nuclear polarization at B± only decreases sublinearly with increasing initialization error, e.g., the nuclear polarization remains slightly above 80% when the optical initialization fidelity decrease to 80%. Since the nuclear polarization is not so sensitive to the NV optical initialization fidelity, hereafter we always assume perfect optical initialization of the NV center.

Now based on the analytical formula in Eq. (2), we discuss in detail the conditions for strongly polarizing distant 13C nuclei with weak HFI , so that the separation between B+ and B− is mT. Since we need to tune the magnetic field to B+ or B−, the experimental control precision δB of the magnetic field should be smaller than 0.08 mT. This is accessible by current experimental techniques, e.g., mT has been reported34. Another obvious condition is that the nuclear depolarization, which always tends to decrease the nuclear polarization, should be negligible, i.e., the maximal DNP rate   or equivalently . This requires that the HFI should not be too weak, e.g., for and s−1, this requires the HFI to be larger than 1 kHz. Hereafter we assume that this condition is satisfied. The intuitive physical picture suggests that the linewidth R (=optical pumping rate) of the NV ground state is essential, so we divide our following discussion into strong optical pumping and weak optical pumping, respectively.

Under strong optical pumping  mT), the resonances at B± are not spectrally resolved. In this case, the nuclear polarization is uniquely determined by the HFI tensor A. For dipolar HFI,

is uniquely determined by the polar angle θ of the 13C location R through . For example, in Fig. 4(c), the resonance of the nuclear spin raising transition at mT (marked by white dotted lines) and the resonance of the nuclear spin lowering transition at mT (marked by black dotted lines) are significantly broadened along the magnetic field axis and are not clearly resolved.

Figure 4
figure 4

Steady state nuclear polarization pss [(a) and (b)] and DNP rate W in units of Hz [(c) and (d)] by dipolar HFI with the NV center, calculated from the analytical formula Eq. (2) with optical pumping rates R = 3.2 MHz [(a) and (c)] and 0.2 MHz [(b) and (d)].The distance of the 13C nucleus from the NV center is  Å in (c) and  Å in (d). The white (black) dotted line indicates the magnetic field B+ (B−).

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The most interesting case is weak optical pumping  mT), where the nuclear spin raising and lowering transitions are spectrally resolved and can be selectively driven into resonance to achieve strong positive (negative) nuclear polarization by tuning the magnetic field to B+ (B−). For example, in Fig. 4(d), the very narrow resonance peak of the raising transition at B+ (marked by white dotted lines) and the resonance peak of the lowering transition at B− (marked by black dotted lines) are clearly resolved. Correspondingly, in Fig. 4(b), these resonances give rise to strong positive (negative) nuclear polarization near B+ (B−), superimposed on the usual dependence on the polar angle θ via [plotted on top of Figs 4(a),(b)]. We notice that a 13C nucleus located at  Å away from the NV center can be highly polarized at a rate ~60 Hz = (2.6 ms)−1, sufficient to overcome the slow nuclear depolarization  s−1.

DNP of multiple 13C nuclei

Having established a complete picture for single-13C DNP, now we generalize the above results to many 13C nuclear spins coupled to the NV center via , where and the excited state HFI is neglected because the induced NV-nuclei flip-flop processes are off-resonant. In the same spirit as that in treating the DNP of a single nuclear spin, we decompose the HFI into the longitudinal part and the transverse part . Then we approximate the longitudinal part with (which is treated exactly) and treat the transverse HFI with second-order perturbation theory, where

is the collective Overhauser field from all the nuclei.

The physical picture of the many-nuclei DNP is as follows. Up to leading order, the flip of different nuclei by the transverse HFI is independent, in the sense that at a given moment, only one nuclear spin is being flipped, while other nuclear spins simply act as “ spectators”. However, the flip of each individual nuclear spin does depend on the states of all the nuclear spins via the collective Overhauser field : each many-nuclei state is the Zeeman eigenstate of the ith nucleus) produces an Overhauser field that shifts the energy of the NV state by an amount −hm. This renormalizes the –  separation from to and hence changes the NV dynamics and NV-induced nuclear spin flip, e.g., the raising and lowering transition rates of the ith nucleus conditioned on the many-nuclei state being is obtained from Eq. (2) by replacing A, B with Ai, , respectively.

The above physical picture is quantified by the following rate equations for the diagonal elements of the many-nuclei density matrix :

which has been derived by adiabatically decoupling the fast electron dynamics from the slow DNP process30,31,32. Here is the same as except that the state of the ith nucleus changes from to . Now we discuss the difference between single-spin DNP and many-spin DNP. In the latter case, the DNP of each individual nucleus occurs in the presence of many “ spectator” nuclei, which produce a fluctuating Overhauser field that randomly shifts the NV energy levels, such that the – separation changes from to a random value . The effect of the Overhauser field is equivalent to a random magnetic field on the NV center, which makes it more difficult to tune the external magnetic field to match the resonance of the nuclear spin raising and lowering transitions. More precisely, a finite mismatch makes the originally resonant raising (lowering) transition off-resonant and hence reduce the resonant DNP rate by a factor . For example, a natural abundance of 13C nuclei gives a typical Overhauser field  MHz. This reduces the typical DNP rate by a factor of 2 for the optical pumping rate , e.g., the maximal DNP rate [Fig. 4(d)] for a 13C nucleus at 2 nm away from the NV center is reduced to , which is still sufficient to overcome the slow nuclear depolarization  s−1. Therefore, under weak pumping , the typical Overhauser field fluctuation does not significantly influence the nuclear polarization.

For N nuclear spin-1/2’s, the number of variables is 2N. When N is small, we can solve Eq. (5) exactly. For large N, however, the exponentially growing complexity prohibit an exact solution to Eq. (5). Here, we introduce a mean-field method (see Methods). It provides a good approximation to the average polarization and average Overhauser field , but does not include any spin-spin correlation effect, such as feedback induced spin bath narrowing12,30,31,32,45,46,47,48.

As shown in Fig. 5(a), for a small number (N = 7) of randomly chosen 13C nuclei coupled to the NV via dipolar HFI, the average polarization from the mean-field approximation agrees well with the exact solution to Eq. (5). For N = 400 randomly chosen 13C nuclei, the exact solution is no longer available and we plot the approximation results in Fig. 5(b). Both Figs 5(a),(b) clearly demonstrate that with decreasing optical pumping rate, the nuclear polarization shows a successively stronger peak (dip) at the resonant magnetic field  mT  mT). The relatively stronger polarization at B− arises from the dependence on the polar angle θ of the nuclear spin location: more nuclear spins have [as plotted on top of Figs 4(a),(b)] and hence favors negative polarization. These results clearly demonstrate the possibility to strongly polarize weakly coupled 13C nuclear spins by using weak optical pumping near B±. The polarization of these weakly coupled 13C nuclei is ultimately limited by nuclear depolarization, which becomes significant when the HFI becomes too weak. This can be clearly seen in the spatial distribution of the nuclear polarization near B+ [Fig. 5(c)] and B− [Fig. 5(d)]. Near B+, strong positive polarization is achieved for 13C nuclei with  Å away from the NV center, corresponding to dipolar HFI strength >6 kHz. For more distant 13C nuclei, the HFI strength is too weak for the DNP to dominate over the nuclear depolarization, so their polarization drops significantly. Similarly, near B−, strong negative polarization can be achieved for 13C nuclei with  Å away from the NV center, corresponding to dipolar HFI strength ~1 kHz.

Figure 5
figure 5

Average polarization of (a) N = 7 and (b) N = 400 randomly chosen 13C nuclei for different optical pumping rates R =  0.2, 0.8 and 3.2 MHz. For R = 0.2 MHz, the spatial distribution of the polarization of N = 400 randomly chosen 13C nuclei in a magnetic field (c) 102.36 mT and (d) 102.45 mT [marked by the arrows in panel (b)]. The depolarization rate  Hz for all panels.

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Finally, we recall that the on-site 15N or 14N nucleus has an isotropic transverse HFI and hence , i.e., they could be completely polarized and thus does not significantly influence the DNP of the 13C nuclei, except for a shift of the  –  separation by ~2.2 MHz (for 14N) or ~3 MHz (for 15N), corresponding to a shift of the resonance magnetic field by ~0.78 G (for 14N) or ~1.1 G (for 15N).

Discussion

In conclusion, we have presented a comprehensive theoretical understanding on the dynamic nuclear polarization induced by an optically pumped NV center near the ground state anticrossing at ambient temperature. Our results not only provide a clearly physical picture for a recently observed24 magnetic field dependence of the polarization direction of first-shell 13C nuclei, but also reveals an efficient scheme to strongly polarize weakly coupled 13C nuclear spins ~25 Å away from the NV center (HFI strength ~1 kHz) by tuning the magnetic field under weak optical pumping. These results provide a clear guidance for optimizing future dynamic nuclear polarization experiments. For example, this scheme could be used to polarize distant 13C nuclei in isotope purified diamond4 to further prolong the NV coherence time. An important limitation of our method is that it requires good alignment of the magnetic field along the N-V axis, because a tilted magnetic field would dramatically decrease the population on and hence degrade the nuclear polarization. For nanodiamonds containing a single NV center, we need to first align this NV center to the magnetic field before using this NV center to polarize weakly coupled 13C nuclear spins.

Methods

Derivation of nuclear spin transition rates

According to the theory30,31,32, the populations of a single nuclear spin-I on its Zeeman sublevels obey the rate equations

Up to second order of the tranvserse HFI (longitudinal HFI treated exactly), the transition rate from to is32

where , the Liouville superoperator and is the NV steady state in the rotating frame conditioned on the nuclear spin state being , i.e., and .

Since we consider the NV center near the GSLAC, we can exclude the energy levels and , which have vanishingly small occupation. Under this approximation, the rotating frame Hamiltonian of the five-level NV model is

where . Neglecting -induced NV spin mixing, the longitudinal HFI reduces to . Neglecting the non-collinear term (as the NV mostly stays in and the NV excited state HFI (which is far off-resonant), the transverse HFI reduces to with . To calculate from Eq. (7), we first determine the NV steady state as , where is the optical pumping rate for . Substituting into Eq. (7) gives

where is the matrix element of the operator , which is a linear combination of , , and .

Now we calculate by taking the , , and matrix elements of

which gives four coupled equations

Here is the energy difference between and are NV states and are nuclear Zeeman states), i.e.,

and is the linewidth of the NV transition , i.e., , , and . Eliminating and gives the “ rate equations”:

from which we obtain the solution

where with and the complex self-energy corrections to and by the optical pumping and , respectively. More precisely, the optical pumping induces an optical Stark shift and an effective dissipation for the NV ground state . Taking as an example, if is much smaller than , then the optical Stark shift reduces to the conventional form of a second-order energy correction in non-degenerate perturbation theroy, while the dissipation takes the semiclassical form of a Fermi golden rule. Substituting Eq. (14) into Eq. (9) immediately gives , which assumes a tedious form as it includes various quantum coherence effects.

For simplification, we use the fact that the NV excited state dephasing GHz other NV dissipation and typical detuning , and and further restrict to weak optical pumping (~26.3 MHz). In this case, the optical Stark shift is negligible and the optical pumping rate simplifies to , so the self-energies only induces NV level broadening. Substituting the resulting expression into Eq. (9) gives

For spin-1/2, Eq. (15) simplifies to Eq. (2) of the main text.

Mean field method for multiple nuclear spins

The essential idea of the mean-field approximation is to assume small inter-spin correlation, so that the N-nuclei density matrix can be factorized as , where the ith nuclear spin state is described by its polarization . This dramatically reduces the number of variables from many-nuclei populations to N single-spin polarizations . Substituting this approximation into Eq. (5) and tracing over all the nuclei except for the ith nucleus give N coupled equations:

where is the spin flip rate of the ith nucleus averaged over the states of all the other nuclear spins:

i.e., depends on the polarizations of all the other nuclear spins. Therefore, Eq. (16) with form N coupled differential equations for . The steady-state solutions are obtained by solving N coupled nonlinear equations with recursive methods.

Additional Information

How to cite this article: Wang, P. et al. Strongly polarizing weakly coupled 13C nuclear spins with optically pumped nitrogen-vacancy center. Sci. Rep. 5, 15847; doi: 10.1038/srep15847 (2015).