Strongly polarizing weakly coupled 13C nuclear spins with optically pumped nitrogen-vacancy center

Enhancing the polarization of nuclear spins surrounding the nitrogen-vacancy (NV) center in diamond has recently attracted widespread attention due to its various applications. Here we present an analytical formula that not only provides a clear physical picture for the recently observed polarization reversal of strongly coupled13C nuclei over a narrow range of magnetic field [H. J. Wang et al., Nat. Commun. 4, 1940 (2013)], but also demonstrates the possibility to strongly polarize weakly coupled13C nuclei. This allows sensitive magnetic field control of the 13C nuclear spin polarization for NMR applications and significant suppression of the 13C nuclear spin noise to prolong the NV spin coherence time.

The atomic nuclear spins are central elements for NMR and magnetic resonance imaging 1 and promising candidates for storing and manipulating long-lived quantum information 2 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) center 3 in diamond, which has an optically polarizable spin-1 electronic ground state with a long coherence time 4 , allowing DNP at room temperature 5,6 . This prospect has attracted widespread interest due to its potential applications in room-temperature NMR, magnetic resonance imaging and magnetometry 7,8 , electron-nuclear hybrid quantum register [9][10][11] , and electron spin coherence protection by suppressing the nuclear spin noise 12 . In addition to the remarkable success in coherently driving spectrally resolved transitions to initialize, manipulate, and readout up to three strongly coupled nuclear spins 10,[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 anticrossing 6,[17][18][19][20][21][22] or ground state level anticrossing (GSLAC) 5,23,24 , driving the NV-nuclear spins into Hartman-Hahn resonance 25,26 or selectively driving certain spectrally resolved transitions between hyperfine-mixed states under optical illumination 27,28 . Successful polarization of bulk nuclear spins in diamond have dramatically enhanced the NMR signal by up to five orders of magnitudes 17,28 and significantly prolonged the NV spin coherence time 25,26 . In particular, near NV excited state level anticrossing, almost complete polarization has been achieved for the on-site 15 N (or 14 N) and the strongly coupled 13 C nuclei in the first shell of the vacancy 6,[20][21][22]29 .
Recently, Wang et al. 24 exploited the GSLAC to achieve near complete polarization of the strongly couled, first-shell 13 C 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 coupled 13  absent. Furthermore, in most of the existing works, only a few strongly coupled nuclear spins (HFI 200 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 imaging 17,27,28 and prolong the NV spin coherence time 25,26 .
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 coupled 13 C nuclei (with HFI ~ 100 MHz) over a narrow range (a few mT) of magnetic field 24 , but also reveals a simple strategy to (i) strongly polarize weakly coupled 13 C 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 13 C 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 |± 〉 1 g and | 〉 0 g separated by zero-field splitting = . i i and Ŝ 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 13 C nuclear spin-1/2 and drop the nuclear spin index i. Since |− 〉 1 g is nearly degenerate with the NV steady state | 〉 0 g under optical pumping, the NV-nucleus flip flop is dominated by the nuclear spin raising transition | 〉 ⊗ | ↓ 〉 → |− 〉 ⊗ | ↑ 〉 0 1 g g and the nuclear spin lowering } are nuclear spin Zeeman eigenstates. The energy mismatch of the raising (lowering) transition is γ ( mT, we can regard Δ Β 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 γ ∆ e B , i.e., off resonance as long as the linewidth of the transition is smaller than γ ∆ e B . This highlights the linewidth (hereafter denoted by R) of the NV ground state as a crucial ingredient for achieving strong nuclear polarization: γ / R e must be smaller than ∆ B , 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 ∝ R optical pumping strength. Therefore, strong negative (positive) nuclear polarization can be achieved under sufficiently weak optical pumping γ ( / ∆ )  R e B 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 13 mT. This gives a simple explanation to the experimentally observed reversal of the polarization direction over a few mT 24 . For weakly coupled 13 C nuclei, ∆ ≈ . 0 08 B 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 where L NV is the Liouville superoperator governing the NV evolution (including various dissipation processes as shown in Fig. 1) in the absence of the nuclear spin, γ =Ĥ BI N N z is the nuclear spin Zeeman Hamiltonian, and ⋅F I 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 theory [30][31][32] . For the optically pumped NV center, this theory is applicable as long as the optical initialization time τ c 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 13 C nuclear spin-1/2, we obtain the rate equation 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 ≡ − ↑ ↓ p p p is given by is the steady-state polarization and ≡ + + − W W W 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: 2 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 illumination 23,33 . 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 ∝ ± +  W A 2 also suggests that the polarization p ss depends strongly on the HFI tensor A, e.g., for the 15  , 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 τ c , the NV center becomes a non-Markovian bath and Eq. (2) becomes inaccurate, e.g., instead of dropping below τ c , the true DNP time would be lower bounded by ~τ c . This can be easily understood: since the nuclear spin dissipation originates from the dissipation of the optically pumped NV center, the timescale /W 1 of the nuclear spin dissipation should be longer than the time sclae τ c 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 minutes 17,34 . The latter can be estimated from the Fermi golden rule, e.g., for a 13 C located at  . Unless explicitly mentioned, the very small leakage from 0 e to |± 〉 1 g is neglected, consistent with the experimentally reported 14,42 high optical initialization probability ~96% into | 〉 0 g . The effect of imperfect NV optical initialization will be discussed shortly.

DNP of strongly coupled 13 C nucleus.
To begin with, we consider the DNP of a strongly coupled 13 C nucleus in the first shell of the NV center under optical pumping near the | 〉 0 g -|− 〉 1 g GSLAC. This configuration has been studied experimentally in ensembles of NV centers 24 , which shows a reversal of the direction of the 13 C 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 13 C 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 τ c , so our analytical formula are not accurate. In this case, we numerically solve Eq. (1) using the experimentally measured ground state HFI tensor 19 MHz. To focus on the intrinsic behavior of the DNP, we set γ = 0 dep . We have verified that due to the strong HFI induced NV-13 C mixing, the 13 C nuclear polarization depends weakly on the optical pumping up to < R 4 MHz. In our numerical calculation, we take = .
R 0 4 MHz. In Fig. 2(a), the numerically calculated nuclear polarization (black solid line) correctly reproduces the sign reversal of the experimentally deduced nuclear polarization 24 (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 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-13 C 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).
For the first-shell 13 C nucleus 24 , 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 /W 1 calculated from Eq. (2) is much longer than τ c , e.g., for weakly coupled 13 C nuclei, our analytical formula  , so that the nuclear spin transition rates ∝ ± +  W A 2 are reduced by a factor η 2 . In Fig. 3(a), the agreement between the analytical results and the exact numerical results improves successively with increasing η, e.g., excellent agreement is reached for η = 125. 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. DNP of weakly coupled 13 C 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 0 g by neglecting the small intersystem crossing from 0 e to ±1 g . 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 13  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., δ = .
B 0 002 mT has been reported 34 . 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  γ 2 dep or equivalently γ + dep . This requires that the HFI should not be too weak, e.g., for = .
R 0 2 MHz and γ = 1 dep 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 γ ( / > ∆ ≈ .  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 ĥ z : each many-nuclei state = ⊗ = m m and hence changes the NV dynamics and NV-induced nuclear spin flip, e.g., the raising and lowering transition rates   . 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 ĥ z that randomly shifts the NV energy levels, such that the . The effect of the Overhauser field is equivalent to a random magnetic field γ / h z e 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 ( ) h z rms makes the originally resonant raising (lowering) transition off-resonant, and hence reduce the resonant DNP rate by a factor ( ) / h R z rms 2 2 . For example, a natural abundance of 13 C nuclei gives a typical Overhauser field ( ) MHz. This reduces the typical DNP rate by a factor of 2 for the optical pumping rate = .
As shown in Fig. 5(a), for a small number (N = 7) of randomly chosen 13 C nuclei coupled to the NV via dipolar HFI, the average polarization p ss from the mean-field approximation agrees well with the exact solution to Eq. (5). For N = 400 randomly chosen 13 C nuclei, the exact solution is no longer available and we plot the approximation results in Fig. 5 and hence favors negative polarization. These results clearly demonstrate the possibility to strongly polarize weakly coupled 13 C nuclear spins by using weak optical pumping near B ± . The polarization of these weakly coupled 13 C 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 13 C nuclei with < R 15 Å away from the NV center, corresponding to dipolar HFI strength > 6 kHz. For more distant 13 C 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 13

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 observed 24 magnetic field dependence of the polarization direction of first-shell 13 C nuclei, but also reveals an efficient scheme to strongly polarize weakly coupled 13 C 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 13 C nuclei in isotope purified diamond 4 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 | 〉 0 g 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 13 C nuclear spins.

Methods
Derivation of nuclear spin transition rates. According to the theory 30  , which assumes a tedious form as it includes various quantum coherence effects.