Local and bulk 13C hyperpolarization in nitrogen-vacancy-centred diamonds at variable fields and orientations

Polarizing nuclear spins is of fundamental importance in biology, chemistry and physics. Methods for hyperpolarizing 13C nuclei from free electrons in bulk usually demand operation at cryogenic temperatures. Room temperature approaches targeting diamonds with nitrogen-vacancy centres could alleviate this need; however, hitherto proposed strategies lack generality as they demand stringent conditions on the strength and/or alignment of the magnetic field. We report here an approach for achieving efficient electron-13C spin-alignment transfers, compatible with a broad range of magnetic field strengths and field orientations with respect to the diamond crystal. This versatility results from combining coherent microwave- and incoherent laser-induced transitions between selected energy states of the coupled electron–nuclear spin manifold. 13C-detected nuclear magnetic resonance experiments demonstrate that this hyperpolarization can be transferred via first-shell or via distant 13Cs throughout the nuclear bulk ensemble. This method opens new perspectives for applications of diamond nitrogen-vacancy centres in nuclear magnetic resonance, and in quantum information processing.


Laser
Selective-π  Figure 5. Optical detection scheme for quantifying the nuclear spin state populations. The detection sequences are comprised of three stages: initialization, nuclear polarization, and readout. The initialization stage is identical in all sequences, and its main purpose is to completely depolarize the nuclear spin due to the strong optical pumping [1,2]. The nuclear polarization stage includes a laser pulse and, optionally, a simultaneous microwave irradiation pulse. The sequences lacking microwave excitation are used for fluorescence calibration, and do not generate nuclear polarization. The readout sub-sequence always comprises of a laser readout pulse, and may be preceded by a selective π-pulse, driving one of the |0, β ↑,↓ ↔ |−1, α ↑ transitions.

A. System Hamiltonian
Here we give a short summary on the quantum mechanical details of the interaction between a single nitrogen-vacancy (NV) center (S) and a single 13 C nuclear spin (I), which is needed for deriving the energy level structure of the coupled system and the microwave (MW) driven polarization scheme. At room temperature the NV-center energy level structure exhibits an electronic triplet as the ground state ( 3 A 2 ) [1,3,4]. The quantum Hamiltonian can thus be described as Here D 0 = 2.87 GHz is the zero-field splitting term, γ e and γ n the electronic and nuclear gyromagnetic ratios, B is the magnetic field vector, and A a hyperfine (HF) tensor that depends on the specific NV and nearby 13 C spin. The important feature to generate the asymmetries of the eigenstates in the different electron spin manifolds is the fact that the electron is a spin 1. Therefore as the hyperfine interaction depends on the quantum magnetic number m S , this naturally provides an asymmetry between the m S = 0 and m S = ±1 subspaces.

B. Hyperfine interaction
The HF coupling between the two spins given in Supplementary Eq. (1), is composed of two terms -a contact term and a dipolar term [5] H HF = −µ e µ n 8π |ψ e (r i )| 2 3 where µ e and µ n are the NV center and the 13 C nuclear magnetic moment respectively, ψ e (r i ) is the electronic wavefunction at the 13 C position r i , n i is the unit vector connecting the electronic wavefunction and the 13 C position. denotes an average over the electronic wavefunction ψ e (r i ). The hyperfine coupling can thus be represented by a 3×3 symmetric tensor A ij , thus expressing the hyperfine Hamiltonian term as approximation For simplicity, we consider first a magnetic field B aligned with the axis of the zerofield tensor D 0 and a secular approximation for the hyperfine interaction with respect to where the axis x was chosen such that A zy = 0 without lack of generality. The feature enabling the magnetization transfers illustrated in Fig. 1, is that in the m s = 0 manifold the eigenstates have a nuclear component |β ↑ = |↑ and |β ↓ = |↓ exhibiting a nuclear Zeeman splitting δ = γ n B 0 ; whereas in the m s = −1 manifold, the |α ↑ and |α ↓ nuclear components of the eigenstates are quantized on a different axis determined by the hyperfine tensor. If A zz γ n B 0 , the latter can be approximated as where tan η = A zx /A zz , and the eigenenergies of these states are split by ∆ ≈ A 2 zz + A 2 zx δ. If A 2 zx + A 2 zz γ n B 0 then A zz has to be replaced by A zz − γ n B 0 (or A zz + γ n B 0 in the m s = +1 manifold), but the overall physics remains the same. The elements of the MW transition matrix are given by in general all non-null, enabling the application of the proposed polarization transfer scheme.
In the Λ-regime referred to in the main text, we opted to describe the system on a basis that is suited to the MW selection rules. The main text thus described this regime (δ Ω ∆) is addressed by the MW. Due to the different HF properties associated with m s = 0 and m s = −1, |−1, α ↑ is an eigenstate but |0, α ↑ and |0, α ↓ are not, and therefore oscillate around the magnetic field at the nuclear Larmor frequency δ = γ n B 0 .

D. Non-secular effects on eigenenergies and eigenstates for an aligned magnetic field
The previous paragraph assumed that in the m s = 0 manifold, the splitting δ of the eigenstates |β ↑,↓ was dominated by the nuclear Zeeman term. If B 0 is very small, it may also be relevant to consider the non-secular corrections of the HF coupling, which may lead to an effective tilt of the nuclear spin quantization away from the magnetic field B. By contrast, the |α ↑,↓ states are not significantly modified by these non-secular HF terms as long as |D 0 ± γ e B 0 | A uv , a condition we will presume fulfilled. To obtain analytical expressions for the energy splittings in this non-secular dominated case, we use second order perturbation theory for evaluating the relevant terms of the HF coupling With this we obtain where |0, β u and |m s , α u are the eigenstates of the secular Hamiltonian (4) and E i are the corresponding eigenvalues with u = ±1/2 for ↑ and ↓ respectively. The 2nd order corrections are more significant within the m s = 0 subspace. In order to provide an order of magnitude for this correction, we consider the case where A xz = A zx = A xx = A yy = A zz . A Taylor expansion to first order of the magnetic field strength gives for the effective Larmor frequency where for the 1st-shell 13 C discussed in the main text, the non-secular contribution overcomes the secular part γ n B 0 by a factor of A 2 mT.
E. Non-secular effects on the eigenenergies and eigenstates for nonaligned magnetic fields So far we considered the magnetic field to be aligned with the D 0 tensor. If this ceases to be the case, the level structure discussed in Fig. 1 remains even if the definitions of |β ↑,↓ and |α ↑,↓ will change. The crucial characteristic to point out, is that these |α ↑,↓ and |β ↑,↓ will still have different quantization axes, and different eigenstate splittings δ and ∆. For weak magnetic fields (γ e B 0 D 0 ), the secular approximation of Supplementary Eq. (4) needs to be rewritten as The secular eigenstates on the m s = 0 manifold will now be defined by the magnetic field direction, i.e. |β ↑ = cos θ 2 |↑ + e iφ sin θ 2 |↓ and |β ↓ = sin θ 2 |↑ − e iφ cos θ 2 |↓ , where θ and φ are the polar and azimuthal angles subtended by the magnetic field in the principal axis system of the zero-field tensor. A 2nd order perturbation theory analysis was considered in Ref. [6], for these misalignment conditions. For weak magnetic fields γ e B 0 D 0 , one can consider the quantization axis of the zero-field tensor to describe the system states. In this frame, the eigenstates of the m s = ±1 manifold are not significantly modified from the ones obtained with the secular approximation. However, the non-secular contribution can turn eventually dominant within the m s = 0 subspace. Therefore, besides the type of corrections considered in Supplementary Eqs. (8) and (9), one now need to consider the non-secular effects due to a non-aligned magnetic field. The quantization axis of the |β ↑,↓ will then strongly depend on the magnetic field orientation; the effective Larmor splitting between these states was approximated in Ref. [6] as Notice that this approximation does not account for the corrections derived in (8), that can be important when the angle θ is small.  Supplementary Fig. 1c for the case Ω β ↑,↓ ,α ↑ = 0, β ↑,↓ | S x |−1, α ↑ . Similar curves are To better appreciate the difference between the eigenstates of the different manifolds, Supplementary Fig. 2 shows the probabilities | m S , m I |E i | 2 for the different eigenstates |E i in the system, where i is the eigenenergy index ordered such that E i ≤ E i+1 . For weak magnetic fields γ e B 0 D 0 the eigenstates are well separated and described by the quantum number m S ; i.e., they are given by |0, β ↑,↓ and |m S , α ↑,↓ . The |α ↑,↓ states are well approximated by Supplementary Eqs. (5) and (6), and are almost independent of the magnetic field strength far way from the level anti-crossings. For the |β ↑,↓ it can be seen that non-secular terms of the HF change the quantization axis with respect to the one of the Zeeman field for this 1st-shell HF strength, but still they are almost independent of the magnetic field strength far away from the level anti-crossings. If the HF strength is reduced by one order of magnitude, mimicking the 2nd shell HF interaction, the non-secular effects on the eigenstates become negligible and thus |β ↑ ≈ |↑ and |β ↓ ≈ |↓ . The same applies for more distant shells.
We now consider the 1st-shell behavior given a misalignment of the magnetic field with respect to the zero-field tensor axis, by a polar angle θ and azimuthal angle φ. Supplementary The reconstruction algorithm is based on mapping the four fluorescence measurements to the three eigenstate populations. We define these two sets with corresponding vectors where Seq # is the fluorescence level at the end of the relevant sequence (see Supplementary Here, F is the fidelity of the selective π-pulse, and X is the fluorescence ratio of the electronic m S = 0 and the m S = −1 states, calibrated by the second sequence: The eigenstate populations V States are thus calculated by inverting the matrix M , An example of the raw fluorescence data arising from the four sequences in Supplementary   Fig. 5 -leading to the eigenstate population results of Fig. 2a in the main text-is given in Supplementary Fig. 6, where F ≈ 0.75. The eigenstate populations are then compared to a master equation simulation (described in Methods: System Hamiltonian) with the same physical parameters of the experiment (Fig. 2). The quantum master equation involving the total 6-level model of a single NV center coupled to a single 13 C nuclear spin is given by whereL is the Lindblad super-operator describing laser pumping, and H is the Hamiltonian where C 0 = γ 0 |0 ±1| using the experimentally determined pumping rate γ 0 = 1/3µs −1 .
The nuclear polarization of Fig. 2e,f is the calculated from this simulated density matrix. The polarization of the nuclear spin at a laser power corresponding to a 3 µsec electronic pumping time, was studied for the selective Ω δ regime. In Supplementary Fig. 7, we plot the nuclear polarization dynamics, as revealed by scanning the difference between the detection sequences #4 and #2 of Supplementary Fig. 5. The optical pumping has two counteracting contributions to the nuclear polarization process. The first is the electronic pumping, which is essential for achieving nuclear hyperpolarization. At low laser power, the electronic polarization build-up proceeds too slowly to efficiently polarize the nuclear spin. Increasing the laser power enhances the electronic polarization rate and increases the nuclear polarization efficiency. A second competing process where the laser induces nuclear depolarization, however, arises due to the difference between the hyperfine tensor in the excited and ground electronic states. This depolarization process was experimentally seen in Ref. [1], and more thoroughly studied in Ref. [2]. Indeed, at high laser powers, the NV center has a large probability of inhabiting the excited state, leading to an acceleration of the nuclear spin depolarization because the dephasing of the nuclear state induced by the different hyperfine tensor. A laser power scan for low MW power (within the selective regime) was thus performed for a single NV center, while monitoring a first-shell 13 C ( Supplementary Fig. 8). As expected, at low laser powers, the NV center is not efficiently polarized and thus neither is the nuclear spin. At the other end, for high laser powers, the nuclear spin depolarization is very rapid and the nuclear polarization drops practically to zero.

C. Dependence on microwave power
In the main text, we studied the dependence of the bulk nuclear spin polarization as a function of the microwave irradiation power. Here we discuss the qualitative behavior of the nuclear polarization versus the MW power. At low MW powers, below one over the coherence rate, the efficiency of the population transfer by the MW is low, and increasing the MW power will increase the polarization transfer efficiency. At high MW powers, above ∆, the Λ-regime is reached, and the nuclear polarization is dependent on the efficiency of the transfer between the dark state |0, α ↓ and the bright state |0, α ↑ . When the MW power is increased the MW radiation dresses the eigenstates and reduces the efficiency of the latter rotation, thereby reducing the nuclear polarization, even for MW powers below the broadband regime. Therefore, a clear optimum for nuclear polarization versus MW power is expected, as witnessed on Fig. 4b of the main text.
Estimating the number of polarized 13 C spins per nitrogen-vacancy center. In the following, we employ a simple phenomenological model to estimate the average number of 13 C spins polarized by a single NV center and the spatial distribution of the polarized 13 C spins. To this effect, we introduce polarization domains (spheres) containing 13 C nuclei around statistically distributed NV centers in the diamond crystal. All estimations are based on a diamond with an assumed NV center concentration of c NV ∼ 5 ppm. With this assumption the average distance between NV centers can be calculated by where N A is the Avogadro constant and ρ diamond and M diamond are the density and the molar mass of diamond. In the special case of polarizing the aligned orientation (Fig. 3c in the main text) only one crystallographic direction, corresponding to 25% of the NV centers, contributes to the electron-nuclei polarization transfer process, increasing the average distance to ∼ 165.5 Å. By neglecting other crystal defects the polarization domain of a single NV defect is simplified to a sphere with a radius of r = 83 Å.
The Boltzmann distribution of a statistical ensemble of 13 C spins at 4.7 T in a fullyrelaxed state corresponds to a population difference between the energy levels of ∼8 ppm (for each million of spins there are 8 spins more to be found in the lower energy level). The experimentally observed enhancement factors of up to ∼ ±250 are equivalent to population differences of ∼ ±1600 ppm. In other words for each million spins there are up to 1600 spins more to be found in the lower/higher energy level. The diamond lattice contains n unit = 8 atoms per unit cell with a lattice parameter of a = 3.57 Å at room temperature. The numbers of 13 C atoms in a sphere with an assumed radius r ∼ 83 Å can be approximated by n13 C = n unit a 3 · 4 3 πr 3 c13 C ≈ 42000 where c13 C =0.1 is the 10% enriched abundance of 13 C spins. This means that a number of 13 C spin flipped by each NV center is in the order of 70. While this back-of-the-envelope calculation is sufficient to give a rough estimation of the polarized spins per the NV domain sphere, we would like to emphasize the approximative and phenomenological nature of this derivation. In addition, the number of spin flips is an average number. It is plausible to expect much higher local enhancement factors around the NV centers, while the contribution of the majority of the 13 C spins in the remainder of the NV sphere to the macroscopic polarization, is probably much lower.