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Optical hyperpolarization of 13C nuclear spins in nanodiamond ensembles

Here we propose and analyse in detail protocols that can achieve rapid hyperpolarization of 13C nuclear spins in randomly oriented ensembles of nanodiamonds at room temperature. Our protocols exploit a combination of optical polarization of electron spins in nitrogen-vacancy centers and the transfer of this polarization to 13C nuclei by means of microwave control to overcome the severe challenges that are posed by the random orientation of the nanodiamonds and their nitrogen-vacancy centers. Specifically, these random orientations result in exceedingly large energy variations of the electron spin levels that render the polarization and coherent control of the nitrogen-vacancy center electron spins as well as the control of their coherent interaction with the surrounding 13C nuclear spins highly inefficient. We address these challenges by a combination of an off-resonant microwave double resonance scheme in conjunction with a realization of the integrated solid effect which, together with adiabatic rotations of external magnetic fields or rotations of nanodiamonds, leads to a protocol that achieves high levels of hyperpolarization of the entire nuclear-spin bath in a randomly oriented ensemble of nanodiamonds even at room temperature. This hyperpolarization together with the long nuclear spin polarization lifetimes in nanodiamonds and the relatively high density of 13C nuclei has the potential to result in a major signal enhancement in 13C nuclear magnetic resonance imaging and suggests functionalized and hyperpolarized nanodiamonds as a unique probe for molecular imaging both in vitro and in vivo.

Dynamical nuclear polarization holds the key for orders of magnitude enhancements of nuclear magnetic resonance signals which, in turn, would enable a wide range of novel applications in biomedical sciences. However, current implementations of DNP require cryogenic temperatures and long times for achieving high polarization. Here we propose and analyse in detail protocols that can achieve rapid hyperpolarization of 13 C nuclear spins in randomly oriented ensembles of nanodiamonds at room temperature. Our protocols exploit a combination of optical polarization of electron spins in nitrogen-vacancy centers and the transfer of this polarization to 13 C nuclei by means of microwave control to overcome the severe challenges that are posed by the random orientation of the nanodiamonds and their nitrogen-vacancy centers. Specifically, these random orientations result in exceedingly large energy variations of the electron spin levels that render the polarization and coherent control of the nitrogen-vacancy center electron spins as well as the control of their coherent interaction with the surrounding 13 C nuclear spins highly inefficient. We address these challenges by a combination of an off-resonant microwave double resonance scheme in conjunction with a realisation of the integrated solid effect which, together with adiabatic rotations of external magnetic fields or rotations of nanodiamonds, leads to a protocol that achieves high levels of hyperpolarization of the entire nuclear-spin bath in a randomly oriented ensemble of nanodiamonds even at room temperature. This hyperpolarization together with the long nuclear spin polarization lifetimes in nanodiamonds and the relatively high density of 13 C nuclei has the potential to result in a major signal enhancement in 13 C nuclear magnetic resonance imaging and suggests functionalized and hyperpolarized nanodiamonds as a unique probe for molecular imaging both in vitro and in vivo.

I. INTRODUCTION
Nuclear magnetic resonance (NMR) [1,2] and magnetic resonance imaging (MRI) [3] have evolved to be powerful techniques to extract molecular-level information in a wide variety of physical, chemical and biological applications. Recently, 13 C based MRI has emerged as a new platform enabling essentially background-free imaging of non-proton nuclei with the additional possibility for the local and permanent destruction of the signal by means of radio frequency (RF) pulses [4]. However, the low signal-to-noise ratio for 13 C based MRI is not sufficient for most clinical and research applications due to the combination of a relatively low gyromagnetic ratio of 13 C and of the low natural abundance of this nucleus. Several strategies have been proposed to enhance the sensitivity of 13 C based MRI. Particularly promising in this context is the hyperpolarization of the 13 C nuclei, i.e. the generating of a large, non-thermal 13 C nuclear spinpolarization. Indeed, hyperpolarized 13 C based MRI has shown exciting potential for in vivo applications, especially for novel metabolic imaging [5][6][7]. One of the most powerful methods for the generation of hyperpolarization is dynamic nuclear polarization (DNP) [8][9][10][11][12], in which a large polarization of electron spins is transferred to nuclear spins, resulting in an enhancement of the MRI signal by several orders of magnitude.
Over the last decade, several breakthroughs have occurred in the manufacturing, surface treatment and use of nanoparticles for biomedical applications [13]. Bio-compatible nanoparticles present an attractive platform for hyperpolarization, as they can be functionalized for molecular specificity, ca exhibit long nuclear spin relaxation times, and contain numerous nuclear spins of a single species (e.g. carbon). Long-lived hyperpolarization has been demonstrated in silica nanopartices [14] and nanodiamonds [15,16], using standard DNP protocols, i.e. transferring the high thermal electron spin polarization at cryogenic temperatures in a strong external magnetic field to nuclei. Nanodiamonds in particular offer exciting possibilities as novel hyperpolarized probes. In addition to excellent biocompatibility [17], the possibility for surface functionalisation and nuclear spin relaxation time of several minutes [18], nanodiamonds contain crystal point-defects with unique optical and magnetic properties. Amongst these the negatively charged nitrogenvacancy (NV) center stands out as its electron spin can be polarized within microseconds by optical pumping while exhibiting a relaxation time in the millisecond range even at room temperature [19]. This establishes the NV center electron spin as promising candidate for achieving nanodiamond hyperpolarization at ambient conditionsroom-temperature optical DNP.
Previous work has demonstrated that in bulk diamond electron spin polarization of the NV center can be generated and subsequently be transferred via hyperfine interactions to nearby nuclei. This polarization can then be detected indirectly via the NV center [15,[20][21][22][23][24][25][26][27] or directly via an NMR scanner [28]. To this end, in bulk diamonds a high magnetic field is aligned with the quan-tisation axis of the NV center to enable optical pumping to a specific electron spin state. Controlled interaction of the electron spin with specific nearby nuclear spins species can then be achieved for example by the application of continuous wave microwave (MW) radiation applied to the NV center electron spin [24]. When the Rabi frequency of the microwave driving field matches the Larmor frequency of a specific nuclear spin species (achieving a Hartmann-Hahn (H-H) resonance [29]), flipflops can occur between the microwave dressed states of the electron spin of the NV center and the surrounding 13 C nuclear spins [30][31][32] as was recently demonstrated experimentally [24].
However, medical MRI applications require large ensembles of nanodiamonds. In such ensembles (powder or solution) the angle between the external magnetic field and the natural orientations of NV centers and therefore their quantisation axis is randomly distributed. This brings about several challenges for achieving high levels of hyperpolarization that need to be addressed by means of carefully designed protocols. First, the energy levels of the NV spin are then distributed over a large energy range of possible values so that the applied microwave field will be resonantly coupled to an exceedingly small fraction of the NV spins. Secondly, in an external magnetic field, additional limitations concern the optical polarization of the NV spins. In particular, the NV spins will be initialized to different states relative to the laboratory frame depending on their angle with the external magnetic field, thus resulting in a small net polarization.
In this work we will show to address these challenges to achieve a radical enhancement of the hyperpolarization in a nanodiamond ensemble. To this end we use an off-resonant driving and the integrated solid effect (ISE) for the NV center spin, resulting in a robust spin polarization transfer between the NV and nuclear spins. A very large fraction of the NV electron spins can be coherently coupled to neighboring nuclear spins. Additionally, adiabatic rotation of the magnetic fields, or the Brownian rotation of the nanodiamonds themselves (in powder or solution respectively), is proposed for extending the polarization scheme to almost the entire nanodiamond ensemble. We also discuss the effect of spin diffusion induced by the nuclear dipole-diploe interactions which will further support the polarization of large volumes.
The paper is organized as follows. In Sec. II we present the coherent coupling and polarization of a single nuclear spin near a randomly oriented NV spin. This serves the introduction of the principal challenges in nanodiamond ensemble polarization, the discussion of the robust initialization and construction of suitable dressed states for the NV center electron spin and the realisation of near resonant coupling with the nuclear spin by using off-resonant driving and the integrated solid effect. In Sec. III, we introduce mechanisms to extend our polarization protocols to the entire nanodiamond ensemble, both for nanodiamond powder and for nanodiamonds in a solution. In Sec. IV, we consider our protocol for multiple 13 C nuclear spins and include the effect of dipolar coupling among the nuclear spins and in particular the benefits of nuclear spin diffusion. In Sec. V we demonstrate the applicability of our polarization schemes for another very broad range of NV center orientations. In Sec. VI we take into account the effect of depolarization processes on the efficiency of our polarization protocol. Finally, the discussion and conclusion parts are given in Sec. VII and Sec. VIII, respectively.

A. Difficulties from the random orientations
The nanodiamond ensembles that we are interested in are realised as powder or solutions. In both cases the spatial orientations of the nanodiamonds and therefore of the NV centers hosted in them is random and uniformly distributed across the full solid angle (see Fig. 1a for an illustration). In the laboratory frame this leads to the lack of a common quantization axis for the NV centers. We begin by describing in more detail the two principal challenges that are being imposed by these random spatial orientations: (1) The direction of the natural quantization axis associated with the crystal-field energy splitting D is not controllable, resulting in a significant variance of the energy levels in the presence of an external magnetic field and (2) optical pumping of the NV center electron spins in an external magnetic field will not initialize all NV center electron spins to the same state.

Zero-field and external magnetic field distribution
The negatively charged NV center, in the following denoted for brevity as the NV center, realises in its ground state an electronic spin triplet (S = 1) which exhibits a zero-field splitting of D = (2π)2.87 GHz which separates the |m s = 0 state energetically from the degenerate |m s = ± manifold. The application of an external magnetic field, in the following assumed to take the value B = 0.36 T, lifts this remaining degeneracy such that the state m s = −1 is shifted below the m s = 0 state (see Fig. 1b). The zero-field splitting and the Zeemann effect due to an external magnetic field of the NV center are described by Herein D denotes the orientation dependent zero-field splitting tensor, γ e = (2π)28.7GHz/T the gyromagnetic ratio, S the electron spin-1 vector operator [33] and B the magnetic field vector. In the principal axis system defined by the NV symmetry axes, the zero-field splitting tensor D is diagonal Here E denotes the strain dependent contribution. It is worth emphasizing that in the scenario that we are considering here, the orientation of the symmetry axis of the NV center relative to the external magnetic field is uniformly distributed over the unit sphere. For the following it will be convenient to conduct the discussion in the laboratory frame whose z-axis we define to take the direction of the externally applied strong magnetic field, γ e B ≫ D. In this frame, the zero-field splitting tensor D will have off-diagonal elements. As γ e B ≫ D these offdiagonal elements are rapidly rotating so that their main effect will be energy shifts of the diagonal element of the Hamiltonian. The Hamiltonian can then be written as (see the Appendix for details of the derivation) where D(θ) = D(1 + 3 cos(2θ)) + 3E(1 − cos(2θ)) 4 , (4) θ is the angle between the magnetic field direction and the NV-axis and the G i are given in the Appendix. Clearly, the random orientations of the NV centers cause a variation of the zero-field splitting D(θ) across the entire interval [−(2π)1.43GHz, (2π)2.87GHz] and δ(θ) across the interval [0MHz, (2π)140MHz] as shown in Fig. 2.
If we were to follow the scheme of Ref. [24], i.e. without the combination of off-resonant drive and ISE technique that we will present in this work, the uncertain detuning of the MW frequency from the electronic resonance can reach the order of GHz, which in turn would prevent effective polarization transfer.
2. Optical initialization of NV spins at the high magnetic field limit A second important difference between a randomly oriented nanodiamond ensemble and a bulk diamond concerns the optical polarization of electron spins of the NV center. For bulk diamonds, the magnetic field can be aligned with the principal axis of the NV center and the electronic spin of the NV center can be optically polarized to the state |m s = 0 by illumination with a 532 nm green laser. However, for an ensemble of randomly oriented nanodiamonds, even in the limit of a strong magnetic field, the NV centers will be optically pumped to the state |m s = 0 θ that is defined by the relative orientation of the NV center with respect to the externally applied magnetic field which defines the laboratory frame.
As discussed in Appendix, these two coordinate systems can be transformed into each other and, employing of S z θ = cos θS z − sin θ(cos φS x − sin φS y ), we can express the eigenstate |m s = 0 θ in terms of the eigenstates |0 , | ± 1 of eq. (3), i.e. in the lab frame, as If θ is large, the eigenstate |0 of the NV center in the laboratory frame as given by Hamiltonian eq. (3) differs significantly from the zero-field eigenstate |0 θ of the NV center. Hence optical initialization of randomly oriented NV centers lead to very different states depending on the orientation of the NV center. Notice though that for moderate misalignment between the NV center and external magnetic field, the initialisation of the NV center is well approximated by |0 . Indeed, for θ < 10 • degrees, | 0|0 θ | 2 > 0.97 and for θ < 20 • we find | 0|0 θ | 2 > 0.88 which implies significant polarization along the quantization axis defined by the external magnetic field. Therefore, for all the orientations of the NV centers that fall into two spherical sectors whose cone angle is 2θ , i.e., the two blue spherical sectors in Fig. 1a, significant optical polarization can be achieved. From now on, we define a deviation by θ to imply that the symmetry axis of the NV spins falls within these two spherical cones.

B. The main ideas
The main idea that we will develop in this section consists of three main ingredients, namely (i) the generation of an energy gap between electronic states that is relatively robust with respect to variations in the relative orientation between the NV center and an external magnetic field and at the same time (ii) a strong coupling between these electronic states and nuclear spins. Finally, (iii) the integrated solid effect is employed to achieve additional robustness against imperfections. All this is achieved by the use of dressed states in a ladder type configuration in which the single quantum transitions are far detuned and the double quantum transition is nearly resonant (see Fig. 3).
In such a setting, that is for a detuning on the single quantum transition that is ∆(θ), the effective Rabi frequency and thus the energy splitting on the double quantum transition is proportional to Ω 2 /∆(θ). In leading order in θ, this suffers a variation that is of the order of Ω 2 /∆ 2 ∂∆ ∂θ θ. This contrasts with the case of detuned driving one single quantum transition only. In that case the effective energy gap is proportional to √ ∆ 2 + Ω 2 which, if ∆ ≫ Ω, suffers a variation that is of order ∂∆ ∂θ θ. Therefore the variation in the former is suppressed by a factor of Ω 2 /∆ 2 compared to the latter. This makes it much easier to achieve and maintain a Hartmann-Hahn resonance.
The second crucial ingredient is related to the fact that the coupling between the electron spin and the nuclear spins is mediated by terms of the form S z ⊗ I z . In a far-detuned single quantum transition the eigenstates involved in the Hartmann-Hahn resonance are |m s = 0 + Ω ∆ |m s = +1 and |m s = +1 − Ω ∆ |m s = 0 . As these states are close to being eigenstates of S z they hardly couple and in consequence the electron-nuclear coupling is suppressed by a factor proportional to Ω ∆ when compared to the coupling between states equally weighted superpositions |m s = 0 ± |m s = +1 which one would have obtained for resonant driving. Crucially, thanks to the small detuning on the double quantum transition we obtain |m s = −1 ± |m s = +1 which lead to a strong coupling between electron and nuclear spins if a Hartmann-Hahn resonance is realised (see r.h.s of Fig.  3). The latter is relatively robust thanks to the strong detuning on the single quantum transitions. A gap is created between the |D = 1 √ 2 (| + 1 + | − 1 ) and the |B = 1 √ 2 (| + 1 − | − 1 ) states. This gap is robust to changes in the one photon detuning (∆) and is also robust to small changes in the two photon detuning(δ). Due to the fact that the coupling to the nuclei is via the dressed states(|D , |B ) and not the bare states the coupling is not decreased due to the off resonant drivings.
So far we have ignored additional energy shifts on the double quantum transition δ(θ) which will lead to a loss of the Hartmann-Hahn resonance. In order to confer additional robustness to our scheme, rather than aiming to maintain a fixed Hartmann-Hahn resonance, we consider an adiabatic sweep of the external magnetic field on the double quantum transition where start and endpoint of the sweep are chosen such that intermittent Hartmann-Hahn resonances are guaranteed. This technique, the integrated solid effect, allows polarization transfer from a large fraction of the NV electron spins to their surrounding nuclear spins.
Therefore, the resonant double quantum transition that is mediated by far-detuned single quantum transitions combines robustness with strong electron-nuclear coupling. It is this idea and its combination with the integrated solid effect that we are going to develop and analyse in detail for realistic experimental parameters and various imperfections in this section.

Double quantum transition
For illustration of the basic idea, let us first consider a system that is composed of an NV center and a single 13 C nuclear spin in a nanodiamond of random orientation. In the laboratory frame the applied magnetic field defines the z-axis, the NV is placed at the origin of the coordinate system and the 13 C nuclear spin, placed near the NV center, is situated at position r. A microwave (MW) field of frequency ω M is applied as off-resonant drive of the spin transitions | − 1 ↔ |0 and |0 ↔ | + 1 while approximately satisfying a double resonance condition, that is 2ω M ∼ = E |+ −E |− . We chose circular polarization but linear polarization would suffice. The Hamiltonian of the whole system is then in which g = µ0 4π γeγn r 3 with r = | r| denoting the distance from the NV spin to the nuclear spin and e r = r/r. I is spin-1 2 vector operator of the nuclear spin with the gyromagnetic ratio γ n and Ω M = √ 2Ω is the Rabi frequency of the driving field. Assuming a point-dipole interaction and neglecting the contact term we obtain In the interaction picture with respect to H 0 = ω M S z , the Hamiltonian is given by in which A = g 1 + 3(e z r ) 2 h = A h is the hyperfine interaction tensor for the nuclear spin, with h determined by e r : h x = 3e x r e z r / 1 + 3(e z r ) 2 , h y = 3e y r e z r / 1 + 3(e z r ) 2 , and h z = (3(e z r ) 2 − 1)/ 1 + 3(e z r ) 2 . The detuning ∆ is given by which is a function of the NV orientation due to the second order energy correction δ(θ).
Since the nuclear coordinate system is arbitrary, we can redefine it by following the method in Ref. [24]. The z-axis is chosen to coincide with the external magnetic field, and we define the y-axis to be perpendicular to both magnetic field direction and h. In the new nuclear coordinate system, the Hamiltonian is given by where I z = I z ′ and a z ′ and a x ′ are the elements of the secular and pseudosecular hyperfine interactions, respectively.
In what follows we assume that (i) |D(θ)| ≫ ∆ which implies that the microwave driving field tends to be far detuned from the single quantum transitions | − 1 ↔ |0 and |0 ↔ | + 1 and (ii) |D(θ)| ≫ Ω such that the Rabi frequency is significantly smaller than the detuning (see Fig. 1b). This will allow us to remove level |0 adiabatically from the dynamics to achieve a simplified effective Hamiltonian eq. (16).
To this end, we consider the dominant parts of eq. (10) by neglecting the nuclear spins as well as ∆S z . In the basis {| + 1 , |0 , | − 1 } we therefore consider The eigenstates and eigenenergies of H N V are and with For |D(θ)| ≫ Ω we have two cases: D(θ) > 0 leads to In both cases, two of the eigenstates are approximately given by |± = 1 √ 2 (| − 1 ± | + 1 ) and form an effective two-level system while the state |0 does not participate in the dynamics because for |D(θ)| ≫ Ω it is far detuned. We will now focus attention on these two states |± and write the Hamiltonian eq. (10) in the subspace spanned by {|+ , |− } to find in which H + and H − corresponding to the cases D(θ) > 0 and D(θ) < 0, respectively. Here . Furthermore | ↓ and | ↓ denote the ground and excited states of the nuclear spin.
For the final step, we now restrict attention to the case where θ ∈ [0 • , 20 • ] in which case D(θ) > 0 and we consider Hamiltonian H + . In matrix notation, the electronic part of the Hamiltonian eq. (16) then takes the form The eigenstates and eigenenergies are in which arctan ζ = −∆ Ω ef f /2 , ∆ and Ω ef f are dependent on the angle θ between NV orientations and applied magnetic field. Defining Pauli operators σx, σỹ, σz in the eigenbasis of H ∆ we find in which the angle ϕ is determined by The Hamiltonian H + can be simplified further under two main assumptions, namely (i) a x ′ ≪ γ n B, that is for weakly coupled nuclear spins and (ii) | γnB 2 − ω ef f | ≪ a x ′ that is we satisfy a Hartmann-Hahn condition. Then the polarization transfer dynamics is described by where 2a z ′ cos ϕσzI z ′ does not affect the flip-flop interaction. Let us briefly highlight the key difference of this offresonant driving in comparison to previously introduced polarization schemes, namely its relatively high robustness with respect to the angle θ between the external magnetic field and the axis of the NV center. If we were to follow the scheme of Ref. [24], even for θ = 1 • , the detuning of the MW frequency from the NV electronic resonance exceeds (2π)1 MHz, which prevents efficient polarization transfer (for θ = 20 • we find (2π)500 MHz detuning). In our scheme, for as long as D(θ) ≫ Ω, the energy difference between the states |− and |+ scales as 2Ω 2 /D(θ) and so does the effective Rabi fre- MHz, the effective Rabi frequency is roughly within [(2π)3M Hz, (2π)3.6M Hz], which narrows the range of the detuning of the resonant frequency.

Integrated solid effect
In the discussion so far, we have assumed that the electron spin of the NV center is held continuously at a Hartmann-Hahn resonance with a specific target nuclear spin to achieve polarization transfer. In practice, however, this will only capture a small fraction of the NV centers and nuclear spins as the resonance condition will depend on ∆ = γ e B + δ(θ) − ω M which in turn is a function of the angle θ via δ(θ) (Fig. 2b). In the range Hz], as shown in Fig. 2b which can lead to a violation of the Hartmann-Hahn resonance for a large fraction of the NV centers. In order to achieve polarization transfer for a larger fraction of the NV centers, we will use our ability to control the strength of the externally applied magnetic field or the frequency of the applied microwave field to implement a sweep of ∆.
In order for us to understand qualitatively the conditions that such a sweep has to satisfy, we briefly examine the energy level diagram of a system consisting of one electron and one nuclear spin as described by eq. (20). This reveals that there are two resonance points A 1 and A 2 for ∆ = ±∆ HH owing to the symmetry of the Hamiltonian. In a fully adiabatic sweep, polarization would exchange twice and result in a vanishing net polarization transfer. Hence the sweep rate needs to be chosen such that around the A i the system becomes non-adiabatic for the two branches including |χ + , ↓ and |χ − , ↑ as this will lead to an approximately 50% probability for polarization transfer. At the same time the sweep has to remain adiabatic with respect to the energy gap for transitions between the {|χ + , ↓ , |χ − , ↑ } and the {|χ + , ↑ , |χ − , ↓ } manifold as transitions between these two manifolds, induced by non-adiabaticity or in fact dephasing events, would lead to depolarization. These requirements set some limitations on the sweep rate that are, fortunately, not too stringent.
The quasi-adiabatic sweep described above realises an instance of the so-called integrated solid effect (ISE) [34] and can achieve polarization transfer between a large fraction of the NV spins and nuclear spins. Therefore, magnetic control of the nuclear spin, and efficient polarization exchange, become possible even for relatively large angles between NV-center axis and external magnetic field. In the following we will examine this idea in more detail.
Let us now examine two regimes of the quasi-adiabatic transfer described above, namely (i) those parts of the sweep parameters that are far from the Hartmann-Hahn resonance points A 1 and A 2 which will lead to an upper bound on the sweep rate and (ii) the behaviour around the Hartmann-Hahn resonances which will provide lower bounds on the sweep rate.
For regime (i), as the coupling between electron and nuclear spin is negligible, it suffices to examine the electronic part of eq. (16), that is in which ∆(t) = γ e B − δ(θ) − ω M . For simplicity we assume the detuning to vary at a constant rate v > 0, i.e. ∆(t) = ∆(t i ) + vt, where ∆(t i ) is the initial detuning at the start of sweep. According to the quantum adiabatic condition we require, at all times, that where E 1m (t) and E 1n (t) are the instantaneous eigenvalues, |E 1m (t) and |E 1m (t) are the eigenstates of effective Hamiltonian H ∆ , respectively. Then the condition for the sweep to be adiabatic is given by Adiabatic evolution implies that the initial eigenstate of this Hamiltonian H ∆ will remain close to the instantaneous eigenstate at any time during the sweep. In particular, during approach to the Hartmann-Hahn resonances and indeed during the entire sweep there will be no transitions of the type |χ − , ↓ to |χ + , ↓ which, as is easily seen by examination of Fig. 4, would have a depolarizing effect in the subsequent sweep.
To gain a feeling for the upper limit on the sweep rates that this implies for typical experimental parameters suppose as in the previous section that Ω = (2π)65 MHz so that in the range θ ∈ [0 • , 20 • ] we find Ω ef f in the range [(2π)3M Hz, (2π)3.6M Hz]. This in turn implies that v < (2π)10 MHz/µs guarantees that the adiabatic condition is satisfied. As shown in Fig. 2b in this range we have δ(θ) ∈ [0, (2π)45M Hz]. Suppose a sweep rate v = (2π)6 MHz/µs, then no more than 10 µs are required to cover all NV centers within θ ∈ [0 • , 20 • ]. Experimentally such sweep rates are easily obtained using arbitrary wave form generators.
Let us now proceed to discuss case (ii), namely the sweep around the Hartmann-Hahn resonances which will provide lower bounds in the sweep rate. Near the resonant points, the effective Hamiltonian eq. (20) in the subspace spanned by {|χ + , ↓ , |χ − , ↑ } is well approximated by the matrix form (24) Here l = 1, 2 corresponds to points A 1 and A 2 , and near these two points we make use of the expansion According to Landau-Zener (LZ) theory [35], transitions are possible between two approaching levels as a control parameter is swept across the point of minimum energy splitting. The asymptotic probability of a LZ-tunneling transition is given by where, by virtue of the Hartmann-Hahn condition γ n B = 4∆ 2 + Ω 2 ef f at resonant pints A 1 and A 2 , we find

This enters the unitary transformation
that is taking place when traversing the avoided crossing. Hereφ s = ϕ s − π/2 is due to the Stokes phase ϕ s . The full sweep takes the system from ∆ ≪ 0 to ∆ ≫ 0 across both A 1 and A 2 . At each resonance point polarization transfer is possible between the electron and the nuclear spins [36]. Consider a sweep starting at ∆(0) ≪ −Ω ef f for which the electron spin is initialized in state |χ − and the adiabatic condition eq. (23) is satisfied. If the nuclear spin is initially found in state | ↓ , the electron-nuclear system will start and finish the sweep in state |χ − , ↓ , i.e. the nuclear polarization is unchanged. If on the other hand the nuclear spin is initially found in state | ↑ , the evolution of the system will start in |χ − , ↑ . In this case the probability for experiencing a polarization transfer at the end of the full sweep, that is finding the system in state |χ + , ↓ , is given by Here Φ St is determined by the initial phase of the system and the phases acquired during the adiabatic evolution and the non-adiabatic transitions. We do not discuss these dynamical phase shifts in detail as they depend on details of the evolution and orientations that vary from NV center to NV center and it is therefore reasonable to assume that Φ St is random [37,38]. Hence, when averaged over many sweeps, the average polarization transfer probability is given by as shown in Fig. 5. The value of P LZ and hence the maximum P = 2P LZ (1 − P LZ ) is determined by the hyperfine coupling strength a x ′ , the rate of the ISE sweep v, and to a lesser extent on the strength of the applied microwave field. It is straightforward to see that P max and hence P vanish for both v ≫ 0 and v → 0, corresponding to the diabatic and adiabatic sweep. The maximum value is achieved for P LZ = 1/2 and the parameters for achieving this maximum are quite flexible, see Fig. 6. P max remains high across a wide range of sweep rates as shown in Fig. 6a and b. Note furthermore, that for a given sweep rate we obtain efficient polarization transfer to the nuclear spins in a broad range of different microwave Rabi frequencies, see Fig. 6. For example, for Ω ef f = (2π)3.6 MHz, a speed v = (2π)6 MHz/µs can obtain P max > 0.2 in the coupling range a x ′ ∈ [(2π)0.18, (2π)0.9]MHz. Furthermore, a slower sweep gives a better polarization transfer for a smaller effective Rabi frequency and a weaker coupling strength between the electron spin and the nuclear spin. In order to polarize the nuclear spins with larger distance from the NV spin, one can slow down the ISE sweep, or increase the Rabi frequency of the microwave field, as shown in Fig. 6c.

C. Preparation of the initial state
So far we have assumed the preparation of a specific initial state in the polarization sequence to be achieved with perfect fidelity. However, due to the random orientations of the NV centers, optical pumping leads to the initialization of the electron spin of the NV center in a wide range of different states. Furthermore, due to the broad distribution of zero-field splittings of the NV center in randomly oriented nanodiamonds, any applied microwave field will experience an uncertain and potentially large detuning from the electronic resonance. This presents challenges to the preparation of the electron spin of the NV center in the initial state that is required for our polarization scheme. We address these challenges with an adiabatic sweep of the frequency of the microwave field which we will show to deliver robust and rapid state preparation.
Let us consider as an example the initialization of the NV center in the state | − 1 . For moderate θ optical pumping will lead to a state that possesses a significant overlap with |0 . In order to map this state to the target | − 1 , we make use of the fact that in the relatively high magnetic fields that we are considering (e.g. B ∼ = 0.36T) the energy gap on the | − 1 ↔ |0 transition is very different from the energy gap of the |0 ↔ | + 1 transition. Hence it is possible to use a circular polarized microwave driving field in which Ω M− = √ 2Ω − is the Rabi frequencies of the MW field, with frequency ω − to drive dominantly the | − 1 ↔ |0 transition. The resulting effective two-level system consisting of the states |0 and | − 1 can be described in the matrix representation by in which ∆ MW (t) = γ e B + δ(θ) + D(θ) − ω − (t). Now we assume that ω − (t) experiences a constant rate sweep such that ∆ MW (t) = ∆ MW (t i ) +∆ − (t − t i ). We set t i = 0 for simplicity. The condition for the sweep to be adiabatic is then Adiabatic evolution implies that the initial eigenstate of this Hamiltonian H 0,−1 will remain close to the instantaneous eigenstate at any time during the sweep. The eigenstates and eigenenergies are . If at the start of the sweep ∆ W M (0) < 0 and |∆ W M (0)| ≫ 2Ω − and the electron spin is prepared in the state |0 , the state will evolve along the path |E + (t) to end in the state | − 1 . As .04 µs to achieve adiabaticity. Hence very high fidelity preparation of the state | − 1 can be achieved within 0.4 µs which is faster than the optical polarization cycle.
So far we have assumed that the optical initial state |0 is well prepared. However, for the relatively high magnetic fields that we are considering, the optical initial state is given by |0 θ = cos θ|0 + sin θ Here S is the surface area of the two sectors within θ deviation and dS is the corresponding area element. The reachable polarization for nearby nuclear spins and the average polarizations are P N1 = 0.91 for θ ∈ [0 • , 20 • ].

D. Polarizing cycles and effective ranges
We can now summarize the polarizing cycles as follows (see also the black box of the flow diagram  advantages: (1) the impact of the random orientations is minimized, and NV spins in a wide range of orientations are still accessible for polarization. (2) Using this method, the coupling strength between the NV and nuclear spins is doubled because it involves the double quantum transitions between the states | − 1 and | + 1 .
In Fig. 8 we simulate many cycles of this polarization protocol for a system of one NV center and a single nuclear spin under the assumption that the NV center is initialized to the state | − 1 as an example. The initial density matrix of the unpolarized nuclear spin is ρ 0 = I/2, with I being the unit matrix. For the state ρ k of the nuclear spin after the k-th iteration of the polarization protocol we have the following relation where U t = e − iHtransdt is time evolution operator, H trans is the Hamiltonian eq. (20), we denote by ∆t the sweep time and by tr e the trace over the electron spin. The polarizing P is defined as P = I z ′ / I z ′ 0 with I z ′ denoting the expectation value of the nuclear spin and I z ′ 0 the expectation value of the completely polarized state.
Finally, as we have discussed in the previous sections, the ISE requires an adiabatic slow passage. For θ ∈ [0 • , 20 • ] we find δ(θ) ∈ [0, (2π)45M Hz] and in order to include the entire range of detunings in the sweep we can estimate the required polarizing time. At a sweep rate of v = (2π)6 MHz/µs we find ∆t ∼ 10 µs. In Fig.  8a, assuming the perfect initialization of the NV spin in the state | − 1 , we take one nuclear spin with the typical coupling a x ′ = (2π)0.6 MHz as an example to demonstrate the polarization transfer (the nuclear spin is about 0.5 nm from the NV spin and a z ′ = (2π)0.64 MHz). Here the polarization transfer after the first cycle, P 1 , does not quite achieve the maximum possible value in Fig. 6 due to the existence of the phase shift Φ St , more precisely P 1 = 0.58 < P max = 0.7 when Ω ef f = (2π)3 MHz, see both Fig. 6b and 8a. Simulations for different hyperfine couplings are shown in Fig. 8b. Again the sweep rate v = (2π)6 MHz/µs and polarization is transferred to the nuclear spins with different coupling to the NV spin, which agree with Fig. 6. Additionally, slowing down the ISE sweep, as shown in Fig. 8c for a sweep rate v = (2π)0.8 MHz/µs we can polarize more distant nuclear spins with a coupling strength a x ′ = (2π)0.1 MHz, which corresponds to a distance of about 1 nm from the NV center. This coupling is approaching the limitation of our polarization scheme, since the rotating-frame spinlattice relaxation time of NV spin in nanodiamonds is estimated to be about 100 µs which places a lower limit of the duration of the full sweep [39].

III. THE BENEFIT OF ADIABATIC ROTATIONS
The NV polarization in our scheme is limited by the adiabatic sweep range and efficiency of the initialization of NV spins, enabling the polarization up to a maximum angle between NV center and external magnetic field. As discussed above, our scheme certainly performs well for NV centers whose orientation forms an angle of less than 20 degrees from the externally applied magnetic field (and may potentially work for somewhat larger angles). It is however desirable to extend the polarization to a larger fraction of the nanodiamond ensemble, ideally covering the entire solid angle. One method for achieving this applies an adiabatic change of the orientation of the magnetic field relative to the NV center, either by rotation of the magnetic field or by rotation of the nanodiamond. For example, following the magnetic field rotation to the Z ′ -direction, another set of NV centers with orientations in a solid angle (shown in red in Fig. 9) around the new magnetic field direction will participate in the polarization dynamics while the NV centers in the original orientations (the blue cone in Fig. 9) will be inactive. Hence previously polarized nuclear spins will not affected by the polarization sequences, but these nuclear spins may use this "idle" time to spread their polariza-

Step2
Another rotation  9. a, Diagram of adiabatic rotating direction of the magnetic field. Blue spherical sectors present θ degrees deviation tolerance for Z-direction of the magnetic field. Magnetic field is rotated by angle ϑ to get red spherical sectors with the halfangle of the cone angle θ ′ to be involved. b, Flow diagrams of polarizing steps of our scheme related to the adiabatic rotations of the magnetic field. The small blue and red arrows are the nuclear spins nearby the different NV spins in different spherical sectors. Four steps are given for our polarizing scheme. We use 2D plotting for simplicity, the sphere is presented by a circle and the areas formed by the dashed lines denote the corresponding spherical sectors. It is shown by the arrows that all the nuclear spins follows the rotated magnetic fields. tion by spin diffusion across larger volumes until further rotations subject them to another polarization cycle.
More specifically, let us consider the rotation of the magnetic field as an example. The Hamiltonian that is describing the rotation of the magnetic field from the Zdirection to the Z ′ -direction is given by where ϑ(t) =θt is the angle between the rotated magnetic field and its initial direction and γ N B Z = (2π)4 MHz.
Similar to the discussion in the initialization scheme, the well-known adiabatic condition requiresθ ≪ 2γ N B Z . For a rotation by ϑ = 180 • this leads to the condition t r ≫ 0.078 µs for the minimal rotation time t r . This implies that t r ∼ = 0.78 µs will ensure adiabaticity and minimal perturbation of nuclear spin polarization for a rotation to an arbitrary orientation. This rotation time is much shorter than even a single polarization cycle. Hence, after running a few polarization cycles that achieve a high net polarization of nuclear spins for NV centers oriented along the Z direction, we can adiabatically rotate the magnetic field to another direction. The polarization of the nuclear spins will then be achieved for this new direction, as shown in Fig. 9. Note though that the rapid rotation of a strong magnetic field is challenging. Hence one needs to resort either to a mechanical rotation of the sample as a whole or, as explained in the next section, make use of the natural Brownian rotation of nanodiamonds in solution.
In addition to the rotation of the externally applied magnetic field for nanodiamond powder, another option consists of the use of random Brownian rotations of nanodiamonds in a solution. If the Brownian rotation is sufficiently slow to satisfy our adiabatic rotation condition, the polarized nuclear spins will maintain their polarization in the direction of the externally applied magnetic field. The time scale of Brownian rotations is determined by [40] where k is the Boltzmann constant, T is the temperature, V H is the hydrodynamic volume of the particle and η the viscosity of the surrounding carrier liquid. Note that the hydrodynamic volume is an effective volume that includes both the true particle volume and the volume of a fluid that is displaced when the particle rotates due to particle-fluid interaction. A typical average hydrodynamic diameter for nanodiamonds of 30 nm in water gives a Brownian relaxation time of τ B ≃ 9 µs, which is comparable with our polarizing cycle time. One can increase the Brownian relaxation time by using nanodiamonds with a larger diameter, a more viscous fluid or lower temperatures. Additionally, surface coating can increase the average hydrodynamic diameter, i.e., after adsorption of ferritin the diameter of nanodiamond was determined to be 85 nm [41].
Here we assume that the hydrodynamic diameter of nanodiamond is 85 nm, and the Brownian relaxation time is τ B ≃ 205 µs. On average after a time τ B , the nanodi-amonds will have rotated to another random direction, and different NV centers will now participate in the polarization dynamics of nearby nuclear spins while other NV centers will be inactive as they are effectively decoupled from their neighboring nuclear spin due to the absence of a Hartmann-Hahn condition. When an NV center "leaves" the coherent exchange range via a random rotation, the polarization of the polarized nuclear spins will still be transferred to the other nuclear spins by spin diffusion. Considering maximum of 20 degrees deviation, 6% percents of the NV spins are involved at each given time, and each NV will be in this range after 16 random rotations on average.

IV. INTERNUCLEAR INTERACTION AND SPIN DIFFUSION
So far, we have considered polarization of an individual nuclear spin and have therefore ignored the effects of inter-nuclear coupling on both the polarization cycle and on the nuclear spin polarization diffusion towards more distant nuclear spins that are not directly interacting with the NV center. This section discusses these effects of nuclear dipolar interactions. Considering our off-resonant driving in the dressed state basis and the suitable rotating frame, the effective Hamiltonian of the whole system consisting of an NV spin and its nearby nuclear spins can be written as Here H N V denotes the NV center effective energy Hamiltonian, H hf the effective electron-nuclear hyperfine interaction, H d−d the dipole-dipole coupling among nuclear spins and H N u is the magnetic field splitting of nuclear spins with a x ′ i and a z ′ i denoting the hyperfine coupling strength between the ith nuclear spin and the electron spin. d ij = µ0 4π γnγn r 3 ij is the coupling strength between the i and j nuclear spins. with r ij = | r ij | denoting the distance between the two nuclear spin and e rij = r ij /r ij . The i and j indices are summed over all nuclei. In order to estimate the effect of internuclear interactions on the polarization transfer from the NV center to the nuclei in the multiple nuclear spins case we consider 5 nuclear spins and one electron spin, including hyperfine and nuclear dipolar couplings. After each polarization cycle, the electron spin is re-initialized in the state | − 1 . The concatenated evolution of the nuclear spin density matrix is determined by an equation analogous to eq. (33). The initial density matrix of the five unpolarized nuclear spins can be written as ρ 0 = I/2 5 , with I being a unit matrix of dimension 2 5 , and the time evolution is given by the operator U t = e −i Htotdt , with H tot denoting the total Hamiltonian eq. (35). The polarization is defined as in section II, P k = I k z ′ / I k z ′ 0 in which the subscript k presents different nuclear spins and P = 1 5 5 k P k . We assume in these simulations only nearest neighbour inter-nuclear spin coupling with rates The simulation results are presented in Fig. 10a and b. We consider the case that the NV is strongly coupled to all five 13 32 MHz, and a x ′ 5 = (2π)0.2 MHz. The polarization is built up for all the five nuclear spins with different coupling to the NV spin, which matches our theory. In this case, the coupling between the nuclear spins is too small to affect the evolution of the polarization dynamical built-up (Fig. 10a). Due to small interaction among the nuclear spins, there is almost no difference between polarization in the presence or absence of the interactions among the nuclear spins, which means our scheme works in the frozen core. Now we would like to exemplify the benefits of internuclear interactions which leads to nuclear spin diffusion and a considerable extension of the range of polarization. Suppose a typical configuration of nuclear spins distributed around the NV. To estimate the polarization efficiencies in a large system, we adopted the simple spin temperature approximation, namely neglecting the nuclear spin coherence [42], which provides a good estimation of polarization transfer efficiency when the sweep time step is sufficiently small. This approximation yields independent rate equations for each individual nuclear spin and should give a conservative estimate for the achieved polarization. For the ∆t sweep times described above, this approximation serves as a lower bound, as the achieved polarization in one cycle with our scheme is higher than the polarization achieved during the same time with small sweep steps.
For the diffusion step, the interaction between the nuclear spins is taken into account, where we use the Gaussian approximation for the nuclear spins. As is typical for paramagnetic centers, the NV center strongly affects the spin diffusion in nearby nuclear spins (the "frozen core") due to the energy mismatch caused by the S z I z hyperfine term. However, when the NV center is in the |m s = 0 state, this energy mismatch is suppressed and diffusion to external nuclear spins is allowed. To give a conservative estimate, we assume that no diffusion in the frozen core takes place when the NV center is in the |m s = ±1 states. For this purpose, we simulate the NV center as a classical spin with the probability of being in each eigenstate given by the optical initialization corresponding to the NV orientation.
As the nanodiamonds are in a solution, we assume MHz: a, The blue and red triangles denote the presence and absence of the interaction among the nuclear spins. Internuclear interactions do not lead to significant changes in the polarization dynamics because the dipole-dipole interaction of nuclear spins is much smaller than the electron-nuclear coupling. b, The polarization dynamics of the individual nuclear spins in the set of five spins represented by different markers within different colors. c, Numerical simulation of the polarization buildup in a large number of polarization cycles in a system consisting of an NV spin and around 548 13 C nuclear spins in a volume of 50000 lattice sites by using Brownian rotation. The internuclear dynamics is treated within the bosonic approximation and we simulate a polarization buildup time of 2 second. The system benefits from polarization diffusion between nuclear spins resulting in a high level of polarization. that the differences in polarization transfer for the different orientations of the NV spins within the 20-degree deviation range can be neglected. Consider a random configuration of nuclear spins, with 548 nuclear spins in the vicinity of an NV center (lattice of 50,000 nuclei per NV center) whose orientation is within the 20-degree deviation range. The net polarization is built up by using Brownian rotations as shown in Fig. 10c. There are two steps in the simulation. First, we assume that the orientation of the NV center satisfies θ ∈ [0 • , 20 • ] and we consider Ω ef f = (2π)2.3 MHz, sweep rate of v = (2π)0.8 MHz/µs and ∆t = 70 µs. The polarization transfer from the NV center to the neighboring nuclear spins is achieved by using our combination of offresonantly driven double quantum transition and the ISE technique. The polarizing cycles are repeated during the Brownian relaxation time τ B = 205 µs. Secondly, after the Brownian relaxation time, the nanodiamond is randomly rotated to another direction. As we discussed in the Sec. III, it takes roughly 3.5 ms for the nanodiamond to rotate back to our maximal deviation range for polarization, and nuclear spin diffusion dominates during this time period (taking into account the NV effect on diffusion in the frozen core). In this step more distant nuclear spins can become polarized by diffusion. These two steps occur sequentially over a 2 second time range. In line with previous sections, the polarization is defined i , M is the number of the involved nuclear spins. Interestingly, the slope of polarization is close to linear. This demonstrates that in the long diffusion time, where the nanodiamond is not involved in the polarization cycles, we do not see a bottleneck caused by slow diffusion and significant polarization is transferred from the NV spin to the nuclear spins within each step 1. The final net polarization reaches 0.2, equivalent to about 110 13 C fully polarized spins are polarized within 2 second in our scheme.

V. A LARGE ANGLE POLARIZATION SCHEME
So far we have concentrated so far on a range of angles θ ∈ [0 • , 20 • ] between NV center and the external magnetic field. Note however, that this is not the only range in which efficient polarization transfer can be achieved. Indeed, we have another interesting range of the NV orientations, concentrated around the direction perpendicular to the magnetic field, i.e., θ ∈ [70 • , 110 • ], which involves 34% the NV spins. Here the condition D(θ) < 0 leads to the Hamiltonian H − in eq. (16). Following steps that are analogous to those leading up to Hamiltonian eq. (20) we find for the subspace {|χ − , ↓ , |χ + , ↑ } Suppose now that the NV center is initialized in state | − 1 by using the off-resonant driving. Then we make use of the ISE technique to sweep adiabatically across the two possible Hartmann-Hahn resonance points of the dressed NV spin and nuclear spin pair. As a result the the population is transferred between |χ − , ↓ and |χ + , ↑ , with the other states being unaffected. As before this allows the nuclear spins to be polarized but now in a direction opposite to the case θ ∈ [0 • , 20 • ]. In the following we discuss the details of this case.
A. Initial polarization of the NV spins We begin with a discussion of the initialization of the NV center spins. As explained in previous sections optical pumping of the NV spins results in the |m s = 0 state in the NV frame, that is where the states |0 , | ± 1 are defined in the laboratory frame. For θ ∈ [70 • , 110 • ] the population in state |0 is very small and the initial state is well approximated by where we ignore any dynamical and geometric phase that has been accumulated in the sweep. It is now crucial to note that the state |0 does not contribute to the polarization dynamics as it does not take part in the far detuned dynamics that is induced by the applied microwave fields. Hence the relevant quantity is the polarization in the subspace spanned by the states {| + 1 , | − 1 }. Normalized by the population that is found in this subspace we find where N r = 1/( sin 2 θ 2 +cos 2 θ) is the normalized coefficient and P r N2 ∼ 0.85.

B. The sweep range for ISE
Owing to the different behaviour of the energy shifts D(θ) and δ(θ) we need to change the sweep range in the ISE. In order to achieve optimum polarization transfer, we choose Ω = (2π)40.5 MHz, which results in a range of effective Rabi frequency Ω ef f ∈ [(2π)2.3M Hz, (2π)3.6M Hz].
The adiabatic slow passage needs to cross the range ∆ ∈ [−(2π)21M Hz, (2π)21M Hz]. For a sweep rate of v = (2π)6 MHz/µs we estimate the required polarizing time as ∆t > 7 µs. Apart from these parameter changes, the principles underlying our discussion of the polarization transfer in Section II are still valid.
If we assume that the NV spin is initial polarized perfectly, we have P max > 0.2 in a coupling range a x ′ ∈ [0.4, 0.9] for |Ω ef f | = (2π)2.27 MHz in Fig. 6c, compared to a x ′ ∈ [0. 18, 0.9] for |Ω ef f | = (2π)3.6 MHz in Fig. 6a. We can say that our off-resonant driving and ISE technique allow polarization transfer for all the NV orientations within θ ∈ [70 • , 110 • ]. Consider the average polarization of the NV spins in this case is about 0.5 and twice effective Rabi frequency range here, applying a singe sweep speed becomes not very efficient for all the NV orientations, especially for the nuclear spins with weak couplings to the NV spin. As we discussed, we have another choice, slow down the ISE sweep can involve the nuclear spins with weaker coupling to the NV spin, i.e., as shown in Fig. 8c, we can use another a speed rate v = (2π)0.8 MHz/µs (∆t ∼ 50 µs) to make it possible to polarized the 13 C spins with coupling strength a x ′ = (2π)0.1 MHz.
In conclusion, off-resonant driving and the ISE technique allow for polarization transfer between the NV spin and its surrounding nuclear spins, when the orientation of the NV spin matches θ ∈ [70 • , 110 • ] but we need to sacrifice half the population to provide a high initial polarization of the dressed NV spins. On the other hand the fact that around 34% of the NV centers participate in the polarization dynamics even without magnetic field rotation is an attractive feature.

VI. DEPOLARIZING EFFECT
Our approach to increase the fraction of NV center that participate in the polarization sequences by means of adiabatic rotations may potentially have an undesired side effect as nuclear spins that have been polarized for one magnetic field orientation may be depolarized again when transferred to a different magnetic field orientation. Consider for example the case shown in Fig. 9 where, after polarization transfer is achieved for the blue spherical sectors, the magnetic field is rotated from Zdirection to different direction Z ′ . If the NV spins in the blue sectors have not been polarized in advance and our combination of off-resonant driving and ISE achieves a Hartmann-Hahn condition at some instant, then the depolarized state of the NV center will be transferred to nuclear spins, whose polarization reduces as a result. In the protocol for the case θ ∈ [0 • , 20 • ] this depolarization is negligible due to the rapid increase of the detuning δ(θ) outside of this range. Only NV centers in the range θ ∈ [20 • , 25 • ] may still experience undesired resonances. As these NV centers are still well initialized by means of optical pumping depolarization is negligible.
The situation is different and indeed more complex for the second protocol that covers the range θ ∈ [70 • , 90 • ]. NV centers with orientations in the range θ ∈ [35 • , 43 • ] are difficult to be polarized to the required state. Unfortunately, they have almost the same energy distributions |D(θ)| and δ(θ) as the NV centers in the range We consider a simple system consisting of one NV and one nuclear spin as an example for studying the depolarizing effect of this case. The depolarizing cycle time is as the same as our polarizing cycle time, since every initialization process is implemented on all the NV spins. Consider a nuclear spin that is initially polarized and in contact with an unpolarized electron spin ρ e0 = I/2. The evolution of the nuclear spin density matrix is then given by where matrix is given by eq. (20) with the choice Ω ef f = (2π)3.5 MHz and δ ′ denoting that a magnetic field in Z ′direction is being considered. The polarization P of the nuclear spin is defined as in Sec. II. As shown in Fig.  11, the nuclear spin is depolarized very rapidly. The depolarization range θ ′ ∈ [35 • , 43 • ] includes 9% of the NV orientations while the range in which the polarization is active includes 34% of the NV centers. Hence, although adiabatic rotation of the magnetic field brings an unexpected depolarizing effect, it is nevertheless quite attractive as the fraction of NV spins that are contributing to the polarization transfer even without magnetic field rotation strongly outweighs the small fraction of 9% of the NV centers that suffer depolarization.

VII. DISCUSSION
To summarize, our polarization scheme for ensembles of nanodiamonds (see Fig. 7 and Fig. 9): Step 1: A moderate to strong magnetic field is applied along Z-direction (of order B = 0.36T ). A polarization cycle consists of optically pumping and initialization of the NV center spins, and the subsequent application of off-resonant microwave driving and the integrated solid effect to realise a robust adiabatic sweep across Hartmann-Hahn resonance points. This achieves the coherent polarization transfer from the NV center spins to the nuclear spins with a range of orientations of the NV center relative to the magnetic field. This polarization cycle is repeated several times.
Step 2: Adiabatic rotations (of the magnetic field or of the nanodiamonds) change the magnetic field orientation relative to the NV centers to a new Z ′ -direction, while the nuclear spins follow the direction of the magnetic field.
Apart from control imperfections, the achievable 13 Cbath spin polarization will be limited by the spin-lattice relaxation time T 1 of the nuclear spins. In recent experiments, by using electron paramagnetic resonance (EPR), solid-state NMR, and DNP techniques, a variety of nanodiamond samples, of varying manufacturing processes and particle sizes, were examined [18]. Several types of nanodiamonds were identified that exhibit very long nuclear spin lifetimes of several minutes. Additionally, this polarization transfer rests on NV spins having a sufficiently long rotating-frame spin-lattice relaxation time T 1ρ in nanodiamonds during the adiabatic passages that are required for the implementation of ISE. Using a strong MW field driving, T 1ρ can be estimate as 100 µs [39,43], which is much longer than our sweep time in one polarization cycle. Assuming a polarization cycle duration of ∆t =10 µs, when rotating the magnetic field, 17 rotations of the magnetic field are required to involve all the NV spins. For a nuclear spin lifetime of more than 2 minutes, this would allow the execution of around 10 5 polarization cycles for each NV spin involved before the nuclear spins start to relax. For nanodiamonds in a solution, each nanodiamond is involved in the polarization cycles roughly once every 17 random Brownian rotations. Assuming τ B ≃ 205 µs, with a pure nuclear diffusion period of roughly 3ms after every Brownian relaxation time τ B , 2 minutes lifetime of the nuclear spin would allow in excess of 10 5 polarization cycles for every NV center.
We can now obtain a rough estimate of the total amount of the net polarization that could be achieved under realistic experimental conditions. Consider a 1 mm 3 nanodiamonds powder, dissolved in a solution. The nanodiamonds are assumed to have a diameter of 85 nm, with an NV center concentration of C = 2 × 10 18 cm −3 (see [44] and references therein for achievable NV concentrations). In the limit of a uniform spatial distribu-tion, this gives about 642 NV spins per nanodiamond, with the total amount of NV spins given by 8.35 × 10 15 NV spins. Consider natural abundance of the 13 C spins (1.1%), the total nuclear spins in this powder of nanodiamond is approximately 8.18 × 10 18 [45]. Note that 13 C enriched nanodiamonds will enable polarization of more nuclear spins with each NV center and hence a larger overall level of polarization. Taking account of the high degree of polarization, in excess of 0.3 as demonstrated in Sec. IV, we find that a total amount of polarization equivalent to 1.6×10 18 nuclear spins should be achievable in such a sample within 2 seconds.

VIII. CONCLUSION
In conclusion, in the high magnetic field limit, we propose a scheme achieving macroscopic levels of 13 C nuclear spin polarization in randomly oriented ensembles of nanodiamonds, realised as powder or solutions. To address the lack of a common quantization axis for the NV centers, an off-resonant microwave drive realises a resonant double quantum transition and this together with the integrated solid effect enable microwave coupling and control of NV spins whose orientation deviates from the external magnetic field by less than 20 degrees. Matching the effective Rabi frequency of the off-resonant double quantum transition of the NV centers to the nuclear spin Larmor frequency, enables near resonant coupling between NV electron spins and the nuclear spins, and the transfer of polarization from the initialized NV spins to the neighbouring 13 C nuclear spins. Additionally, the effect of nuclear dipole-dipole interactions allows the weakly coupled nuclear spins to be polarized by spin diffusion. Adiabatic rotation (of the magnetic field or sample) can then extend the polarizing scheme to more NV spins, achieving a net polarization of all nuclear spins via our scheme. By using our polarization scheme, roomtemperature, optical based polarization of the nuclear spins is made possible.
These results introduce several exciting opportunities for the application hyperpolarized nanodiamonds, especially in the biomedical sciences. Diamond nanoparticles are biocompatible, exhibit no in vivo toxicity [46], and can be attached to a wide range of specific proteins, peptides or antibodies [17] while maintaining long T1 times. Thus, hyperpolarized nanodiamonds present an exciting platform as MRI probes for molecular imaging.

APPENDIX: EFFECTIVE HAMILTONIAN OF AN NV CENTER AT THE HIGH MAGNETIC FIELD LIMIT
In our work we assume that in the laboratory frame of reference a strong magnetic field is applied along the z-direction, γ e B z ≫ D. In the following we derive the representation of the NV center Hamiltonian eq. 3 in the FIG. A1. Two related rotations of the coordinate system are used to map from the natural orientation z θ -axis of the NV spin to z-axis which is defined by the high magnetic field. laboratory frame of coordinates. To this end we need to transform the zero-field splitting tensor, which is diagonal in the natural coordinate system defined by the NV center axes, to the laboratory frame of reference. We assume that the angle between the magnetic field, i.e. the z-direction, and the NV-axis, the z θ -direction is given by θ and that the angle between the laboratory frame ydirection and the NV center y θ directions is given by φ in the x-y plane, as shown in Fig. A1. The spin operators in the NV system of coordinates, S x θ , S y θ and S z θ , are then related to the spin operators S x , S y and S z in the laboratory coordinates by S x θ = sin θS z + cos θ(cos φS x − sin φS y ), S y θ = cos φS y + sin φS x , S z θ = cos θS z − sin θ(cos φS x − sin φS y ).

The Hamiltonian of the electron spin
can then be rewritten in the basis defined by the eigenstates of the S z operator of the electron spin, {| + 1 , |0 , | − 1 } as There exist two uncontrollable components, the angles θ and φ as well as the local strain E, which modifies the spin level structure.
As we know that a high magnetic field suppresses the effect of the off-diagonal terms in H ′′ N V . Assume that the Hamiltonian is of the form H ′′ N V = H ′′ N V 0 + ǫV , where H ′′ N V 0 is the diagonal part of the Hamiltonian and ǫV the off-diagonal part. In our case the magnetic field is sufficiently large for us to assume that the off-diagonal part is a weak perturbation. By using the Schrieffer-Wolff transformation in condensed matter, the second order corrections due to the off-diagonal terms ǫV can be obtained as in which H ′′ M is defined as the corrected Hamiltonian. By simple calculation, we find After shifting the zero of energy, the effective Hamiltonian in the laboratory frame is given by Therefore a strong magnetic field suppresses the first order effect of the off-diagonal terms in H ′′ N V but results in a second order modification of the diagonal elements δ(θ). This is the Hamitonian as in eq. (3).
In order to estimate the validity of this approximation in the high magnetic field limit we have prepared the system in state |m s = 0 and observed the subsequent time evolution of its population under Hamiltonian H ′′ N in Fig.  A2 (see caption for simulation parameters). It is evident, that the population of the initial state m s = 0 remains essentially constant which implies in turn that the effect of the off-diagonal elements in Hamiltonian H ′′ N V are indeed negligible for the moderate to high magnetic field case that is relevant to our work.