Method for the hyperpolarisation of nuclear spins

Abstract

A method of hyperpolarisation of nuclear spins in one or more particle(s) moving relatively to a polarisation structure, wherein a polarisation of electron spins in the polarisation structure is transferred to the nuclear spins in the particle(s), wherein for one or more of the moving particle(s) within 20 nm from a surface of the polarisation structure, the correlation time of the interaction with the nearest polarisation structure electron spin due to the molecular motion is larger than the inverse of the nuclear Larmor frequency; the electron spins in the polarisation structure are polarised above thermal equilibrium; and the polarisation transfer is performed resonantly.

Claims

1. A method of hyperpolarization of nuclear spins in one or more particle(s) moving relatively to a polarisation structure, wherein a polarisation of electron spins in the polarisation structure is transferred to the nuclear spins in the particle(s), wherein a. for one or more of the moving particle(s) within 20 nm from a surface of the polarisation structure, the correlation time of the interaction with a nearest polarisation structure electron spin due to the molecular motion is larger than the inverse of the nuclear Larmor frequency; b. the electron spins in the polarisation structure are polarised above thermal equilibrium; and c. the polarisation transfer is performed resonantly.

2. The method according to claim 1, wherein a diffusion coefficient D of the moving particle(s) which are not at a distance shorter than 100 nm from a surface of the polarisation structure is greater than 10.sup.12 m.sup.2/s.

3. The method according to claim 1, wherein the particles are part of a liquid, and wherein the particles are dissolved in a solvent or suspended in a suspension agent.

4. The method according to claim 1, wherein the polarisation structure comprises a diamond, and wherein the electron spins to be polarised are those of colour centres.

5. The method according to claim 1, wherein the electron spins of the polarisation structure, the polarisation of which is to be transferred to the nuclear spins of the particle(s), are located within less than 1 m of at least one surface of the polarisation structure.

6. The method according to claim 1, wherein the electron spins in the polarisation structure are polarised by means of optical pumping.

7. The method according to claim 1, wherein the transfer step involves an application of a microwave field or a radio frequency (RF) field.

8. The method according to claim 7, wherein a magnetic flux density of an external magnetic field is smaller than 3 T.

9. The method according to claim 1, wherein the transfer step is performed by an interaction involving at least two electron spins of the polarisation structure and one nuclear spin of a particle.

10. The method according to claim 1, wherein the transfer of polarisation is aided by spins of mediator electrons and, wherein the polarisation transfer from the electron spins of the polarisation structure to the nuclear spins of the particle(s) occurs in two steps such that first the mediator electron spin is polarised by the electron spins of the polarisation structure and then the nuclear spins are polarised by the mediator electron spin.

11. The method according to claim 1, wherein the optical pumping step and the transfer step are repeated cyclically.

12. The method according to claim 10, wherein after each cycle of the polarisation transfer, a pause allows for the nuclear spin polarisations to spread throughout the polarisation structure.

13. The method according to claim 1, wherein at the polarisation transfer the polarisation structure has a temperature of above 70 K.

14. The method according to claim 1, wherein the one or more particle(s) are used in a nuclear magnetic resonance (NMR) imaging application.

15. The method according to claim 1, wherein the polarisation structure is composed of nanoparticles in a packed-bed-type configuration or in a suspension.

16. The method according to claim 1, wherein the polarisation structure is composed of a microfluidic structure, with colour centres in at least one of the channel walls.

17. The method according to claim 2, wherein the transfer of polarisation is aided by spins of mediator electrons and, wherein the polarisation transfer from the electron spins of the polarisation structure to the nuclear spins of the particle(s) occurs in two steps such that first the mediator electron spin is polarised by the electron spins of the polarisation structure and then the nuclear spins are polarised by the mediator electron spin.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 shows the experimental setup with a microfluidic polarization structure in a conceptual representation;

(2) FIG. 2 depicts a nitrogen vacancy centre in a diamond lattice, and shows the relevant energy levels for the initialization and polarization;

(3) FIG. 3 shows a schematic of the interface between the polarization structure and the particles with the nuclear spins;

(4) FIG. 4 presents an image of a microfluidic channel etched in a diamond containing NV centres;

(5) FIG. 5 presents the microfluidic channel enclosed with a PMMA covering;

(6) FIG. 6 presents three different possible configurations of the polarisation structure;

(7) FIG. 7 shows the nuclear polarization achieved in a bulk diamond (221 mm) using the NOVEL protocol. The polarization is observed with a single-shot read out in an NMR scanner;

(8) FIG. 8 shows the polarisation transfer per unit volume achieved per time, with dependence on the diffusion coefficient;

(9) FIG. 9 illustrates the sequence of the polarisation;

(10) FIG. 10 shows the experimental setup with a polarization structure composed of nanoparticles in a suspension in a conceptual representation;

(11) FIG. 11 illustrates the main concepts of overcoming random orientation of colour centresthe solid angle in which the colour centres transfer polarization, the integrated solid effect and the detuned driving schemes;

(12) FIG. 12 demonstration of polarisation transfer for different orientations in nanodiamonds;

(13) FIG. 13 depicts the dependence on the angle to the external magnetic field in the high magnetic field regime of the zero field splitting and second order corrections to the energy levels; and

(14) FIG. 14 shows the efficiency of the polarisation transfer with a detuned driving and the integrated solid effects, with dependence on the total coupling strength and the frequency sweep velocity.

DESCRIPTION OF SPECIFIC EMBODIMENTS OF THE INVENTION

(15) Microfluidic Channel

(16) Experimental Setup

(17) FIG. 1 conceptually details the experimental setup with a solution of pyruvate particles 1 flowing through a microfluidic channel 2, placed in the magnetic field of an electromagnet 3. The microfluidic channel 2 contains multiple colour centres. A laser 4 serves to excite the colour centre electrons. A microwave source 5 and amplifier, in combination with a resonator 6 allows facilitation of polarisation transfer from the colour centre spins to the .sup.13C nuclei in the pyruvate molecules.

(18) The polarisation structure can be composed of different materials containing colour centres. While the experimental details below will be given for NV centres in a diamond substrate, the setup and polarisation transfer for other materials and colour centres are similar, e.g. for divacancies and silicon vacancies in silicon carbide or optically active defects in zinc-oxide or other wideband gap materials. The differences are only in the wavelength of the optical irradiation, the zero-field splitting and relaxation rates of the colour centre and the rate of polarisation achieved. The same protocols can be adapted with different parameters for optimizing polarisation depending on the colour centre and material.

(19) FIG. 2 illustrates the interaction interface between the colour centre spin and the particles in the solution containing nuclear spins.

(20) Material

(21) Diamond material for this protocol should contain NV centres close to the diamond surface. Material itself can be grown using CVD technique or HPHT method. NV centres close to the diamond surface can be introduced by injection of nitrogen gas during CVD growth or by implantation of nitrogen after the growth. In the latter case post-irradiation of sample can be employed to increase the formation yield of NV centres.

(22) Inside the diamond layer approximately 20 nm from the surface, denoting the z axis as perpendicular to the diamond surface, the diamond should ideally contain one NV centre every 100 nm.sup.2 the x,y dimensions. FIG. 2 depicts an NV centre in a diamond lattice, and the corresponding level structure.

(23) FIG. 4 depicts a microfluidic channel in a diamond, which can then covered by e.g. a Poly(methyl methacrylate) (PMMA) layer to create a closed structure (see FIG. 5).

(24) For silicon-carbide, the crystal can be grown by sublimation techniques. The colour centres can be generated by irradiation with neutrons or electron (see for example Kraus, H et al (2014) Room-temperature quantum microwave emitters based on spin defects in silicon carbide, Nat Phys 10, 157-162; The relevant portions of the publication are incorporated into the present disclosure by way of reference.)

(25) Solution Properties

(26) Many different solvents can be used in the solution. Pyruvate could be dissolved in a mixture of water and glycerol, for example, the concentration of the two solvents determining the pyruvate self-diffusion coefficient.

(27) Polarisation of Electron Spin

(28) Electron spins associated with NV centres can be polarised either by the application of short laser 4 pulses, or by continuous irradiation. Optical pumping is achieved by excitation of the NV centre into an excited electronic state. The decay of this state occurs predominantly into the m.sub.s=0 level of the ground state. Several cycles of excitation and decay produce a >95% polarization of the NV centre.

(29) A similar spin-selective recombination by optical excitation and decay through a metastable state happens for colour centres in silicon-carbide (e.g. to the m.sub.s=0 state in divacancies) and in the other host materials mentioned above.

(30) Dynamical Polarisation Transfer from Electron Spin to Nuclear Spin

(31) A proposed experimental realization of a DNP protocol for the polarisation transfer is achieved by establishing a Hartmann-Hahn condition between the electron and nuclear spin. This is achieved by driving the electron spin transitions between m.sub.s=0 and m.sub.s=1 state by means of a microwave field whose intensity is chosen to match the energy difference between dressed electronic spin eigenstates and the nuclear spins in an external magnetic field.

(32) The dynamics of the NV electronic spin and an additional nuclear spin, in the presence of a continuous driving microwave field have been theoretically analysed in Cai, J M et al, Diamond based single molecule magnetic resonance spectroscopy, New Journal of Physics, 2013, 15, 013020, and the article's supplementary information; the relevant portions of the publication and the supplementary information are incorporated into the present disclosure by way of reference. The Hamiltonian describing the NV centre electronic m.sub.s=0, 1 states and an additional .sup.13C nuclear spin, in the presence of an external magnetic field B and a resonant microwave field is
H=.sub.z.Math.1+.sub.N1.Math.|B.sub.eff|.sub.z+.sub.NA.sub.hyp.sub.x.Math.(sin .sub.x+cos .sub.z)(1)
where is the Rabi frequency of the driving field and a are the spin-1/2 operators, defined in the microwave-dressed basis

(33) $.Math..Math.=\frac{1}{\sqrt{2}}\left(.Math.0.Math..Math.-1.Math.\right)$
for the electronic basis, and in the (|.sub.z,|.sub.z) basis for the nuclear spins, where z is defined along the direction of B.sub.eff. B.sub.eff is an effective magnetic field and is given by B.sub.eff=B()A.sub.hyp, where A.sub.hyp is the hyperfine vector which characterises the coupling between the two spins. In equation (1), .sub.N is the gyromagnetic ratio of the nuclear spin and cos =.Math.{circumflex over (b)}, where and {circumflex over (b)} are the directions of the hyperfine vector A.sub.hyp and the effective magnetic field B.sub.eff, respectively. The first two terms in the Hamiltonian form the energy ladder of the system ( for the dressed NV spin, and .sub.N|B.sub.eff| for the Larmor frequency of the nuclear spin), whereas the last two terms are responsible for electron-nuclear spin interaction. Here, the former represents mutual spin-flips, or coherent evolution of the electron-nuclear pair, and the latter is the nuclear spin dephasing caused by electron flips. When the driving field is adjusted properly, an energy matching condition (known as the Hartmann-Hahn condition) given by
=.sub.N|B.sub.eff|=.sub.N|B()A.sub.hyp|,(2)
can be engineered, equalising the first two terms in Hamiltonian (1). Then, the coupling term in the Hamiltonian becomes dominant, and the time evolution of the system is a coherent joint evolution of the electron nuclear pair. For instance, starting in the |+, state, the system evolves according to |==|+, cos(Jt).sup.+|, sin(Jt), with J given by

(34) $\begin{array}{cc}J=\frac{1}{4}N.Math.A_{\mathrm{hyp}}.Math.\sin .& \left(3\right)\end{array}$

(35) Thus, at time t=/2J the two spins become maximally entangled, and after a t=/J a full population transfer occurs and the states of the two spins are in effect swapped.

(36) In the NOVEL sequence, a short microwave pulse at the electron spin frequency is first applied in order to rotate the spin to the X-Y plane (e.g. to the I+> state), termed a /2 pulse. The resonant microwave frequency described above is than applied, with a /2 phase shift from the /2 pulse. This resonant microwave pulse is termed the spin locking pulse. A spin locking pulse of length 10-100 s would efficiently transfer the polarization to the nuclear spins in the ensemble. FIG. 7 shows the hyperpolarisation of .sup.13C nuclear spins inside the diamond achieved via the NOVEL sequence, and compare with thermal polarisation.

(37) However, for polarising nuclear spins in a solution, the molecular motion has to be taken into account, causing time-varying terms in the Hamiltonian. In a change of notation, the time-varying Hamiltonian used to describe the coupled spin system in the secular approximation, similar to eq. (1) is given by
H(t).sub.z.sup.e+.sub.NB.sub.eff(t).Math..sup.N.sub.x.sup.e[A(t).Math..sup.N](4)
where A(t) is the hyperfine vector, B.sub.eff(t)=B0.5/.sub.N A(t) is the effective magnetic field and .sub.N is the nuclear gyromagnetic ratio, S the NV centre spin operator and the nuclear spin operator vector. A(t) stems from the fluctuating position of the nuclear spin r(t) relative to the NV centre. Specifically, A(t) and r(t) are related by

(38) $A\left(r\right)=-\frac{{}_0{}_e{}_N}{4{.Math.r.Math.}^3}\left(3{\hat{r}}_x{\hat{r}}_z,3{\hat{r}}_y{\hat{r}}_z,3{\hat{r}}_z^2-1\right),$
as usual.

(39) To analyse the efficiency of the polarisation transfer via resonant conditions in the presence of molecular motion, we simulate the Hamiltonian of eq. (4) by a Monte-Carlo simulation. The molecular motion of the particle is calculated by the Ito equations (details of the method provided in C W Gardiner. Stochastic methods. Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1985; the relevant portions of this publication are incorporated into the present disclosure by way of reference.) The coupling of the nuclear spin in the particle to the NV is then calculated at very small intervals (under 10 nanoseconds). The solution to the Liouville-von Neumann equation,

(40) $i\frac{}{t}=\left[H,\right],$
with the time-dependent Hamiltonian of eq (4) is obtained via a Runge-Kutta method (for details, see e.g. Press, W et al (2007), Section 17.1 Runge-Kutta Method, Numerical Recipes: The Art of Scientific Computing (3rd ed.), Cambridge University Press; The relevant portions of the publication are incorporated into the present disclosure by way of reference.) As no coherent effects involving several spins are expected in the parameter range of interest, and the nuclear interaction of the moving particles is negligible, the polarisation transfer rate can be extended to numerous particles with nuclear spins in the region of interest. FIG. 8 shows the polarisation transfer dependent on time for a NV centre 5 nm from the diamond surface, interacting with the nuclear spins in a (5 nanometer).sup.3 box external to the diamond. Different diffusion coefficients are plotted in FIG. 8, and as can be seen, the polarisation transfer rate via resonant coupling dramatically increases for smaller diffusion coefficients.

(41) Thus, the polarisation method is summarized in FIG. 9.

(42) As many other colour centres in other materials (e.g. silicon vacancies in silicon carbide) exhibit the same or similar photodynamics and Hamiltonian, the polarisation transfer will be achieved by the same protocols, with modifying the microwave drive frequency to the corresponding resonance (e.g. due mainly to a different zero field splitting).

(43) Mediating Electron Spins

(44) As mediating electron spins external to the polarisation layer, the diamond layer can be coated with TEMPO, Trityl radicals or other paramagnetic species. Alternatively, dangling bonds already on the diamond surface can be used as the mediating electron spins. In a third option, radicals can be added to the solution, to serve as mediating electron spins. Trityl radicals for example are typically used in dynamic nuclear polarisation setups, and can be used for mediating the polarisation transfer to further away nuclear spins.

(45) Nanoparticles Suspended in a Solution

(46) Another attractive setup for the polarisation structure is the use of nanoparticles suspended in a solution. While many elements are similar to the microfluidic channel setup, key differences, mainly dealing with the random orientation of the nanoparticles, will be highlighted below.

(47) Experimental Setup

(48) FIG. 10 conceptually details the experimental setup with a solution of pyruvate particles with a suspension of nanodiamonds containing NV centres. The solution is placed in the magnetic field of an electromagnet. A laser serves to excite the colour centre electrons. A microwave source, in combination with a resonator allows facilitation of polarisation transfer from NV centre spins to the .sup.13C nuclei in the pyruvate.

(49) Diamond Material

(50) The nanodiamonds are preferably produced by high-pressure high temperature (HPHT) method. For the direct polarisation of external spins via NV centres, ultra-small nanodiamonds, i.e. diamonds with volume between 1 nm.sup.3 and 1000 nm.sup.3, are preferable, though nanodiamonds with diameter up to 50 nm or more are also possible. Polarisation transfer will be enabled by dipolar interactions between NV centre spins and external nuclear spins.

(51) Polarisation of Electron Spin

(52) Electron spins associated with NV centres can be polarised either by the application of short laser pulses, or by continuous irradiation. Optical pumping is achieved by excitation of the NV centre into an excited electronic state. For NV centres aligned in parallel to the magnetic field, the decay of this state occurs predominantly into the m.sub.s=0 level of the ground state. Several cycles of excitation and decay produce a >95% polarization of the NV centre. For small deviations in the angle, the optical polarisation depends on the external magnetic field strength.

(53) For weak magnetic fields, .sub.EB<<D, With .sub.EB denoting the electron Larmur frequency, D the zero field splitting (D=2.87 GHz), the optical polarisation will be very weakly affected by the angle between the NV crystal axis and the magnetic field. Thus, we expect high fidelity optical polarisation in this regime for all NV angles.

(54) For strong magnetic fields (.sub.EB>>D), there is a stronger dependence on the angle. For given spherical angles , , of the NV centre crystal axis in relations to the external magnetic field, the NV is initialized to the state

(55) ${.Math.0.Math.}_{,}=\cos .Math.0.Math.+\sin \left[\frac{1}{\sqrt{2}}\left(-e^i.Math.+1.Math.+e^{-i}.Math.-1.Math.\right)\right]$

(56) For angle variations up to =10 from =0, the optical polarization works extremely well, |0|0.sub.,|.sup.2>0.97. =/2, one gets

(57) ${.Math.0.Math.}_{,}=\left[\frac{1}{\sqrt{2}}\left(-e^i.Math.+1.Math.+e^{-i}.Math.-1.Math.\right)\right].$

(58) While the |1> terms oscillate very quickly in the strong magnetic field, the amplitude of the |0> state remains zero. Thus, on average we get a strong average polarization P.sub.NV=|<1|(t)>|.sup.2|<0|(t)>|.sup.20.5. Moreover, unlike the usual case where P.sub.NV=0.5, as |<0|(t)>|.sup.2=0, one can achieve 95% polarization with additional polarization cycles.

(59) Dynamical Polarisation Transfer from Electron Spin to Nuclear Spin

(60) The same simulations and methods applied in the microfluidic channel setup apply in the suspended nanodiamonds setup as well. The one key difference is the method used for achieving a resonance condition (e.g. Hartmann Hahn condition) for performing the transfer, detailed below.

(61) Exchange of polarisation between optically pumped electron spin of NV centre and nuclear spins can be performed using several established dynamic nuclear polarisation protocols, in combination with our schemes based on Microwave frequency sweeps. For example, the integrated solid effect. Moreover, these protocols can be combined with the polarization sweep and the detuned driving scheme, for optimal polarization transfer. Most DNP protocols either involve interactions between electron spins or are based on two underlying physical mechanisms: fulfilling the Hartmann-Hahn condition and excitation of selective transitions. The DNP protocols differ in the configurations for achieving these conditions and by the usage of pulses or continuous waves. We will first detail the effective microwave frequency sweeps in nanodiamonds for the high magnetic field and low magnetic field regimes, and then describe the DNP protocols modified for combination with a microwave frequency sweep, and applied for the effective regimes in the nanodiamond ensemble.

(62) For the above DNP protocols, the experimental setup is similar, with the differences being found in the specific choice of microwave frequency, the specific type of the frequency sweep (continuous or discrete), the specific choice of the pulse sequence and/or magnetic field strength. The same equipment can be used for all protocol examples detailed below, as they are performed in similar regimes.

(63) In the high magnetic field regime, the main challenge in performing the frequency sweep is the very large deviations of the energy gap. Moreover, for many NV orientations, optical polarization not feasible. However, two angle ranges enable a short frequency sweep which includes a large portion of optically polarizable NV centres.

(64) In the laboratory coordinate system, we define the z-axis to coincide with the applied magnetic field, and the NV to be placed at the origin of the coordinate system. In the laboratory frame the Hamiltonian of a single NV is then

(65) $H_{\mathrm{eff}}^{}=\left({}_{e}B+\left(\right)\right)S_z+D\left(\right)S_z^2,D\left(\right)=\frac{D\left(1+3\cos \left(2\right)\right)+3E\left(1-\cos \left(2\right)\right)}{4},\left(\right)=\frac{{}_e{\mathrm{BG}}_1^2}{{\left({}_{e}B\right)}^2-{\left[D\left(\right)\right]}^2}+\frac{G_2^2}{2{}_{e}B}$$\mathrm{With}$$G_1=\frac{\left(D-E\right)\sin \cos e^i}{\sqrt{2}},G_2=\frac{D+3E+\left(E-D\right)\cos 2e^{2i}}{4}.$

(66) With .sub.EB denoting the electron Larmur frequency, D the zero field splitting (D=2.87 GHz), E the strain of the diamond (usually over 2 orders of magnitude smaller than D), and is the angle between the NV crystal orientation and the external magnetic field orientation. The range of the energy gap between the NV spin |0> level and |1> is approximately .sub.EB+D() (as () is roughly 2 orders of magnitude smaller), which contains a large uncertainty dependent on the angle as D()(2)1.43 GHz, (2)2.87 GHz.

(67) The energy gap varies slowly for =0 or =/2. =/2 (D()=D/2) is a particularly interesting case, as a microwave sweep corresponding to =/2 to =/2 corresponds to a solid angle of 4 sin . For example, for =10, this area encompasses over 17% of the possible NV orientations (when taking into account the symmetry between and ).

(68) For an NV in an orientation with angle from the external magnetic field, defining the x-axis so that the crystal orientation and magnetic field are on the x-z plane, the Hamiltonian of the NV centre spin can be written as

(69) $H=\frac{2}{3}{\mathrm{DS}}_z^2-\frac{1}{3}{\mathrm{DS}}_x^2-\frac{1}{3}{\mathrm{DS}}_y^2+{}_{e}B\cos \left(\right)S_z+{}_{e}B\sin \left(\right)S_x.$

(70) Of important note is the fact that the zero-field splitting term does not differentiate between m=+1, 1 states. Therefore, the splitting between these states is determined solely by the magnetic field. While it seems that for >/2, the energy gap between m.sub.s=+1,1 changes sign, this is just a consequence of choice of the z direction. Denoting {circumflex over (b)} as the magnetic field direction, and defining z=sign({circumflex over (b)}.Math.{circumflex over (z)})z, we get that the in this new orientation E.sub.+1,1=2|.sub.EB cos |>0.

(71) We now focus on the {m.sub.s=0,+1} subspace, and apply as usual a microwave field perpendicular to the magnetic field 2 cos(t), one gets in the interaction picture (with the RWA approximation)
H=(D+|.sub.eB cos()|).sub.z+ cos().sub.x.

(72) Unfortunately, different crystal orientations with the same value of will have different values of (the angle from the microwave field).

(73) Thus, NV orientations with the same value of , when choosing =D+|.sub.eB cos()| will have different Rabi frequencies, depending on their angle from the MW field orientation.

(74) This challenge can be solved by aligning the microwave field along the magnetic field axis. In this case, the Hamiltonian in the interaction picture becomes
H=(D+|.sub.eB cos()|).sub.z+ sin().sub.x.

(75) Now, all NV orientations with the angle from the magnetic field will also have the same Rabi frequency sin(). The frequency sweep in this regime needs to be combined with a corresponding change in the microwave amplitude, as sin() should be kept at a relatively constant value. The frequency sweep can now be performed in any range [.sub.min,.sub.max]. It is usually preferable for .sub.min to start at a value where sin .sub.min is not too small, which would require the microwave driving amplitude to be quite large. .sub.min>10 is usually enough.

(76) The integrated solid effect is a modification to the solid effect scheme by combining it with a microwave sweep. It is usually used for instances where the electron spin resonance (ESR) line is broad compared to the nuclear Larmor frequency, and thus cannot be captured with a single resonant frequency. A rigorous theoretical treatment is given in Henstra, A, and W Th Wenckebach. Dynamic nuclear polarisation via the integrated solid effect I: theory. Molecular Physics 112.13 (2014): 1761-1772. The relevant portions of the publication are incorporated into the present disclosure by way of reference.

(77) At the first stage the laser polarises the NV centre by optical pumping, as described above. Next, the microwave frequency is swept across the NV centre spin resonance point. In the tilted rotating frame of the nanodiamond relative to the external magnetic field, the Hamiltonian of the system is

(78) $H_{\mathrm{trans}}=2{}_{\mathrm{eff}}{\stackrel{~}{}}_z+{}_n{\mathrm{BI}}_{z^{}}+\frac{a_{x^{}}\sin }{2}\left(\stackrel{~}{}+I_{-}^{}+h.c.\right).$

(79) With 2.sub.eff denoting the NV centre effective Rabi frequency, .sub.NB denoting the nuclear Larmor frequency, .sub.x the dipolar hyperfine interaction, and sin the angle of the tilting. At the beginning of the sweep, 2.sub.eff is negative, and an adiabatic frequency sweep across the Hartmann Hahn condition 2.sub.eff=.sub.NB polarizes the nuclear spins in a given direction. Then, an adiabatic crossing of 2.sub.eff=0 flips the NV centre electron spin. Thus, when later crossing the 2.sub.eff=.sub.NB H.H. condition, the polarization is again performed in the same direction. It is important to note that if the NV centre is not optically repolarized between the two H.H. conditions, the sweep frequency needs to maximize the probability of only one adiabatic (Landau-Zener) transition being performed in the two H.H. conditions. This is due to the fact that the second Lanau-Zener transition in the same sweep will cancel the polarization transfer effect of the first Landau-Zener transition. Thus, also a very slow sweep will not achieve efficient polarization.

(80) FIG. 11(b) depicts the polarization transfer method with the ISE technique. The resonance occurs twice during the ISE sweep, at the points A1 and A2, where Landau-Zener transitions are possible. Polarisation is achieved when one, and only one, of the Landau-Zener transitions is crossed. An efficient polarization then requires a sweep velocity in an intermediate range.

(81) Larmor frequencies of .sup.13C nuclear spins were approximately 5 MHz for magnetic fields used in our experiments, though stronger magnetic fields can be used for larger Larmor frequencies. For an expected average total coupling between the NV centre and external nuclear spins in particles of .sub.x[0.01 MHz,0.1 MHz], the frequency sweep velocity should be between 0.1-10 MHz/s.

(82) After a single polarization sweep, the NV centres will be repolarized by optical pumping. Polarisation transfer is then enabled by continuous laser 4 optical pumping combined by microwave frequency sweeps. FIG. 12 demonstrates the efficient polarization transfer between an NV centre and a nuclear spin for various angles in these two ranges using the integrated solid effect. Thus, for a 10 degree deviation, the polarization step can be performed in 130 microseconds. The solid angle given by the 10 degree deviation for =0 is shown on FIG. 11(a).

(83) The polarization transfer can be further greatly improved by using a detuned driving scheme. The polarization transfer from a NV centre to coupled nuclear spins, aided by a detuned driving scheme has been theoretically analysed in Chen Q et al, Optical hyperpolarization of .sup.13C nuclear spins in nanodiamond ensembles, E-print arxiv 1504.02368 (2015); the relevant portions of the publication are incorporated into the present disclosure by way of reference. As noted before, the random orientation of NV center spins in relations to the magnetic field induces a significant variance of the energy levels, as the direction of the natural quantization axis associated with the crystal-field energy splitting is not controllable. Especially, the random orientations cause a large range of the zero-field splitting
D()[(2)1.43 GHz,(2)2.87 GHz],
which makes it difficult for polarization transfer. Applying a circularly polarized microwave field, assuming a point-dipole interaction, and neglecting the contact term we obtain

(84) 0$H_e{}_M\left(S_x\cos {}_{M}t+S_y\sin {}_{M}t\right)+\left({}_{e}B+\left(\right)\right)S_z+D\left(\right)S_z^2+{}_n{\mathrm{BI}}_z-{\mathrm{gS}}_z\left[3e_r^z\left(e_r^x{I}_x+e_r^y{I}_y\right)+\left(3{\left(e_r^z\right)}^2-1\right)I_z\right].$

(85) Where .sub.M=2 is the Rabi frequency of the driving field and .sub.M is its frequency, .sub.eB (.sub.nB) is the NV centre (nuclear) spin coupling to the external magnetic field, and () is a correction to the energy levels due to second-order corrections of the misaligned zero field splitting. Consider the off resonant case, .sub.M=.sub.eB+(.sub.M), with () shown in FIG. 13b, where .sub.M is the specific chosen angle, the driving is detuned from the |m.sub.s=1.Math.|m.sub.s=0 and |m.sub.s=0.Math.|m.sub.s=+1 by the misaligned zero-field splitting D(). In the interaction picture, we get

(86) $H^{}=\left(.Math.-1.Math..Math.0.Math.+.Math.0.Math..Math.+1.Math.+h.c.\right)+D\left(\right)S_z^2+{}_n{\mathrm{BI}}_{z^{}}+S_z.Math.\left(a_{x^{}}{I}_{x^{}}+a_{z^{}}{I}_{z^{}}\right).$

(87) Here a.sub.z and a.sub.x are the elements of the secular and pseudosecular hyperfine interactions, respectively. We proceed with diagonalizing the NV spin Hamiltonian. In the basis {|+1, |0, |1} for the electron spin, the Hamiltonian is

(88) $H_{\mathrm{NV}}=\left(\begin{array}{ccc}D\left(\right)& & 0\\ & 0& \\ 0& & D\left(\right)\end{array}\right),$

(89) While the eigenvalues and eigenstates of the Hamiltonian can be solved exactly, we will provide here an approximate answer which provides more insight on the detuned driving concept. As D()>>, the eigenstates of this Hamiltonian are approximately

(90) $.Math..Math.=\frac{1}{\sqrt{2}}\left(.Math.-1.Math..Math.+1.Math.\right),$
and |0. The |+, | states have very similar energies, with an energy gap of

(91) $2{}_{\mathrm{eff}}=\frac{{}^2}{D\left(\right)}D\left(\right),.$
The |0 state is far detuned with an energy gap of roughly D() from the {|+, |} states. Let's examine the dependence of 2.sub.eff on .

(92) $\frac{\left(2{}_{\mathrm{eff}}\right)}{}=\frac{2{}_{\mathrm{eff}}}{D\left(\right)}\frac{D\left(\right)}{}$

(93) As D()>>2.sub.eff, this is a radical decrease in the dependence of the energy gap on the angle from the magnetic field. Thus, when tuning the driving microwave amplitude so that 2.sub.eff=.sub.nB, almost all the angle dependence is due to () (second order corrections to the energy levels), which adds an S.sub.z detuning term to the Hamiltonian. As seen on FIG. 13, () changes on the order of tens of MHz when varying the magnetic field, compared with several GHz for D(). FIG. 11(c) illustrates the concept of the detuned driving.

(94) A potential polarization scheme can combine the detuned driving, optimal frequency sweep and the integrated solid effect. Focusing on 10 degrees deviation from the perpendicular crystal axis (90 degree deviation from magnetic field orientation): 1. Optically polarize the NV centres to the {|+, |} subspace 2. Apply a very fast frequency sweep of the |0.Math.| transition (can be applied with strong driving amplitude. Sweep duration can be under 1 s for >10 MHz). Thus, in the {|+, |} subspace, the NV centres are polarized to the | state 3. Apply the integrated solid effect protocol with detuned driving for achieving polarization transfer. The sweep should be performed from .sub.eB+91 MHz to .sub.eB+97 MHz 4. Repeat steps 1-3

(95) FIG. 14 shows the dependence of the polarisation transfer based on the sweep velocity and total coupling to the NV centre spin.

(96) The effect of molecular motion on a detuned driving scheme for achieving the Hartmann-Hahn resonance would be nearly identical to that seen for the NOVEL sequence above.