Optical hyperpolarization of ${}^{13}\mathrm{C}$ nuclear spins in nanodiamond ensembles

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 analyze in detail protocols that can achieve rapid hyperpolarization of ${}^{13}\mathrm{C}$ 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 ${}^{13}\mathrm{C}$ 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 ${}^{13}\mathrm{C}$ 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 ${}^{13}\mathrm{C}$ nuclei has the potential to result in a major signal enhancement in ${}^{13}\mathrm{C}$ nuclear magnetic resonance imaging and suggests functionalized and hyperpolarized nanodiamonds as a unique probe for molecular imaging both in vitro and in vivo.

Figure 1

Schematic of dressed-state resonant coupling of NV spins and nuclear spins in a nanodiamond ensemble and energy-level diagrams. (a) The random orientations of the NV spins are uniformly distributed over the unit sphere. Small yellow circles represent examples with orientations indicated by red arrows which represent the unit vector pointing from the nitrogen to the vacancy of an NV center (see the example of such an NV center in the dashed circle, the axis pointing along “N-V” forms the natural quantisation axis). (b) Ground electronic spin states of an NV center in a strong external magnetic field. Microwave driving fields are applied off-resonantly achieving an effective resonant double quantum transition between the state $m_s=+1$ and $m_s=-1$ in the large detuning regime. (c) The effective coupling provides a dressed state basis which then permits energy conserving flip-flops between the dressed states and external resonant nuclear spins.

Figure 2

(a) Zero-field distribution $D\left(\theta \right)$ of the NV spins in nanodiamonds with $D=\left(2\pi \right)2.87$ GHz and $E=\left(2\pi \right)20$ MHz. (b) The second-order corrections which induce an energy distribution $\delta \left(\theta \right)$.

Figure 3

The level structure of the three-level Hartmann-Hann. A gap is created between the $|+\rangle =\frac{1}{\sqrt{2}}\left(|+1\rangle +|-1\rangle \right)$ and the $|-\rangle =\frac{1}{\sqrt{2}}\left(|+1\rangle -|-1\rangle \right)$ states. This gap is robust to changes in the one photon detuning $\left[D\right(\theta \left)\right]$ and is also robust to small changes in the two-photon detuning $\left[\delta \right(\theta \left)\right]$. Due to the fact that the coupling to the nuclei is via the dressed states $\left(|+\rangle ,|-\rangle \right)$ and not the bare states, the coupling is not decreased due to the off resonant drivings.

Figure 4

Eigenenergies of Eq. (20) for $B=0.36$ T in which $|\frac{a_{x^{\prime }}sin{\phi }_l}{2}|=\left(2\pi \right)0.5$ MHz, ${\mathrm{\Omega }}_{\mathrm{eff}}=\left(2\pi \right)2.2$ MHz (dot-dashed red line), ${\mathrm{\Omega }}_{\mathrm{eff}}=\left(2\pi \right)3$ MHz (dotted green line), ${\mathrm{\Omega }}_{\mathrm{eff}}=\left(2\pi \right)3.6$ MHz (dashed blue line). The level diagram shows the progression of the eigenvalues of the Hamiltonian during the ISE sweep. Energies of the states $|{\chi }_{+},\uparrow \rangle$ and $|{\chi }_{-},\downarrow \rangle$ (solid blue line) for ${\mathrm{\Omega }}_{\mathrm{eff}}=\left(2\pi \right)2.2$ MHz are included. Transitions from states represented by solid blue lines to state represented by dashed lines may lead to depolarization of the nuclear bath.

Figure 5

Double-passage transition. A schematics of a pair of the adiabatic energy levels as in Fig. 4. At each crossing there is a probability of $P_{LZ}$ for a Landau-Zener transitions. The lines with one (two) arrows show the two paths where the transition to the upper level happens during passage of the first (second) resonance point. The average probability for a polarization transfer at the end of the completed passage is given by $2P_{LZ}(1-P_{LZ})$.

Figure 6

The maximal polarization transfer $P_{\max }$ as a function of the effective coupling ${\mathrm{\Omega }}_{\mathrm{eff}}$. (a) ${\mathrm{\Omega }}_{\mathrm{eff}}=\left(2\pi \right)3.6$ MHz (b) ${\mathrm{\Omega }}_{\mathrm{eff}}=\left(2\pi \right)3$ MHz; (c) ${\mathrm{\Omega }}_{\mathrm{eff}}=\left(2\pi \right)2.3$ MHz.

Figure 7

A simple flow diagram of the entire polarization scheme: For a given direction of magnetic field, the large box includes numerous polarization cycles (each polarization cycle includes optical initialization of the NV spin, adiabatic preparation of the state $|-1\rangle$, and then polarization transfer by matching a H-H resonance condition by using off-resonant microwave driving and the ISE technique). After the adiabatic rotation of the magnetic field (the thick black arrow represent the magnetic field orientation which is rotated to another direction denoted by the dashed blue arrow) we repeat our polarizing cycles to achieve the polarization of additional nuclear spins.

Figure 8

Exact polarization dynamics with $B=0.36$ T, $v=\left(2\pi \right)6\text{MHz}/\mu \text{s}$ and $\mathrm{\Delta }t=10$ $\mu \text{s}$: (a) The coupling strength is given as $a_{x^{\prime }}=\left(2\pi \right)0.6$ MHz, ${\mathrm{\Omega }}_{\mathrm{eff}}=\left(2\pi \right)3$ MHz (blue circles), and ${\mathrm{\Omega }}_{\mathrm{eff}}=\left(2\pi \right)3.6$ MHz (red triangles). (b) We have ${\mathrm{\Omega }}_{\mathrm{eff}}=\left(2\pi \right)3.5$ MHz, $a_{x^{\prime }}=\left(2\pi \right)0.7$ MHz (red triangles), $a_{x^{\prime }}=\left(2\pi \right)0.5$ MHz (green cubes), and $a_{x^{\prime }}=\left(2\pi \right)0.3$ MHz (blue circles). (c) Exact polarization dynamics with $v=\left(2\pi \right)0.8\text{MHz}/\mu \text{s},\mathrm{\Delta }t=50$ $\mu \text{s}$, and ${\mathrm{\Omega }}_{\mathrm{eff}}=\left(2\pi \right)2.3$ MHz, $a_{x^{\prime }}=\left(2\pi \right)0.3$ MHz (red triangles), and $a_{x^{\prime }}=\left(2\pi \right)0.1$ MHz (blue circles).

Figure 9

(a) Diagram of adiabatic rotating direction of the magnetic field. Blue spherical sectors present ${\theta }^{\circ }$ deviation tolerance for $Z$ direction of the magnetic field. Magnetic field is rotated by angle $\vartheta$ to get red spherical sectors with the half angle of the cone angle ${\theta }^{\prime }$ 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 two-dimensional 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.

Figure 10

The polarization buildup for five nuclear spins for ${\mathrm{\Omega }}_{\mathrm{eff}}=3.23$ MHz. The nuclear spins are strongly interacting with the NV center spin, $a_{x_1^{\prime }}=\left(2\pi \right)0.7$ MHz, $a_{x_2^{\prime }}=\left(2\pi \right)0.5$ MHz, $a_{x_3^{\prime }}=\left(2\pi \right)0.4$ MHz, $a_{x_4^{\prime }}=0.32$ MHz, and $a_{x_5^{\prime }}=\left(2\pi \right)0.2$ 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 $548{}^{13}\mathrm{C}$ 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 sec. The system benefits from polarization diffusion between nuclear spins resulting in a high level of polarization.

Figure 11

The depolarization of the previously polarized nuclear spin with $B=0.36$ T, ${\mathrm{\Omega }}_{\mathrm{eff}}=\left(2\pi \right)3.5$ MHz, $v=\left(2\pi \right)6\text{MHz}/\mu \text{s}$, and $\mathrm{\Delta }t=10$ $\mu \text{s}$. The coupling strength is given as $a_{x^{\prime }}=\left(2\pi \right)0.3$ MHz, (blue circles) and $a_{x^{\prime }}=\left(2\pi \right)0.6$ MHz (red triangles).

Figure 12

Two related rotations of the coordinate system are used to map from the natural orientation $z_{\theta }$ axis of the NV spin to $z$ axis which is defined by the high magnetic field.

Figure 13

Validation of neglecting off-resonant coupling with $D=\left(2\pi \right)2870$ MHz, $\theta ={10}^{\circ },E=\left(2\pi \right)20$ MHz. The dashed blue line and solid red line present the time evolutions of the probability of the state $|0\rangle$ corresponding to $B=0.36$ T and $B=0.54$ T, respectively.