Enhancement of nuclear spin coherence times by driving dynamic nuclear polarization at defect centers in solids

The hyperpolarization of nuclear spins can enable powerful imaging and sensing techniques provided the hyperpolarization is sufficiently long lived. Recent experiments on nanodiamond ${}^{13}\mathrm{C}$ nuclear spins demonstrate that relaxation times can be extended by three orders of magnitude by building up dynamic nuclear polarization (DNP) through the driving of electron-nuclear flip-flop processes at defect centers. This finding raises the question of whether the nuclear spin coherence times are also impacted by this hyperpolarization process. Here, we theoretically examine the effect of DNP on the nuclear spin coherence times as a function of the hyperpolarization drive time. We do this by developing a microscopic theory of DNP in a nuclear spin ensemble coupled to microwave-driven defect centers in solids and subject to spin diffusion mediated by internuclear dipolar interactions. We find that, similarly to relaxation times, the nuclear spin coherence times can be increased substantially by a few orders of magnitude depending on the driving time. Our theoretical model and results will be useful for current and future experiments on enhancing nuclear spin coherence times via DNP.

Figure 1

(a) A schematic representation of polarization transfer via DNP. The electron center constitutes an ESR line at a frequency ${\omega }_{0s}$. The nuclear spins in a radius of $r<a$ around the electron center are strongly polarized while the nuclei within $a<r<b$ participate in the DNP process. The spin polarization is transferred further away to $b<r<c$ via nuclear spin diffusion mediated by dipole-dipole interactions between the nuclei. In red is shown a possible nuclear process where there is a spin flip-flop between pairs of nuclei, leading to local fluctuations in the nuclear field. (b) The corresponding energy level diagram. The ESR line is at frequency ${\omega }_{0s}$ and the NMR line is at frequency ${\omega }_{0I}$. Driving the system at a microwave frequency ${\omega }_m={\omega }_{0s}-{\omega }_{0I}$ results in a spin polarization transfer between electron and nuclei (DNP).

Figure 2

Spatial profile $\langle P_z(r,\theta ,t)\rangle$ of the microwave driven nuclear polarization around the electron center at the origin as obtained from the solution of the Liouville-von Neumann equation at various times. (a) At time $t=0.1\mathrm{s}$. (b) At time $t=1\mathrm{s}$. We use the parameters ${\omega }_{0s}=10\mu \mathrm{eV}$, $B_e=0.1\mu \mathrm{eV}$, ${\omega }_{0I}=0.0004{\omega }_{0s}$. The relaxation parameters for the Lindblad operator used were ${\gamma }_1=1\mu \mathrm{s}$ and ${\gamma }_2=1\mathrm{s}$. Note that this so far does not account for nuclear diffusion (which is the subject of Sec. 3) but rather provides an initial condition to that problem.

Figure 3

Time evolution of DNP around the electron center at the origin as obtained from the solution of the Liouville-von Neumann equation at various distances from the origin. We use the parameters ${\omega }_{0s}=10\mu \mathrm{eV}$, $B_e=0.1\mu \mathrm{eV}$, ${\omega }_{0I}=0.0004{\omega }_{0s}$, $\theta =\pi /4$. The relaxation parameters for the Lindblad operator used were ${\gamma }_2=1\mu \mathrm{s}$ and ${\gamma }_1=1\mathrm{s}$. Also note that this solution so far does not account for nuclear spectral diffusion (which is the subject of Sec. 3) but rather serves as its initial condition.

Figure 4

Logarithm (natural) of the numerically evaluated $\alpha$ (in the units ${\mathrm{m}}^{-1}$) as a function of the logarithm (natural) of the driving time $T$ (in the units s) using the Gaussian approximation along a particular direction ($\theta =\pi /4$). The plot shows the data (in blue dots) for $log\left(\alpha \right)$ as a function of $log\left(T\right)$. The data shows a linear behavior shown in the red line (at least for the timescales we are concerned with) and thus can be extrapolated to obtain $\alpha$ for arbitrary driving times. The parameters chosen are the same as in Fig. 2. We find that for those parameters, $log\left(\alpha \left(T\right)\right)=a_{1}logT+a_2$, with $a_1=-0.16632$, and $a_2=18.8096$.

Figure 5

Numerically evaluated $\alpha$ (in the units ${\mathrm{nm}}^{-1}$) as a function of $\theta$ for ten different driving times $T$. The black diamond on each curve indicates the average value ${\langle \alpha \rangle }_{\theta }$ (obtained by averaging over all $\theta$ values for a particular $T$). The dotted line indicates that the average value for each driving time $T$ collapses on to the value at a particular $\theta$. We find that ${\langle \alpha \rangle }_{\theta }/{\alpha }_{\theta =\pi /4}\approx 1.24$. We also note that the value of $\alpha$ between $\theta =\pi /4$ and $\theta \approx 0$ differs by just a factor of two.

Figure 6

Numerically obtained normalized $C\left(\omega \right)$ for a particular drive time of $s=15$ s. The inset shows the analytically evaluated correlation function $C\left(t\right)$.

Figure 7

Decoherence under various dynamical decoupling pulse sequences (free induction decay, spin echo, and CPMG) for a driving time of $T\sim 0.01\mathrm{s}$. The other parameters used are $a_1=-0.16632$, $a_2=18.8096$, $x_0=10\mathrm{nm}$, $D=25{\mathrm{nm}}^2/\mathrm{s}$, ${\beta }^{-1}=1\AA$, $\eta \sim {10}^{-20}{\left(\mathrm{eVs}\right)}^2/{\mathrm{m}}^3$, $I_0=\sqrt{\pi }\hslash /\left(0.2\mu {\mathrm{m}}^3\right)$. The initial DNP distribution is taken to have $\alpha ={\alpha }_{\theta =\pi /4}$.

Figure 8

Logarithm of the nuclear $T_2$ time as a function of the logarithm of the driving time for various different values of the noise strength $\eta$. A clear enhancement (up to $\sim 3$ orders of magnitude) of the nuclear $T_2$ time with driving time is seen for $-3<{log}_{10}\left(T\right)<0$. We also note the suppression of $T_2$ time with increasing noise strength. Further the $T_2$ time saturates after the DNP drive time is increased beyond $T\sim 1\mathrm{s}$. We use $a_1=-0.16632$, $a_2=18.8096$, $x_0=10\mathrm{nm}$, $D=25{\mathrm{nm}}^2/\mathrm{s}$, ${\eta }_0\sim {10}^{-20}{\left(\mathrm{eVs}\right)}^2/{\mathrm{m}}^3$, ${\beta }^{-1}=1\AA$, and use the $n=4$ CPMG pulse sequence. The initial DNP distribution is taken to have $\alpha ={\alpha }_{\theta =\pi /4}$.

Figure 9

Logarithm of the nuclear $T_2$ time as a function of the logarithm of the driving time for various different values of the diffusion constant $D$. For larger $D$ values the coherence time saturates more quickly as a function of driving time. We use $a_1=-0.16632$, $a_2=18.8096$, $x_0=10\mathrm{nm}$, $\eta \sim {10}^{-20}{\left(\mathrm{eVs}\right)}^2/{\mathrm{m}}^3$, ${\beta }^{-1}=1\AA$, $D_0=0.5{\mathrm{nm}}^2{\mathrm{s}}^{-1}$, and use the $n=4$ CPMG pulse sequence. The initial DNP distribution is taken to have $\alpha ={\alpha }_{\theta =\pi /4}$.

Figure 10

Density plot for the nuclear $T_2$ time (in seconds) as a function of the noise parameter and diffusion constant for a constant driving time of $T=10\mathrm{s}$. The parameters used were $D_0=25{\mathrm{nm}}^2/\mathrm{s}$, ${\eta }_0\sim {10}^{-23}{\left(\mathrm{eVs}\right)}^2/{\mathrm{m}}^3$, $a_1=-0.1662$, $a_2=18.4374$, $x_0=10\mathrm{nm}$, ${\beta }^{-1}=1\AA$, and the $n=4$ CPMG pulse sequence was used. The initial DNP distribution is taken to have $\alpha ={\alpha }_{\theta =\pi /4}$. An increase in noise strength leads to a higher suppression of $T_2$ compared to lowering the diffusion constant $D$. The different order of the $T_2$ obtained here is due to our different choice of parameters compared to Figs. 8 and 9.

Figure 11

Plot of the logarithm of the nuclear $T_2$ time as a function of logarithm of the DNP polarization time for experimentally relevant parameters for nanodiamond [56], i.e., $D=5{\mathrm{nm}}^2/\mathrm{s}$, ${\omega }_{0s}\sim 300\mu \mathrm{eV}$. Note that for this plot we perform the ensemble averaging to obtain the $T_2$ time over a wide range of radii from the center $0.5\mathrm{nm}<x_0<10\mathrm{nm}$ and also over the angle $\theta$. For the DNP Gaussian model for each $\theta$ we have calculated $\alpha$ separately from the exact DNP solution. The top inset shows the saturation of the $T_2$ time for arbitrarily higher DNP driving times and the bottom inset shows the decay of the signal with time for increasing DNP driving times. The other parameters we chose were ${\beta }^{-1}=1\AA$, $\eta \sim {10}^{-23}{\left(\mathrm{eVs}\right)}^2/{\mathrm{m}}^3$ [76], and the $n=4$ CPMG pulse sequence.