High-resolution spectroscopy of single NV defects coupled with nearby ${}^{13}$C nuclear spins in diamond

We report a systematic study of the hyperfine interaction between the electron spin of a single nitrogen-vacancy (NV) defect in diamond and nearby ${}^{13}\text{C}$ nuclear spins, by using pulsed electron-spin resonance spectroscopy. We isolate a set of discrete values of the hyperfine coupling strength ranging from 14 MHz to 400 kHz and corresponding to ${}^{13}\text{C}$ nuclear spins placed at different lattice sites of the diamond matrix. For each lattice site, the hyperfine interaction is further investigated through nuclear-spin polarization measurements and by studying the magnetic field dependence of the hyperfine splitting. This work provides information that is relevant for the development of nuclear-spin-based quantum register in diamond.

Figure 1

(a) Energy-level diagram of the NV defect as a function of the amplitude of a static magnetic field $B$ applied along the NV defect axis. (b) Hyperfine structure of the $m_s=0$ and $m_s=-1$ ground-state manifolds for a NV defect coupled to its ${}^{14}\mathrm{N}$ (nuclear-spin projection $m_{I^{\left(\mathrm{N}\right)}}$) and a single ${}^{13}\mathrm{C}$ (nuclear-spin projection $m_{I^{\left(\mathrm{C}\right)}}$). The energy level scheme is given for a positive ${}^{13}\mathrm{C}$ hyperfine splitting, i.e., ${\mathcal{A}}_{zz}>0$, and considering that off-diagonal components of the ${}^{13}\text{C}$ hyperfine tensor are negligible. ESR spectra exhibit six nuclear-spin conserving transitions (red arrows). All notations are defined in the main text. (c)–(e) ESR spectra recorded for a single NV defect coupled to its ${}^{14}$N and (c) zero, (d) one, and (e) two nearby ${}^{13}\mathrm{C}$. The blue values indicate the corresponding hyperfine splittings and solid lines are data fit with Gaussian functions. For this experiment, a magnetic field $B\approx 20$ G is applied along the NV defect axis in order to lift the degeneracy of the $m_s=\pm 1$ electron-spin sublevels. For a magnetic field amplitude $B\approx 510$ G, a level anticrossing (LAC) in the excited-state [highlighted in (a)] induces nuclear-spin polarization such that only two nuclear-spin conserving transitions are observed, as indicated with bold arrows in (b) (see Fig. 5).

Figure 2

Distribution of ${}^{13}\text{C}$ hyperfine splittings observed for a set of roughly 400 single NV defects at low magnetic field ($B\approx 20$ G). The well-documented[20, 22] 130 MHz splitting linked to a ${}^{13}\text{C}$ placed in the nearest neighbor lattice sites of the vacancy is not represented. The capital letters denote the lattice sites corresponding to a given hyperfine splitting. The dashed red lines represent the mean value of each ${}^{13}\text{C}$ family (A to O) and error bars represent estimated errors in ESR spectra fit with a 95$\%$ confidence interval. For a given ${}^{13}\text{C}$ family, the small dispersion of the hyperfine splittings is attributed to local paramagnetic defects, which could shift the nuclear-spin levels. The mean of the hyperfine splitting associated with each ${}^{13}\text{C}$ family are summarized in Table .

Figure 3

(a) Typical ESR spectrum recorded at low magnetic field ($B=20$ G) for a single NV defect coupled with a single ${}^{13}\text{C}$ nuclear spin belonging to family I (${\mathcal{A}}_{\mathrm{C}}^{hs}=1.12$ MHz). (b) Zoom on the hyperfine transitions associated with a ${}^{14}$N nuclear-spin projection $m_{I^{\left(\mathrm{N}\right)}}=+1$. Forbidden electron-spin transitions ($T_{23}$,$T_{14}$) appear in the ESR spectrum when the magnetic field magnitude is increased. (c) Energy level structure of the $m_s=0$ and $m_s=-1$ ground-state manifolds for a ${}^{14}$N nuclear-spin projection $m_{I^{\left(\mathrm{N}\right)}}=1$. Solid (resp. dashed) arrows indicate allowed (resp. forbidden) electron-spin transitions. All notations are defined in the main text.

Figure 4

(a) Frequencies of forbidden ($T_{23}$,$T_{14}$) and allowed ($T_{13}$,$T_{24}$) electron-spin transitions as a function of the magnetic field magnitude. Solid lines are data fits using Eqs. (12, 13, 14, 15) with ${\mathcal{A}}_{zz}$ and ${\mathcal{A}}_{\mathrm{nd}}$ as fitting parameters. (b) $T_{23}-T_{13}$ as a function of the magnetic field magnitude. The black dashed line is data fit with a linear function, from which the ${}^{13}\text{C}$ nuclear-spin gyromagnetic ratio ${\gamma }_n=1.07\left(1\right)$ kHz.G${}^{-1}$ is extracted. (c) Relative intensity $\mathcal{R}$ between forbidden and allowed electron-spin transitions as a function of the magnetic field. The dashed line is the theoretical expectation $\mathcal{R}={\tan }^2\left(\frac{\theta }{2}\right)$, where $\theta$ is defined by Eq. (11) with ${\mathcal{A}}_{zz}=1.02$ MHz and ${\mathcal{A}}_{\mathrm{nd}}=0.51$ MHz. The slight deviation of experimental data points is attributed to an imperfect microwave $\pi$-pulse in the pulsed-ESR spectroscopy scheme.

Figure 5

(a)–(d) Typical pulsed-ESR spectra recorded at low magnetic field ($B\approx 20$ G) and at the excited-state LAC ($B\approx 510$ G) for single NV defects coupled by hyperfine interaction with a single ${}^{13}\text{C}$ belonging to family (a) D, (b) H, (c) J, and (d) K. The ${}^{14}$N nuclear spin is perfectly polarized while the polarization efficiency of nearby ${}^{13}\text{C}$ strongly depends on the lattice site. (e) Polarization efficiency of a set of single NV defects coupled with different families of ${}^{13}$C. We note that these results are obtained by using pulsed-ESR spectroscopy, i.e., with an optical pumping duration of 300 ns. Positive polarization (resp. negative) indicates a positive (resp. negative) hyperfine splitting. Within each ${}^{13}\text{C}$ family, the variation of polarization efficiency is attributed to misalignment of the magnetic field along the NV defect axis. The mean polarization efficiency for each ${}^{13}\text{C}$ family in given in Table . (f) Free induction decay recorded for the same NV defect as in (d) at the excited-state LAC and for an optical pumping duration of 3 $\mu$s. A single line is observed in the Fourier transform, which is the signature of nearly perfect polarization of the ${}^{13}\text{C}$ nuclear spin.