Multidimensional Spectroscopy of Nuclear Spin Clusters in Diamond

Optically active spin defects in solids offer promising platforms to investigate nuclear spin clusters with high sensitivity and atomic-site resolution. To leverage near-surface defects for molecular structure analysis in chemical and biological contexts using nuclear magnetic resonance (NMR), further advances in spectroscopic characterization of nuclear environments are essential. Here, we report Fourier spectroscopy techniques to improve localization and mapping of the test bed ${}^{13}\mathrm{C}$ nuclear spin environment of individual, shallow nitrogen-vacancy centers at room temperature. We use multidimensional spectroscopy, well-known from classical NMR, in combination with weak measurements of single-nuclear-spin precession. We demonstrate two examples of multidimensional NMR: (i) improved nuclear spin localization by separate encoding of the two hyperfine components along spectral dimensions and (ii) spectral editing of nuclear-spin pairs, including measurement of internuclear coupling constants. Our work adds important tools for the spectroscopic analysis of molecular structures by single-spin probes.

Figure 1

(a) Experiment schematic. An NV electron spin (blue) interacts with proximal ${}^{13}\mathrm{C}$ nuclear spins (yellow) in the diamond (gray) via the dipolar hyperfine interaction. Optical preparation and readout of the NV spin state is achieved using laser pulses and photoluminescence intensity measurements. Arrows indicate microwave ($\mu \mathrm{w}$; blue, red) and radio-frequency (rf; purple) tones for electron and nuclear spin manipulation, respectively. (b) Energy-level diagram for the NV ground-state manifold, including the bath of ${}^{13}\mathrm{C}$ nuclei (inset). The ${}^{15}\mathrm{N}$ spin is neglected for clarity. (c) Canonical sequence for 1D NMR spectroscopy using the weak-measurement detection scheme. (d)–(f) Detail of the “polarize,” “manipulate,” and “detect” blocks. In (d)–(f), green denotes laser pulses, blue denotes microwave pulses ($\mu {\mathrm{w}}_1$), and purple denotes the rf pulse. Horizontal axis is time.

Figure 2

Two-dimensional hyperfine spectroscopy. (a) Periodic inversion of the NV spin leads to modulation of the hyperfine interaction. Blue (red) shows the direction of the transverse hyperfine coupling parameter $a_{\perp }$ when the NV spin is in the $|-1⟩$ ($|+1⟩$) state. (b) Corresponding pulse sequence. Periodic inversion of the NV spin creates an effective hyperfine magnetic field of amplitude $B_1$ experienced by ${}^{13}\mathrm{C}$ spins. The modulation period $T=2\pi /{\omega }_0$ corresponds to one nuclear Larmor period, where ${\omega }_0={\gamma }_{\mathrm{n}}{B}_0$. (c) 2D Fourier power spectrum with respect to $t_1$ and $t_2$. $f_1$ and $f_2$ are the corresponding frequency axes. The horizontal axis $f_1$ encodes the parallel hyperfine component $a_{||}$, while the vertical axis $f_2$ encodes the transverse hyperfine component $a_{\perp }$ [Eq. (3)]. The “bare” Larmor frequency ${\omega }_0/2\pi$ is indicated by a dashed vertical line. The time increment and number of points are $11.36\mathrm{\mu }\mathrm{s}$ and 1925 for the $t_1$ dimension, and $11.2\mathrm{\mu }\mathrm{s}$ ($=22T$) and 126 for the $t_2$ dimension, respectively. (d) 1D projection of the 2D Fourier spectrum. (e),(f) Numerical simulation of the five main peaks using the density matrix method.

Figure 3

Two-dimensional $J$ spectroscopy of nuclear spin pairs. (a) Illustration of an NV spin coupled to a nuclear spin pair. (b) Pulse sequence consisting of a nuclear spin echo formed by two rf $\pi /2$ and $\pi$ pulses, respectively. The single echo could be extended to multiple echoes using a Carr-Purcell-Meiboom-Gill sequence. Indirect evolution time $t_2$ is incremented in steps of $300\mathrm{\mu }\mathrm{s}$ (NV-B) and $388\mathrm{\mu }\mathrm{s}$ (NV-C), respectively. No microwave pulses are applied. (c),(d) 2D Fourier spectra for NV-B and NV-C. The horizontal axis $f_1$ encodes the full free precession Hamiltonian. The vertical axis $f_2$ only encodes nuclear spin-spin coupling ${\omega }_{\mathrm{D}}$ [Eq. (4)]. Horizontal lines indicate frequencies of expected spin-pair configurations. Atomic arrangements for identified pairs (1), (2), and (2’) are shown as insets. (e),(f) 1D projections of the respective 2D spectra. In (e), satellite peaks are marked by asterisks. (g) Expected peak patterns for a spin pair with weak, intermediate, and strong ${\omega }_{\mathrm{D}}$. $\mathrm{\Delta }{a}_{||}$ is the frequency difference between the nuclear spins’ $a_{||}$ coupling constants.