Figure 1

Concept of isolating and controlling weakly coupled spins. (a) Surrounding ${}^{13}\mathrm{C}$ nuclear spins precess about axes that depend on the NV electron spin state. For $m_s=0$, all ${}^{13}\mathrm{C}$ spins precess about ${\mathit{\omega }}_L$ set by the applied magnetic field ${\mathbf{B}}_0$. For $m_s=-1$, each spin precesses about a distinct axes $\tilde{\mathit{\omega }}$ due to the hyperfine interaction ${\mathit{\omega }}_h$. (b) Calculated probability $P_x$ to preserve the initial electron spin state after a decoupling sequence with $N=32$, for two ${}^{13}\mathrm{C}$ spins with $\theta =\pi /4.5$, Nucleus $I$: ${\omega }_h=2\pi \times 40\mathrm{kHz}$, Nucleus $II$: ${\omega }_h=2\pi \times 20\mathrm{kHz}$, $B_0=293\mathrm{G}$. Each spin can be selectively addressed by tuning the interpulse delay $2\tau$ into resonance with its dynamics. (c,d) Bloch spheres showing the nuclear spin dynamics for $\tau$ resonant with nucleus $I$ (arrow). (c) For nucleus $I$, the net result is a rotation around antiparallel axes ($\widehat{{\mathbf{n}}_{\mathbf{0}}}$ and $\widehat{{\mathbf{n}}_{\mathbf{1}}}$) for the two electron states, resulting in entanglement. (d) Nucleus $II$ is decoupled: its rotation is independent of the electron state.Reuse & Permissions

Figure 2

Resolving individual weakly coupled ${}^{13}\mathrm{C}$ nuclear spins. (a) $P_x$ as function of $\tau$ for a decoupling sequence with $N=32$ and a magnetic field $B_0=401\mathrm{G}$. The sharp resonances in the echo signal correspond to the coherent interaction with individual ${}^{13}\mathrm{C}$ atoms. (b) Magnification of the section marked in (a) indicating resonances associated with six nuclear spins. (c) Positions ${\tau }_k$ of resonances with order $k$ observed in (a) relative to the Larmor period $T_L=2\pi /{\omega }_L$, $\Delta \tau ={\tau }_k/T_L-(2k-1)/4$. The five sets of equally-spaced resonances correspond to the spins numbered in (b). Lines are fits to Eq. (3). (d) Close up for nuclear spin 6 ($\tau \approx 8.57\dots 8.59\mu \mathrm{s}$) with $N=96$. Line: fit based on Eq. (2). Errors are $\pm 1$ standard deviation (s.d.).Reuse & Permissions

Figure 3

Comparison of the measured signal with the prediction based on the parameters in Table  (orange line). We observe good agreement for the positions and amplitudes of multiple resonances for magnetic fields of both $B_0=401\mathrm{G}$ and $B_0=505\mathrm{G}$. Error bars are $\pm 1$ s.d.Reuse & Permissions

Figure 4

Coherent conditional rotations of weakly-coupled nuclear spin 3. (a) $P_x$ with different numbers of pulses $N$ for the resonance centered at $\tau \approx 8.4\mu \mathrm{s}$. Lines: calculations based on the values in Table . Error bars: $\pm 1$ s.d. (b) $P_x$ at resonance as function of $N$. A fit using the values in Table  (line) yields a decay constant for the oscillation of 1.8(3) ms (supplementary information). The top horizontal axis shows the total length of the gate. Uncertainties are 2 s.d.Reuse & Permissions