Figure 1

Effective nuclear spin-spin interaction. (a) Schematic diagram of the electron-mediated interaction between the nuclear spins, which exploits the magnetic dipolar interaction and the local hyperfine coupling. (b) Diagram of the energy levels of the two NV centers. Since $D_j$ is the largest energy scale, the energies are clustered in manifolds determined by the electronic spin. The transverse part of the hyperfine coupling $A_j^{\perp }$ induces transitions between different manifolds and mediates an effective $XX$ interaction between the nuclear spins. (c) Schematic diagram of the Zeeman splitting for the electronic and nuclear energy levels. By carefully selecting the microwave frequencies, we drive a particular electronic and nuclear transition. (d) Energy levels of the driven Hamiltonian. For very strong driving ${\Omega }_e$, the electronic spin in the lowest manifold is $|-⟩\propto |0⟩-|-1⟩$. The hyperfine coupling $A_j^{\parallel }$ induces virtual transitions to the excited manifold, split by the dipolar interaction and the inhomogeneous broadening, and leads to an effective $ZZ$ interaction.Reuse & Permissions

Figure 2

Nuclear quantum gate between two NV centers. Scheme for the initialization, evolution, and readout of the effective $J_{\mathrm{eff}}^{xx}$ interaction in (a), and $J_{\mathrm{eff}}^{zz}$ interaction in (b). Here, $X_{j,\varphi }=\exp (i\varphi {\tau }_j^x/2)$,$Y_{j,\theta }=\exp (i\theta {\tau }_j^y/2)$ represent finite pulses, and $\mathcal{P}$ ($\mathcal{M}$) stand for the electron (nuclear) spin polarization. (c) Comparison between the effective dynamics under the Hamiltonian (3) and the exact time evolution under Eqs. (1, 2) for the $J_{\mathrm{eff}}^{xx}$ nuclear interaction. The expectation values represented correspond to the nuclear spin $⟨I_j^z⟩$ and electronic spin $⟨S_{\mathrm{tot}}^z⟩=⟨S_1^z+S_2^z⟩$. (d) Comparison between the exact (5) and effective (6) dynamics for the $J_{\mathrm{eff}}^{zz}$ nuclear interaction, together with an echo scheme that allows us to get rid of the fast single-nuclei dynamics. We represent the nuclear expectation values $\delta ⟨{\tau }_j^x⟩=⟨{\tau }_j^x{⟩}_{t_f}-⟨{\tau }_j^x{⟩}_{t_0}$. The dotted lines correspond to $J_{12}=0$, where there is no interaction induced on the nuclei. (e) Performance of the $ZZ$ gate in the presence of different strengths of the electron dephasing noise $b_j=\{5,15,25,35,50,55\}\mathrm{kHz}$, where the nuclear noise is $B_j=0.1b_j$. The corresponding Ramsey decoherence times are roughly $T_{2e}\approx \{0.2,0.07,0.04,0.03,0.02,0.018\}\mathrm{ms}$, $T_{2n}=0.1T_{2e}$.Reuse & Permissions