Coherent Control of Two Nuclear Spins Using the Anisotropic Hyperfine Interaction

We demonstrate coherent control of two nuclear spins mediated by the magnetic resonance of a hyperfine-coupled electron spin. This control is used to create a double-nuclear coherence in one of the two electron spin manifolds, starting from an initial thermal state, in direct analogy to the creation of an entangled (Bell) state from an initially pure unentangled state. We identify challenges and potential solutions to obtaining experimental gate fidelities useful for quantum information processing in this type of system.

Figure 1

(a), upper: Schematic of the electron-${}^{13}\mathrm{C}\mathrm{\text{−}}{}^1\mathrm{H}$ system on the malonic acid radical. The electron occupies a $\pi$ orbital oriented along the $\widehat{a}$ axis, perpendicular to the ${}^{13}\mathrm{C}\mathrm{\text{−}}{}^1\mathrm{H}$ bond along the $\widehat{c}$ axis of the Cartesian $(a,b,c)$ coordinate system shown ($\widehat{b}$ and $\widehat{c}$ are in the plane of the three carbon atoms). (a), lower: Table of Hamiltonian parameters for the crystal orientation used in this work, with external magnetic field direction $\widehat{B_0}=(-0.55,0.27,0.79)$. Larmor frequencies appear along the diagonal, and hyperfine coupling coefficients are off-diagonal, with all frequencies in MHz. The two columns on the far right list observed $T_1$ and $T_2$ relaxation times in microseconds. (b) Field-swept ESR spectrum with experimental data (dotted black line) and best-fit simulation (solid red line). Additional peaks near the central region of the spectrum are due to another defect present in the crystal and may be ignored. The ${}^{13}\mathrm{C}$ and ${}^1\mathrm{H}$ splittings are ${\Omega }_C=122\mathrm{MHz}$ and ${\Omega }_H=48\mathrm{MHz}$, respectively. (c) Upper panel shows three-pulse ESEEM data recorded with hard pulses resonant with the transition at $+37\mathrm{MHz}$; in the lower panel, its Fourier transform shows the two ${}^1\mathrm{H}$ transition frequencies ${\omega }_{\uparrow H}=11.99\pm 0.01\mathrm{MHz}$ and ${\omega }_{\downarrow H}=36.35\pm 0.04\mathrm{MHz}$ corresponding to electron spin-up and spin-down states, respectively.

Figure 2

(a) Energy level diagram for the three-spin system showing transition frequencies (in MHz) between eigenstates. $|\uparrow ⟩$ and $|\downarrow ⟩$ label the electron manifolds. Note that the nuclear quantization axes depend on the electron spin state, so that nuclear eigenstates in the two electron spin manifolds are different. (b) Schematic of the experimental sequence for creating and detecting double-nuclear coherence in the electron spin-down manifold. Steps 1 and 4 refer to driving direct electronic transitions, while steps 2 and 3 involve an operation similar to a Hadamard gate (labeled “GRAPE $\pi /2$ pulse”) between the two nuclear sublevels $|01⟩$ and $|10⟩$.

Figure 3

(a) The double-nuclear coherence echo signal versus delay time in experiment (red crosses, solid line) and simulation (blue dashed line) for the 22 MHz transition in the electron spin-down manifold. (b) The Fourier transform of the oscillatory signal for the 22 MHz transition experiment (left) and for the analogous experiment performed on the 52 MHz transition in the electron spin-up manifold (right).