Universal control of nuclear spins via anisotropic hyperfine interactions

We propose a method for completely controlling a nuclear spin subsystem using only microwave irradiation of resolved anisotropic hyperfine interactions with a single electron spin. This paradigm of control has important applications for spin based solid-state quantum information processing. In particular we argue that indirect addressing of the nuclear spins via an electron spin acting as a spin actuator allows for nuclear spin quantum logic gates whose operational times are significantly faster than either gates based on rf fields resonant with nuclear spin flips or nuclear-nuclear dipolar interactions. We demonstrate experimental aspects of this method with one electron and one nuclear spin of a single crystal of irradiated malonic acid.

Figure 1

(Color online) (a) The electron spin state is aligned (or antialigned) with the quantization axis given by $\vec{B}=B_0\widehat{z}$, while the nuclear spin state aligns along the local field. When the electron spin is $\mid \uparrow ⟩$, the nuclear spin quantization axis, $\mid {\alpha }_0⟩$ is a vector sum of $A_{zz}$, ${\omega }_I$ along $\widehat{z}$ and $A_{\perp }$ along $\widehat{x}$ drawn as double arrows. $(A_{\perp }=\sqrt{A_{zx}^2+A_{zy}^2})$. When the electron spin is $\mid \downarrow ⟩$, the sign of $A_{zz}$ and $A_{\perp }$ is negative, yielding a different nuclear quantization axis $\mid {\beta }_0⟩$. (b) Since $⟨{\beta }_m\mid {\alpha }_l⟩\ne 0$, electron spin flips induced by the ${\widehat{S}}_x$ operator drive many transitions (dashed arrows) between energies of ${\mathcal{H}}_0$. This allows for universal control of the entire spin system. In our experimental setup $(N=1)$ the energy differences are ${\omega }_{12}∕2\pi =7.8\mathrm{MHz}$, ${\omega }_{34}∕2\pi =40\mathrm{MHz}$, ${\omega }_{14}∕2\pi =12.005\mathrm{GHz}$, ${\omega }_{23}∕2\pi =11.954\mathrm{GHz}$.

Figure 2

The connectivity of energy levels of $1e\text{−}2n$ system. When the control Hamiltonian is applied resonant with a single transition (bold line) it excites detuned transitions between the levels of the electron spin up and spin down manifolds (double line). An edge is drawn between two eigenstates if the control Hamiltonian operator has a nonzero matrix according to $[⟨m\mid {\widehat{S}}_x\mid l⟩\ne 0]$. (a) If one of the hyperfine tensors between a nuclear spin is isotropic, universality does not occur. The control operator induces transitions that join two sets of four levels (gray and black), but does not join all levels. (b) Full connectivity is achieved when both hyperfine interactions are anisotropic.

Figure 3

(Color online) Schematic pulse sequence for measuring Ramsey fringes. $U_{\mathrm{p}}$ creates a nonequilibrium population difference between levels 1 and 2, and $U_{\mathrm{c}}$ creates a coherence of the nuclear spin in the $S=-1∕2$ manifold. During $\tau$, this coherence evolves under ${\mathcal{H}}_0$, acquiring an observable phase. The coherence is transformed back to a population difference between nuclear spin levels and then to electron spin levels. A pair of short unmodulated pulses are used to detect an electron spin echo whose height is proportional to the resultant electron spin population. The wave form used to implement $U_{\mathrm{p}}{U}_{\mathrm{c}}$ is shown in the inset. Note that all pulses have a carrier frequency resonant with the 1–4 transition $(\Omega ∕2\pi =11.909\mathrm{GHz})$ and induce transitions between 1–4, 2–4, 1–3, and 2–3 via time-dependent amplitude modulations.

Figure 4

(Color online) Measurements of the electron spin echo as a function of the time $\tau$ between coherence transfer indirectly reveal the nuclear precession rate. Numerical simulations of the experiments (solid line) show agreement with the observed signal (dashed). The Ramsey fringe experiment $(\times )$ reveals a clear precession of the coherence between the $\mid 1⟩$ and $\mid 2⟩$ states at roughly $8\mathrm{MHz}$.