Switched control of electron nuclear spin systems

We study control of electron-nuclear spin dynamics at magnetic field strengths where the Larmor frequency of the nucleus is comparable to the hyperfine coupling strength. The quantization axis for the nuclear spin differs from the static $B_0$ field direction and depends on the state of the electron spin. The quantization axis can be switched by flipping the state of electron spin, allowing for universal control of nuclear spin states. We show that by performing a sequence of flips (each followed by a suitable delay), we can perform any desired rotation of the nuclear spin, which can also be conditioned on the state of the electron spin. These operations, combined with electron spin rotations, can be used to synthesize any unitary transformation of the coupled electron-nuclear spin system. We discuss how these methods can be used for design of experiments for transfer of polarization from the electron to the nuclear spins.

Figure 1

(a) depicts the field vectors ${\mathbf{B}}_{\alpha }$ and ${\mathbf{B}}_{\beta }$ when $\mid {\omega }_I\mid >\frac{A}{2}$ and $A>0$. (b) depicts the corresponding energy-level diagram for the electron-nuclear spin system. Note ${\omega }_{\beta }>{\omega }_{\alpha }$.

Figure 2

(Color) (A) shows the trajectories of nuclear spin on the Bloch sphere, as it precesses around the ${\mathbf{d}}_{\alpha }$ and ${\mathbf{d}}_{\beta }$ directions. This trajectory plot aids in visual construction of rotation between points on the sphere by alternation between rotations around ${\mathbf{d}}_{\alpha }$ and ${\mathbf{d}}_{\beta }$. (B) depicts one such construction (projected on the plane of axes ${\mathbf{d}}_{\alpha }$ and ${\mathbf{d}}_{\beta }$) that rotates unit vector ${\mathbf{d}}_{\alpha }$ to the unit direction $\mathbf{n}$. The initial rotation is around axis ${\mathbf{d}}_{\beta }$ for time ${\tau }_1$, followed by rotation around ${\mathbf{d}}_{\alpha }$ for time ${\tau }_2$. The times ${\tau }_1$, ${\tau }_2$ can be explicitly computed from the figure based on the algorithm described above.

Figure 3

(Color) (A) depicts the trajectory executed by nuclear spin magnetization, initially oriented along the direction ${\mathbf{d}}_{\beta }$, when the pulse sequence in (B) is implemented. The transformation is the inversion of the nuclear spin resulting in the transformation $\mid \beta +⟩\to \mid \beta -⟩$. (C) shows that the desired rotation (shown with solid arrow) $\mid \beta +⟩\to \mid \beta -⟩$ is implemented by switching the electron between $\alpha$ and $\beta$ manifolds.

Figure 4

Depiction of the geometry of vectors $\mathbf{n}$, $\left({\mathbf{n}}_c\right)$, ${\mathbf{n}}_{\perp }$, ${\left({\mathbf{n}}_c\right)}_{\perp }$, and ${\mathbf{d}}_{\alpha }$.