Single Spin Measurement Using Cellular Automata Techniques

We analyze a conceptual approach to single-spin measurement. The method uses techniques from the theory of quantum cellular automata to correlate a large number of ancillary spins to the one to be measured. It has the distinct advantage of being efficient: under ideal conditions, it requires the application of only $O(\sqrt[3]{N})$ steps (each requiring a constant number of rf pulses) to create a system of $N$ correlated spins. Numerical simulations suggest that it is also, to a certain extent, robust against pulse errors, and imperfect initial polarization of the ancilla spin system.

Figure 1

Cube lattice: a crystal with two types of nuclei $A$ and $B$, one represented as light gray spheres, the other dark gray. Each species $A$ is neighbored by only $B$ type nuclei and vice versa. The lines connecting the spheres represent the nearest-neighbor couplings. The white sphere represents the spin we wish to measure. This spin is coupled to the dark gray nucleus in the top-left vertex.

Figure 2

Pyramid lattice: a different view on the same structure as Fig. 1. The top gray sphere is layer 1 and corresponds to the top-left vertex spin in Fig. 1. The three light gray spheres directly below it are layer two, and so on. The lines connecting the spheres represent the nearest-neighbor couplings, and are exactly the same as in Fig. 1.

Figure 3

Ideal NMR spectrum: this is the ideal spectrum for a spin on the second layer. There are 5 distinct peaks, one for each of the five neighbor field values.

Figure 4

Suppressing homonuclear coupling: this is the spectrum for a spin on the second layer of a rhombohedral lattice with angles $\alpha =\beta =\gamma =\pi /3$, with the homonuclear couplings suppressed. Note the similarity to the ideal case.

Figure 5

Pulse Sequence: rf power as a function of time. By choosing $\tau =1/(10{\omega }_d)$ and $T=1/(2.1{\omega }_d)$ the frequency profile of this pulse sequence (obtained by a Fourier transform) is an excitation of the frequencies ${\omega }_{\pm 6}\to {\omega }_{\pm 4}$. By choosing the total rotation angle of these pulses to be $\pi$, we can rotate just spins $B$ which have frequencies ${\omega }_{\pm 3}\to {\omega }_0$ as required by the algorithm. This pulse sequence is presented only as an example, as more complex pulse sequence can be designed to improve further these results.