Dynamics and control of a quasi-one-dimensional spin system

We study experimentally a system comprised of linear chains of spin-1/2 nuclei that provides a test bed for multibody dynamics and quantum-information processing. This system is a paradigm for a class of quantum-information processing devices that can perform particular tasks even without universal control of the whole quantum system. We investigate the extent of control achievable on the system with current experimental apparatus and methods to gain information on the system state, when full tomography is not possible and in any case highly inefficient.

Figure 1

(Color online) Comparison of the spectra when the polarization is retained by all spins in the chain (dashed line) and for the excitation of the extremities only (solid line). The full width at half maximum is $\approx 19\mathrm{kHz}$ and the distance between peaks $\approx 8\mathrm{kHz}$. The experimental data were obtained with sequence (4) and a solid echo readout. The wait time $t_1$ was $t_1=0.5\mathrm{\mu }\mathrm{s}$ for the all-spin spectrum and $30.5\mathrm{\mu }\mathrm{s}$ for the chain-end excitation. A small Teflon signal (from the tuning capacitors) can be seen in the end-of-chain signal in the $10$–$20\mathrm{kHz}$ region.

Figure 2

(Color online) Zero- and double-quantum intensities as a function of the evolution time under the double-quantum Hamiltonian. Nearest-neighbor couplings only are assumed, with equal strength as given by the fitting to experimental data (see Fig. 3). In particular notice the clear differences in the behavior for the two initial states. Also the even-odd spin number dependence of the MQC intensities is interesting: while this tends to go to zero for large number of spins in the collective initial-state case, this difference is observed even for very large number of spins for the other initial state.

Figure 3

(Color online) Experimental results. Top: the initial state is the collective thermal state ${\sum }_k{\sigma }_z^k$. The experimental points have been fitted (solid line) to the theoretical curves for nearest-neighbor coupling only, with the dipolar coupling as fitting parameter. The number of spins was varied to find the best fit, which results in being $N=11$ spins. Bottom: MQC intensities for the initial state ${\rho }_0={\sigma }_z^1+{\sigma }_z^N$. Also plotted are the theory predictions for the same dipolar coupling and 11 spins (solid line) or 10 spins (dotted line). The experimental behavior observed (a constant behavior also for longer time) indicates that a simple model with 11 spin chains is not an accurate description at longer times (see text).