Ensemble controllability of the Bloch equations

Finding control fields (pulse sequences) that can compensate for the dispersion in the parameters governing the evolution of a quantum system is an important problem in coherent spectroscopy and quantum information processing. The use of composite pulses for compensating dispersions in system dynamics is widely known and applied. In this paper, we make explicit the key aspects of the dynamics that make such a compensation possible. We highlight the role of Lie algebras and non-commutativity in the design of a compensating pulse sequence. Finally, we investigate three common dispersions in NMR spectroscopy, the Larmor dispersion, rf inhomogeneity and the strength of couplings between the spins.