The robustness of composite pulses elucidated by classical mechanics. II. The role of initial state distribution

In nuclear magnetic resonance (NMR), composite pulses (CPs) are widely used to correct for pulse imperfections, e.g., RF field inhomogeneity and resonance offset. In previous work, we developed a classical canonical framework to perform stability analysis and used this as a measure of population inversion robustness. In that work, a single initial condition was allowed to evolve under various pulse imperfections. The current work extends this approach to 2D distributions of initial conditions on the Bloch sphere; the objective is to minimize the area in order to preserve coherence, while maximizing population inversion of the entire distribution. While a body of work discusses the inversion of arbitrary initial states via the full unitary propagator, the systematic error introduced by an ensemble of initial conditions in point-to-point population inversion pulses has not been addressed with geometrical tools. As a case study, we first investigate Levitt's 90(x)180(y)90(x) pulse sequence, when there is a spread in initial conditions. The canonical framework enables us to assess the projected-area robustness of Levitt's pulse sequence, and we find that it is maintained to a great extent even when considering a spread of initial conditions. Nevertheless, by conducting a numerical optimization, we have identified several variants of Levitt's pulse sequence that produce a larger coherent population inversion when there is a spread in initial conditions. We then apply the methodology to constant-rotation pulses to give geometrical insight into their robustness and to contrast their mechanism with that of the distributed point-to-point dynamics.