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

Abstract

In nuclear magnetic resonance (NMR), Composite Pulses (CPs) are widely used to correct for pulse imperfections, e.g., RF field inhomogeneity and resonance offset. Although robust pulse sequences have been developed throughout the years, the imperfection of the initial state has not been widely discussed in the literature as an additional systematic error. In previous work, we developed a classical canonical framework to perform stability analysis and used this as a measure of CP 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. As a case study, we 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 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.

Figures (8)

• Figure 1: Preservation of volume in the extended phase space $(\phi,\eta,w)$ is not necessarily accompanied by the preservation of projected area: (a) two projections of the initial domain, each evolves under a single Hamiltonian, $H_{A}$ and $H_{B}$. The shadow area defines the union of both domains; (b) a continuous version of (a) where many sets are stacked together. The left side shows the initial domain as a cube and its projected area $A_{\text{i}}$, which consists of many square layers that perfectly overlap with each other. The right side shows the final domain as a parallelepiped, and its projected area $A_{\text{f}}$ consists of many parallelograms that do not perfectly overlap with each other, but are slightly shifted. $A_{\text{i}}$ and $A_{\text{f}}$ can be also seen as the integrations of infinitesimal areas (white-colored) $a_{\text{i}}=d\phi_{\text{i}}d\eta_{\text{i}}$ and $a_{\text{f}}=d\phi_{\text{f}}d\eta_{\text{f}}$. Overall, the projected area of the final domain is larger.
• Figure 2: A visual representation of the co-area formula. The upper panel shows a scalar function $g(x)$ with $x=(x_{1},x_{2})$, where $g$ is the color scale. The lower panel shows $g(x)$ through its level sets $u(x)=y$. The total integral $\int g(x)|\mathcal{J}_{2}|dx$ is decomposed into contributions with boundary lengths $dH_{1}$ of these level sets. The sequence shows how a continuous surface emerges from an increasingly dense stack of level-set slices, i.e. the Riemann sum is refined and the sum gets closer to the continuous co-area integral. Courtesy of Gabriel Peyré (CNRS).
• Figure 3: The time-evolution of the projected area for the ensemble of RF field inhomogeneity. The initial distribution is the black strip at the top. The first pulse segment defocuses the distribution in the $\eta$-direction and focuses it in the $\phi$-direction; the second pulse segment primarily changes the center without significant reshaping; the third pulse segment refocuses the distribution in the $\eta$-direction and defocuses it in the $\phi$-direction. The overall effect on a single IC is largely preserved when applied to the entire spread of ICs. The dashed lines mark the alpha shapes for each distribution. Inset: the same distributions presented on the Bloch Sphere.
• Figure 4: The distributions of shear coefficients $\mathcal{G}_{\text{fi}}$ for the ensemble of RF field inhomogeneity.
• Figure 5: Four sets of parameters from the $3D$ numerical scan that outperform Levitt's pulse sequence for an ensemble of ICs subject to RF field inhomogeneity. The subfigures display the time-evolution of the ensemble at $t=0,\tau_{1},\tau_{1}+\tau_{2},T$ with the same color-coding as shown before. The optimized parameters, as detailed above each subfigure, lead to a ratio coefficient $R_{30}$ smaller than with Levitt's pulse sequence.