The robustness of composite pulses elucidated by classical mechanics: Stability around the globe

TL;DR

This paper introduces a classical mechanics-based framework using stability matrices to explain the robustness of composite pulses in quantum control, providing new insights into their focusing behavior on the Bloch Sphere.

Contribution

It offers a novel classical mechanical justification for composite pulse robustness by mapping Bloch Sphere dynamics to stability matrices, revealing directionality and ensemble width effects.

Findings

The stability matrix approach clarifies ensemble refocusing mechanisms.

Different ensembles refocus along different directions.

The method explains the effectiveness of Levitt's $90(x)180(y)90(x)$ composite pulse.

Abstract

Composite Pulses (CPs) are widely used in Nuclear Magnetic Resonance (NMR), optical spectroscopy, optimal control experiments and quantum computing to manipulate systems that are well-described by a two-level Hamiltonian. A careful design of these pulses can allow the refocusing of an ensemble at a desired state, even if the ensemble experiences imperfections in the magnitude of the external field or resonance offsets. Since the introduction of CPs, several theoretical justifications for their robustness have been suggested. In this work, we suggest another justification based on the classical mechanical concept of a stability matrix. The motion on the Bloch Sphere is mapped to a canonical system of coordinates and the focusing of an ensemble corresponds to caustics, or the vanishing of an appropriate stability matrix element in the canonical coordinates. Our approach highlights the directionality of the refocusing of the ensemble on the Bloch Sphere, revealing how different ensembles refocus along different directions. The approach also clarifies when CPs can induce a change in the width of the ensemble as opposed to simply a rotation of the axes. As a case study, we investigate the $90(x)180(y)90(x)$ CP introduced by Levitt, where the approach provides a new perspective into why this CP is effective.