Composite Pulses Enhance Precision in Magnetic Resonance and Computing

The precise manipulation of quantum systems relies heavily on techniques that maintain coherence despite external disturbances, a challenge addressed by composite pulses. These carefully designed pulses refocus ensembles of quantum particles, mitigating the effects of field imperfections and resonance offsets, and are fundamental to applications ranging from nuclear magnetic resonance spectroscopy to quantum computing. Jonathan Berkheim from the Weizmann Institute of Science, David J. Tannor, and colleagues now present a classical mechanical justification for the robustness of these composite pulses, detailed in their article, “The robustness of composite pulses elucidated by classical mechanics: Stability around the globe”. Their work maps the motion of quantum systems onto the Bloch sphere, a geometrical representation of quantum states. It demonstrates how refocusing corresponds to the behaviour of stability matrices within a canonical coordinate system, offering new insight into the directionality and behaviour of these pulses.

Scientists at the Institute present a novel approach to understanding the robustness of composite pulses, framing the analysis within established principles of classical mechanics and, specifically, through the concept of stability matrices. Composite pulses are sequences of electromagnetic pulses designed to manipulate the quantum states of systems like nuclear spins, and their robustness—the ability to perform reliably despite imperfections—is crucial for applications in magnetic resonance imaging and quantum computing. Researchers map the quantum mechanical evolution of these systems onto the Bloch sphere—a geometrical representation of a qubit’s state—and then translate this onto a canonical system of coordinates, effectively creating a classical mechanical analogue of quantum dynamics. This allows for a new perspective on complex quantum phenomena.

The study demonstrates a direct link between the stability of the spin ensemble’s trajectory in phase space—the space defined by the possible states of the system—and the successful refocusing achieved by composite pulses. Refocusing is the process of bringing a dispersed ensemble of spins back into a coherent state, enhancing signal strength. This is characterised by the formation of caustics, points where trajectories converge, and linked to the vanishing of specific elements within a stability matrix defined in the canonical coordinate system. Crucially, this framework elucidates the directional nature of ensemble refocusing, demonstrating that different ensembles refocus along distinct trajectories, a nuance often overlooked in traditional analyses. This provides a more complete picture of pulse behaviour and allows for more precise control.

Researchers demonstrate that this classical mechanical perspective clarifies when composite pulses induce a change in the ensemble’s width—its spread in state space—rather than simply rotating its orientation. This distinction is critical for understanding the limitations and potential of composite pulses in maintaining signal integrity. A change in width indicates a loss of coherence, while rotation preserves it. By understanding which effect dominates, scientists can optimise pulse sequences for specific applications.

The analysis moves beyond traditional quantum mechanical treatments by reformulating the behaviour of composite pulse sequences within a classical mechanical framework. Leveraging concepts from Hamiltonian mechanics—the study of energy in classical systems—stability analysis, and the theory of caustics, the research offers a complementary perspective on composite pulse robustness and effectiveness. The central finding establishes a direct link between the stability of the spin ensemble’s trajectory in phase space and the successful refocusing achieved by composite pulses, providing a new way to assess and optimise pulse sequences for specific applications.

Future research will focus on extending this framework to more complex pulse sequences and exploring its potential applications in other areas of quantum control, potentially leading to the development of more robust and efficient quantum technologies. Scientists plan to investigate the limitations of this classical mechanical analogy and explore ways to incorporate quantum effects into the model, further refining the understanding of composite pulse behaviour. They also aim to develop computational tools that can efficiently simulate the dynamics of spin ensembles under the influence of composite pulses, enabling the design of optimised pulse sequences for specific applications.