Stereographic projection to and from the Bloch sphere: Visualizing solutions of the Bloch equations and the Bloch–Riccati equation

Abstract

Stereographic projection mapping is typically introduced to explain the point at infinity in the complex plane. After this brief exposure in the context of complex analysis, students rarely get an opportunity to fully appreciate stereographic projection mapping as an elegant and powerful technique on its own with many fruitful applications in the physical sciences. Here, using a classical description of nuclear magnetic resonance in the rotating frame, I show how stereographic projection mapping to and from the Bloch sphere can be used for visualizing solutions to Bloch's equation and the Bloch–Riccati equation, respectively. After developing the fundamentals of stereographic projection mapping using examples drawn from nuclear spin precession in the rotating frame, the method is then applied to visualizations of composite pulse excitation of a spin-1/2 system and to radiation damping in a system of isolated spins-1/2. In the case of the radiation-damped system, these visualizations provide particularly vivid illustrations of loxodromic Möbius transformation dynamics.