Visualization of Spin Dynamics in Coupled Nuclear Multispin Systems

Abstract. Since the dawn of quantum mechanics ways to visualize spins and their interactions attracted attention of researchers and philosophers of science. Angular momentum probability (AMP) surfaces are known for visualizing density matrices; a plotted surface represents probability of finding maximum projection of angular momentum along any direction in space. While AMP surfaces visually reveal symmetries of density matrices which directly translate to measured properties, they focus solely on one total-angular-momentum manifold, neglecting interaction between different manifolds often encountered in the NMR of multispin systems. In this work, we extend applicability of this visualization method and introduce its extension, angular momentum coherence (AMC) surface approach. Three examples of nuclear spin dynamics in two-spin systems are presented and visualized using the AMP/AMC surfaces: (I) spin-lock induced crossing (SLIC) sequence with initial state being the singlet state between two spins; (II) interplay between chemical exchange and spin dynamics during the high-field signal amplification by reversible exchange (SABRE) experiment; (III) zero- to ultralow-field (ZULF) NMR experiment in the presence of magnetic field applied perpendicular to the sensitive axis of the detector. The presented visualization technique extends applicability of AMP surfaces to coupled multispin systems and will facilitate intuitive understanding of spin dynamics during complex NMR experiments as exemplified here by the considered cases. The temporal sequences (“the movies”) of such surfaces show phenomena like interconversion of polarization moments (Auzinsh et al., 2010) and, as particularly relevant in NMR and demonstrated here, polarization transfer between different spins. Such effects are difficult to grasp by looking at (time-dependent) numerical values of density-matrix elements.