From theory to practice: Floquet-Magnus and Fer expansions in triple oscillating field NMR

Abstract

The Floquet-Magnus expansion (FME) and Fer expansion (FE) schemes were introduced in solid-state nuclear magnetic resonance (NMR) in 2011 and 2006, respectively. Key features of the Floquet-Magnus expansion are its ability to account for the calculations developed in a finite-dimensional Hilbert space instead of an infinite-dimensional space within the Floquet theory as well as its use of its distinguishable function, ${\Lambda }_n\left(t\right),n=1,2,3,\dots$, not available in other concurrent theories such as average Hamiltonian theory, Floquet theory, and Fer expansion. The distinguishable function facilitates the evaluation of the spin behavior in between the stroboscopic observation points. This paper provides an in-depth analysis of both the FME and FE methods and integrates them with the Triple Oscillating Field Technique (TOFU) in solid-state NMR. This is a significant and novel contribution as it presents a unified framework for explaining spin dynamics. The use of both FME and FE provides new theoretical insights and extends the applicability of these methods beyond traditional approaches. The application to the TOFU technique, which circumvents the dipolar truncation problem, indicates substantial practical implications for distance measurement in solid-state NMR, a critical aspect of molecular structure determination. We take advantage of the interaction frequencies and the time modulation arising from the TOFU pulse sequence, which allows selective recoupling of specific terms in the Hamiltonian that fulfill determined specific conditions. The work presented unifies and generalizes the results of the FME and FE and delivers illustrations of novel insights that boost previous applications that are based on the classical information. We believe that the revisited approaches in this work and the derived expressions can serve as useful information and numerical tools for time evolution in spin dynamics, time-resolved spectroscopy, quantum control, and quantum dynamics[81,82].