  The Floquet-Magnus and Fer expansion 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 from other theories such as average Hamiltonian
theory, Floquet theory, and Fer expansion, which facilitates the evaluation of
the spin behavior in between the stroboscopic observation points. This article
focuses on revisiting the Floquet-Magnus and Fer expansion approaches and
applying both methods to calculate the effective Hamiltonians and propagators,
which control the spin system evolution during the Triple Oscillating Field
Technique radiation experiment (TOFU). The TOFU pulse sequence is an important
technique that was shown to avoid the dipolar truncation problem and form a new
basis for accurate distance measurement by solid-state NMR. We take advantage
of the interaction frequencies and the time modulation arising from the TOFU
pulse sequence allowing selective recoupling of specific terms in the
Hamiltonian that fulfill determined specific conditions. The work presented
unifies and generalizes existing results of the Floquet-Magnus and Fer
expansions and delivers illustrations of novel springs that boost previous
applications that are based on the classical information. We believe that the
revisited approaches of this work and the derived expressions can serve as
useful information and numerical tools for time evolution in time-resolved
spectroscopy, quantum control, and open system quantum dynamics.
