Floquet Hamiltonian theory

The efficient combination of RF pulses with MAS requires theoretical tools. While the initial multiple-pulse sequences were based on the average Hamiltonian theory (AHT) [14,25,26,32,33], some of the later schemes were more effectively formulated on the basis of Floquet theory [52], combinations of both AHT and Floquet theory [51], synchronisation arguments [42,43,46,55], and numerical methods [50]. A combination of RF pulses with MAS was also visualised in terms of certain symmetry conditions based on AHT [77]. An attempt to present all these in a unified picture is perhaps not out of place for a better understanding of the underlying phenomena. 

The models for the control processes start with the Schrodinger equation for the molecule in interaction with a laser field that is treated either as a classical or as a quantized electromagnetic field. In Section II we describe the Floquet formalism, and we show how it can be used to establish the relation between the semiclassical model and a quantized representation that allows us to describe explicitly the exchange of photons. The molecule in interaction with the photon field is described by a time-independent Floquet Hamiltonian, which is essentially equivalent to the time-dependent semiclassical Hamiltonian. The analysis of the effect of the coupling with the field can thus be done by methods of stationary perturbation theory, instead of the time-dependent one used in the semiclassical description. In Section III we describe an approach to perturbation theory that is based on applying unitary transformations that simplify the problem. The method is an iterative construction of unitary transformations that reduce the size of the coupling terms. This procedure allows us to detect in a simple way dynamical or field induced resonances—that is, resonances that... 

Mananga presents the possibility of applying the Floquet-Magnus expansion to the most useful interaetions known in solid-state NMR using the magic-echo scheme. The results of the effective Hamiltonians of these theories and average Hamiltonian theory are presented. ... 

Floquet theory principles, 35—36 single-surface nuclear dynamics, vibronic multiplet ordering, 24—25 Barrow, Dixon, and Duxbury (BDD) method, Renner-Teller effect tetraatomic molecules, Hamiltonian equations, 626-628 triatomic molecules, 618-621 Basis functions ... 

In this section we introduce bimodal Floquet theory (BMFT), based on which we represent the spin Hamiltonians and analyse the various homonuclear dipolar decoupling sequences. 

Evaluation of the response of the spin system to a time dependent Hamiltonian requires an appropriate mathematical framework. This framework must deal with Hamiltonians that are periodically time dependent with at least two characteristic frequencies, and Wc, that are not necessarily commensurate. We choose bimodal Floquet theory (BMFT) towards this, and in this Section we will set the basis of this theory. The approach is very similar to the single mode Floquet theory (SMFT) approach adapted by others to NMR spectroscopy [91]. 

Floquet theory provides a generalised form for the propagators of systems with periodically time dependent Hamiltonians [90,91]. The propagator for a doubly periodic Hamiltonian in the BMFT representation maybe written as [92, 93]... 

Since in the Floquet representation the Hamiltonian K defined on the enlarged Hilbert space is time-independent, the analysis of the effect of perturbations (like, e.g., transition probabilities) can be done by stationary perturbation theory, instead of the usual time-dependent one. Here we will present a formulation of stationary perturbation theory based on the iteration of unitary transformations (called contact transformations or KAM transformations) constructed such that the form of the Hamiltonian gets simplified. It is referred to as the KAM technique. The results are not very different from the ones of Rayleigh-Schrodinger perturbation theory, but conceptually and in terms of speed of convergence they have some advantages. 

An introduction to Floquet theory has been presented. The potential of this theoretical approach has been demonstrated using explicit calculations of the sideband patterns in MAS NMR. It has been shown that the Floquet theory works by expanding the periodic (due to sample spinning) Hamiltonian into a Fourier series, and that, regardless of the complexity of the time dependence of the Hamiltonian, the Floquet approach is the same. 

The use of van Vleck s contact transformation method for the study of time-dependent interactions in solid-state NMR by Floquet theory has been proposed. Floquet theory has been used for studying the spin dynamics of MAS NMR experiments. The contact transformation method is an operator method in time-independent perturbation theory and has been used to obtain effective Hamiltonians in molecular spectroscopy. This has been combined with Floquet theory to study the dynamics of a dipolar coupled spin (I = 1/2) system. 

A theoretical treatment of the DREAM adiabatic homonuclear recoupling experiment has been given using Floquet theory. An effective Hamiltonian has been derived analytically and the time evolution of the density operator in the adiabatic limit has been described. Shape cycles have been proposed and characterized experimentally. Application to spin-pair filtering as a mixing period in a 2D correlation experiment has been explored and the experimental results have been compared to theoretical predictions and exact numerical simulations. 

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