The Fourier Space Representation

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] 

The fact that we can express the propagator in this form allows us also to expand the reduced density operator in the RF interaction frame as 

At this stage in our discussion it becomes convenient to represent the Hamiltonian and the density operator in a Fourier matrix representation defined by a set of dressed Fourier states n,k In this infinite representation we 

These operators enable a compact Fourier matrix representation of the Hamiltonian and the density matrix  

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