Operator-based triple-mode Floquet theory in solid-state NMR

Many solid-state NMR experiments exploit interference effects between time dependencies in the system Hamiltonian to design an effective time-independent Hamiltonian with the desired properties. Effective Hamiltonians can be designed such that they contain only selected parts of the full system Hamiltonian while all other parts are averaged to zero. A general theoretical description of such experiments has to accommodate several time-dependent perturbations with incommensurate frequencies. We describe an extension of the analytical operator-based Floquet description of NMR experiments to situations with three incommensurate frequencies. Experiments with three time dependencies are quite common in solid-state NMR. Examples include experiments which combine magic-angle spinning and radio-frequency irradiation on two nuclei or asynchronous multiple-pulse sequences on a single spin species. The Floquet description is general in the sense that the resulting effective Hamiltonians can be calculated without a detailed knowledge of the spin-system Hamiltonian and can be expressed fully as a function of the Fourier components of the time-dependent Hamiltonian. As a prototype experiment we treat the application of two continuous-wave (cw) radio-frequency fields under magic-angle spinning. Experiments that are included in such a description are Hartmann-Hahn cross polarization or rotary-resonance recoupling experiments with simultaneous cw decoupling.