Closed solution to the Baker-Campbell-Hausdorff problem: exact effective Hamiltonian theory for analysis of nuclear-magnetic-resonance experiments.

A closed solution to the Baker-Campbell-Hausdorff problem is described. The solution, which is based on the Cayley-Hamilton theorem, allows the entanglement between exponential operators to be described by an exact finite series expansion. Addressing specifically the special unitary Lie groups SU(2), SU(3), and SU(4), we derive expansion formulas for the entangled exponential operator as well as for the effective Hamiltonian describing the net evolution of the quantum system. The capability of our so-called exact effective Hamiltonian theory for analytical and numerical analysis is demonstrated by evaluation of multiple-pulse methods within liquid- and solid-state nuclear-magnetic-resonance spectroscopy. The examples include composite pulses for inversion, decoupling, and dipolar recoupling, as well as coherence-order- and spin-state-selective double- to single-quantum conversion, homonuclear dipolar decoupling, finite rf excitation for quadrupolar nuclei, heteronuclear coherence transfer, and gates for quantum computation.