Analytically solvable two-level quantum systems and Landau-Zener interferometry

A simple algorithm is presented based on a type of partial reverse engineering that generates an unlimited number of exact analytical solutions to the Schr\"odinger equation for a general time-dependent two-level Hamiltonian. I demonstrate this method by deriving exact solutions corresponding to fast control pulses that contain arbitrarily many tunable parameters. It is shown that the formalism is naturally suited to generating analytical control protocols that perform precise nonadiabatic rapid passage and Landau-Zener interferometry near the quantum speed limit. A general, exact formula for Landau-Zener interference patterns is derived.