Hamiltonian design to prepare arbitrary states of four-level systems

We propose a method to manipulate, possibly faster than adiabatically, four-level systems with time-dependent couplings and constant energy shifts (detunings in quantum-optical realizations). We inversely engineer the Hamiltonian, in ladder, tripod, or diamond configurations, to prepare arbitrary states using the geometry of four-dimensional rotations to set the state populations, specifically we use Cayley's factorization of a general rotation into right- and left-isoclinic rotations.