Dynamical symmetry reduction and discrete tomography of a Ξ atom

When implemented using a reasonable Hamiltonian, the tomography of a three-level 4 atom is complicated by the equidistant energy levels of the atom. This restricts the possible transformations to those in the SO(3) subgroup of SU(3). Although complete reconstruction is possible for a single 4 atom using a continuous set of tomograms, the discrete optimal set of tomograms, related to mutually unbiased bases in dimension 3, are not accessible by time evolution. We discuss here the search for an optimal set of discrete basis states compatible with the reduced SO(3) symmetry of the system.