Tomography vs quantum control for a three-level atom

We investigate the possibilities of controlling and reconstructing the state of a single three-level atom. We propose a physical scheme where information about the atomic state is extracted by measuring the total number of excitations after successive application of electromagnetic field pulses. We show that, in the non-degenerate case (different transition frequencies for different atomic transitions), a three-level atom is completely controllable and its state can be completely reconstructed. In the degenerate case (when both atomic transitions are identical), we consider two dynamically inequivalent configurations, Λ and Ξ. In this case, we show that the density matrix can always be completely reconstructed whereas their respective system cannot be completely controlled. We explain why this last incompatibility between control and tomography arises.