On Controllability of Λ- and V-Atoms and Other Three-Level Systems with Two Allowed Transitions

Lobachevskii Journal of Mathematics. 2023. Vol. 44. P. 2101–2108.

Kuznetsov S. A., Pechen A. N.

A three-level Λ-atom (resp., V-atom) is a three-level quantum system with a forbidden direct transition between the two lowest (resp., highest) energy states. These systems are particular examples of three-level systems with only two allowed transitions and are important in quantum optics, e.g., for modeling electromagnetically induced transparency, etc. For generality, we consider not only Λ- and V-atoms, but all other types of three-level systems with only two allowed transitions between basis states. For the most general case of arbitrary non-zero complex values of the interaction Hamiltonian, we combine in a unified way the description of the structure of the dynamical Lie algebra as well as controllability properties for such systems. It turns out that for non-degenerate transition frequencies the system is completely controllable for any non-zero interaction Hamiltonian, while for the degenerate case the system is either uncontrollable or controllable, depending on the exact values of matrix elements of the free and interaction Hamiltonians.

Keywords: controllability, quantum control, three-level system, Λ-atom, V-atom