The driven quantum asymmetric top is an important paradigm in molecular physics with applications ranging from quantum information to chiral-sensitive spectroscopy. A key prerequisite for these applications is the ability to completely control the rotational dynamics. The inherent degeneracy of quantum rotors poses a challenge for quantum control since selecting a particular rotational state cannot be achieved by spectral selection alone. Here, we prove complete controllability for rotational states of an asymmetric top belonging to degenerate values of the orientational quantum number M. Based on this insight, we construct a pulse sequence that energetically separates population in degenerate M-states. Introducing the concept of enantio-selective controllability, we determine the conditions for complete enantiomer-specific population transfer in chiral molecules and construct pulse sequences for the example of propanediol and carvone molecules for population initially distributed over degenerate M-states. Our work shows how to leverage controllability analysis for the solution of practical quantum control problems.

In this paper we study the controllability problem for a symmetric-top molecule, both for its classical and quantum rotational dynamics. As controlled fields we consider three orthogonally polarized electric fields which interact with the electric dipole of the molecule. We characterize the controllability in terms of the dipole position: when it lies along the symmetry axis of the molecule neither the classical nor the quantum dynamics are controllable, due to the presence of a conserved quantity, the third component of the total angular momentum; when it lies in the orthogonal plane to the symmetry axis, a quantum symmetry arises, due to the superposition of symmetric states, which has no classical counterpart. If the dipole is neither along the symmetry axis nor orthogonal to it, controllability for the classical dynamics and approximate controllability for the quantum dynamics are proved to hold. The controllability properties of the classical rotational dynamics are analyzed by applying geometric control theory techniques. To establish the approximate controllability of the symmetric-top Schrödinger equation we use a Lie-Galerkin method, based on block-wise approximations of the infinite dimensional systems.