Classical and quantum controllability of a rotating symmetric molecule

TL;DR

This paper investigates the controllability of a symmetric-top molecule's classical and quantum rotational dynamics under electric fields, revealing conditions for controllability and approximate controllability through Lie-Galerkin methods.

Contribution

It characterizes controllability conditions based on dipole orientation and introduces a Lie-Galerkin approach for quantum approximate controllability analysis.

Findings

Classical and quantum controllability depend on dipole orientation.

Controllability is lost when dipole aligns with the symmetry axis.

Approximate quantum controllability is achieved using Lie-Galerkin methods.

Abstract

In this paper we study the controllability problem for a symmetric-top molecule, both for its classical and quantum rotational dynamics. The molecule is controlled through three orthogonal electric fields interacting with its electric dipole. 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 approximate controllability of the symmetric-top Schroedinger equation is established by using a Lie-Galerkin method, based on block-wise approximations of the infinite-dimensional systems.