Spectra and dynamics of driven linear quantum rotors. Symmetry analysis and algebraic methods

The development of laser technologies and techniques to cool atoms and molecules has opened new possibilities to manipulate and control quantum matter by tailored external fields. In particular, quantum rotors subject to external fields have enjoyed considerable interest of both the physics and chemistry communities. Furthermore, experimental as well as theoretical investigations of the dynamics of driven quantum rotors have provided insight into fundamental aspects of quantum dynamics, such as revival structures, quantum-classical correspondence, and quantum chaos. Herein, we investigate, mostly analytically, the spectral properties as well as the dynamics of a two-dimensional (planar) linear quantum rotor exposed to external orienting and aligning fields. Supplemented by numerical computations that reach beyond what is achievable analytically, we paint quite a complete picture of the physics of the driven planar rotor. The conditionally quasi-exact solvability of the corresponding eigenproblem, termed the generalized planar pendulum, enables obtaining the lower part of the generalized pendulum's spectrum analytically. However, this is only possible under certain conditions imposed on the parameters that characterize the strengths of orienting and aligning interactions. By making use of Lie-algebraic and group theoretical methods, we calculated all algebraically obtainable eigenvalues and eigenfunctions of the generalized pendulum's eigenproblem based on the solutions that had been obtained previously via supersymmetric quantum mechanics. Furthermore, an analogous analysis of the hyperbolic counterpart of the generalized planar pendulum, the Razavy problem, confirmed that the normal and hyperbolic pendula are anti-isospectral, but only in the analytic part of their spectra. Moreover, our analysis of the generalized pendulum problem revealed a relationship between its conditional quasi-exact solvability and the topology of eigenenergy surfaces: the intersections (genuine and avoided) of the eigenenergy surfaces spanned by the dimensionless aligning and orienting interaction parameters arise for certain ratios of these parameters. These ratios coincide with those for which analytic solutions to the generalized pendulum eigenproblem exist. Building on our knowledge of the analytic stationary states and aided by our numerical calculations, we investigated the dynamics that arises from the interaction of a rigid rotor with aligning and orienting fields that have particular temporal shapes: an ultra-short pulse ($\delta$-kick) and rapidly switched-off or switched-on pulses. For these types of pulses, we found analytic expressions for the resulting wavefunctions, kinetic energies, and the evolution of orientation and alignment. In addition, the revival structure of the resulting wavepackets and the quantum-classical correspondence is captured by the space-time dependence of the probability density known as quantum carpets. In the sudden limit, i.e., when the duration of the pulse is much shorter than the rotational period of the rotor, a combination of orienting and aligning interactions produces a unique pattern in post- pulse populations of the rotational states, captured by diagrams termed population quilts. Accordingly, for each of the orienting, aligning and combined interactions, the time- evolutions of the orientation and alignment differ considerably. The analysis conducted in the case of rapidly switched fields revealed a significant influence of the spectral properties and eigensurface topology of the generalized pendulum on the post-switch populations as well as the time-evolution of the expectation values and the corresponding revival structure.