Non-equilibrium topological phases of periodically driven molecules and quantum rotors

Rotations constitute one of the fundamental symmetries in physics, characterized by their intricate group structure and infinite dimensional representations. In contrast to classical rotations, quantum mechanics unveils the SO(3) symmetry group structure, manifesting in phenomena without classical counterparts, from angular momentum quantization to non-trivial addition of angular momenta. While most studies of topological physics have focused on two-band systems, the SO(3) symmetry group of quantum rotors offers an inherently more complex platform with unprecedented possibilities for exploring topological phenomena. Despite their ubiquity in nature– from molecules to nanorotors– their potential for hosting topological phases has remained largely unexamined. In this thesis, we mainly focus on periodically driven linear molecules as a prototype for studying topological phenomena in quantum rotors. Recent technological advances in coherent control of molecules, particularly through precisely shaped laser pulses, have made it possible to investigate linear rotors in the context of topology. While planar rotors have received some attention in recent years, threedimensional rotors–particularly linear molecules–harbor substantially richer topological phenomena due to their non-abelian nature and their additional angular degrees of freedom. We demonstrate that these systems can host novel edge states and topological features fundamentally impossible in planar systems. We begin by establishing a theoretical bridge between periodically kicked rotors and "crystalline" lattices in angular momentum space. Using non-interacting linear molecules as our primary example, we show how quantum interference and revival patterns lead to the possibility to simulate band models with arbitrary number of bands N. While our framework applies to various quantum rotors, including nanorotors and kicked Bose-Einstein condensates, linear molecules provide an ideal experimental platform due to their abovementioned precise controllability. The core of this work examines adiabatic dynamics of 3D quantum rotors, establishing a geometric framework based on the Euler class to characterize its non-abelian topology. The non-Hermitian nature of the system enables novel braiding behaviors and topological transitions impossible in static systems, leading to an anomalous Dirac string phase with edge states in each gap, even though the Berry phases are all zero. These features can be directly observed through molecular alignment and rotational level populations. These findings establish quantum rotors as an alternative platform for studying multi-band topological physics, while suggesting practical implementations for quantum computation where topological protection could offer natural resilience against decoherence. The rich structure of three-dimensional rotation groups, combined with the tunability of topological features through driving parameters, makes this platform particularly valuable for exploring fundamental physics and developing quantum technologies.