This presentation explores the role of topological invariants in the non-equilibrium dynamics of periodically-driven quantum rotors. Under generic driving, quantum rotors exhibit dynamical localization, a prominent example of quantum chaos[1]. Utilizing Floquet theory, we analyze the non-linear response of these systems, transitioning from static eigenstates to non-equilibrium Floquet states. In a recent publication[2], we have generalized the model to include three-dimensional rotations and diverse laser pulses, inspired by experiments[3] on closed-shell diatomic molecules driven by periodic, far-off-resonant laser pulses. This approach uncovers a complex phase space with both localized and delocalized Floquet states. We demonstrate that the localized states are topological in nature, originating from Dirac cones protected by reflection and time-reversal symmetry. These states can be modified through laser strength adjustments, making them observable in current experiments through molecular alignment and observation of rotational level populations. Notably, in scenarios involving higher-order quantum resonances leading to multiple Floquet bands, the topological charges become non-Abelian. This results in the remarkable finding that the exchange of Dirac cones across different bands is non-commutative, enabling non-Abelian braiding. This phenomenon is linked to the recently identified non-Abelian topological Euler invariant[4], paving the way for the study of controllable multi-band topological physics in gas-phase experiments with small molecules, as well as for classifying dynamical molecular states by their topological invariants.
[1] Casati, G., & Chirikov, B. (Eds.). Quantum Chaos: Between Order and Disorder. Cambridge University Press (1995).
[2] Karle, V., Ghazaryan, A., & Lemeshko, M. Topological Charges of Periodically Kicked Molecules. Physical Review Letters, 130, 103202 (2023).
[3] Bitter, M., & Milner, V. Control of quantum localization and classical diffusion in laser-kicked molecular rotors. Physical Review A, 95, 013401 (2017). 
[4] Bouhon, A., Bzdušek, T., & Slager, R. J. Geometric approach to fragile topology beyond symmetry indicators. Physical Review B, 102, 115135 (2020).