Harnessing Nonlinear Dynamics for Time-Driven Berry Phase in Classical Systems

Phases arising from cyclic processes are fundamental in physics, bridging quantum and classical domains and providing deeper insights into the topology and dynamics of physical systems. This study investigates the accumulation of a time-driven Berry phase in a classical nonlinear system comprised of two spherical granules and introduces a method in which gauge variants naturally evolve over time without altering internal or external conditions. We develop a perturbation-based model to map the system's elastic characteristics to Bloch states and confirm the theoretical predictions of the frequency-dependent Berry phase through experiments. Our findings reveal that the Berry phase can exhibit trivial and nontrivial values, influenced by external driving forces and static precompression. Our results demonstrate a rich array of vibrational modes, capable of displaying identical Berry phase signatures across different frequencies-a significant departure from previous studies that identified a single topological resonance. Multiple nontrivial Berry phases emerge in highly nonlinear settings, whereas more linear regimes exhibit a singular nontrivial phase. Notably, the behavior of the Berry phase in our system mirrors fundamental quantum mechanics concepts, such as path-dependent state evolution. This study highlights the potential of classical mechanical systems to mimic quantum phenomena, opening new pathways for quantum-inspired topological computation and offering fresh perspectives on using time-driven Berry phase accumulation to investigate topological properties in nonlinear media.