Topological Interpretation of Classical Granular Network Through Realization of Berry Phase and Elastic Bit

The study of the Berry phase in classical and quantum systems has opened new possibilities in the field of quantum information science and technology. The Berry phase, which emerges from an adiabatic cyclic process, can create a relation with the dynamics of an elastic system. In this study, we show the formation of a Berry phase in a quantum analogue two-level system, an elastic bit. We investigate the formation of the elastic bit by harmonically driving the system in a classical nonlinear system that consists of two single-point contact granules, which creates a Hertz-type nonlinearity. The formation of different coherent superpositions of states are generated by tuning the frequency or amplitude separately or simultaneously. First, we show the effect of the driving parameters in forming elastic bits in a linearized system, where the superposition of states is time-independent. When mapped onto the Bloch sphere, the state vector’s trajectory of this elastic bit in parameter space can be precisely manipulated using the external drivers’ amplitude, phase, and frequency, resulting in a specific Berry phase. The quantized Berry phase observation indicates that the elastic bit exhibits trivial and nontrivial topologies. An equal superposition of states of the elastic bit yields the nontrivial Berry phase of . In contrast, the zero Berry phase corresponds to pure states, and any superpositions of states can take values different from or . We also show the nonlinearity effect in the Berry phase’s experimental formation, where the coherent states are time-dependent. Using the orthonormal basis for nonlinear responses and mapping the displacement coefficients in Bloch states, we show how time affects the manipulation of the elastic bit and its states. Our analytical and experimental studies reveal the Berry phase’s involvement in exposing numerous topological properties of the classical granular network. These properties are important in topological computing, especially in the non-abelian computation, which acquire a Berry phase when braided around each other. This phase encodes quantum information in a way that is inherently protected from local noise and perturbations, providing robustness against decoherence. In non-Abelian computing, holonomic quantum gates are implemented using the Berry phases acquired by the system.