Observation of Berry’s Phase in Elastic Bits: Quantum Analogues in Classical Granular Systems

Geometric and Berry phases play pivotal roles in classical and quantum mechanics, offering deep insights into the essence of physical systems. By establishing the classical analogue of quantum coherent superposition of states in an externally driven granular network, we demonstrated the controlled buildup of a geometric phase, i.e., the Berry phase, in an elastic bit—the classical counterpart of a quantum bit. 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 elastic bit displays trivial and nontrivial topologies, signified by the quantized Berry phase. An equal superposition of states of the elastic bit yields the nontrivial Berry phase of π. In contrast, the zero Berry phase corresponds to pure states, and any superpositions of states can take values different from 0 or π. Moreover, such phases correlate with the structure's eigenmode vibration: the trivial phase aligns with distinct in-phase or out-of-phase eigenmodes, whereas the nontrivial phase connects to a unique coupled vibration, where energy shuttles between the granules, and the granules alternate between oscillation and rest. We further delved into generalizations of Berry's phase for non-cyclic evolutions. By employing and controlling the Berry phase (and even the geometric phase), this work paves the way for quantum analogue sensing and computing.