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The Berry phase, a concept of significant interest in quantum and classical mechanics, illuminates the dynamics of physical systems. Our current study explores this phenomenon within a classical granular network, employing an "elastic bit" that serves as a classical counterpart to the quantum bit. This approach establishes a connection between classical and quantum mechanics. By adjusting external forces, we generate an elastic bit within the granular network and map its behavior onto a Bloch sphere, akin to operating quantum logic gates. Varied manipulations of these external drivers yield a spectrum of Berry phases, from trivial (0) to nontrivial (π), unveiling the topological nature of the elastic bit. Crucially, this topological behavior is governed by external manipulations rather than the material or geometric properties of the medium. The nontrivial Berry phases, in particular, highlight energy localization within the granule vibrations, marking a significant insight into system dynamics. This research bridges the gap between the quantum and classical realms and paves the way for designing novel materials with unique properties. 

In quantum computing and information technology, the coherent superposition of states is an essential topic for realizing the physical state of data processing and storage. The fundamentals of current technology, a quantum bit, have limitations due to the collapse and decoherence of wave function, which hinders the superposition of states. We eliminate the limitations by introducing the elastic bit generated through the Hertz-type nonlinearity of granular beads. This study shows the experimental formation of the elastic bit in a coupled granular network manipulated by external harmonic excitation. The excitation generates a phase-dependent dynamic movement, and mapping onto the energy states of the linear vibration modes forms the coherent superposition of states. This state vector component comes from the amplitude of the coherent states, which is projected into the Hilbert space through time dependency. The coherent states represent an actual amplitude, which makes the elastic bit susceptible to decoherence. The elastic bit also demonstrates quantum operation, showcasing the Hadamard gate, which maps one superposed state to another. These characteristics of the elastic bit pave the way for sustainable quantum computation and data storage. 

This study investigates the Berry phase, a key concept in classical and quantum physics, and its manifestation in a classical system. We achieve controlled accumulation of the Berry phase by manipulating the elastic bit (a classical analogue to a quantum bit) in an externally driven, homogeneous, spherical, nonlinear granular network. This is achieved through the classical counterpart of quantum coherent superposition of states. The elastic bit's state vectors are navigated on the Bloch sphere using external drivers' amplitude, phase, and frequency, yielding specific Berry phases. These phases distinguish between trivial and nontrivial topologies of the elastic bit, with the zero Berry phase indicating pure states of the linearized granular system and the nontrivial π phase representing equal superposed states. Other superposed states acquire different Berry phases. Crucially, these phases correlate with the structure's eigenmode vibrations: trivial phases align with distinct, in-phase, or out-of-phase eigenmodes, while nontrivial phases correspond to coupled vibrations where energy is shared among granules, alternating between oscillation and rest. Additionally, we explore Berry's phase generalizations for non-cyclic evolutions. This research paves the way for advanced quantum-inspired sensing and computation applications by utilizing and controlling the Berry phase. 

Phi-bits are classical mechanical analogues of qubits. Comprehending the nonlinear phenomena that underlie the control and relationships between phi-bits is of utmost importance for advancing phi-bit-based quantum-analogue computing systems. Phi-bits are acoustic waves in externally driven nonlinearly coupled arrays of waveguides, that can exist in a coherent superposition of two states. Tuning the frequency, amplitude, and phase of external drivers is a means of controlling the phi-bit states. We have developed a discrete element model to analyze and predict the nonlinear phi-bit response to external drivers that may result from different types, strengths, and orders of nonlinearity due to the presence of (i) intrinsic medium (epoxy) coupling the waveguides and (ii) external factors such as signal generators/transducers/ultrasonic couplant assembly. Key findings include the impact of nonlinearity type, strength, and order as well as damping on the modulus and phases of the complex amplitudes of the phi-bit coherent superposition of states. This research serves as an exploration for control of design parameters in the creation of phi-bits, which will enable the preparation and manipulation of superpositions of states essential for developing phi-bit-based quantum analogue information processing platforms.