Quantum geometry embedded in unitarity of evolution: revealing its impacts as geometric oscillation and dephasing in spin resonance and crystal bands

Quantum Hall effects provide intuitive ways of revealing the topology in crystals, i.e., each quantized "step" represents a distinct topological state. Here, we seek a counterpart for "visualizing" quantum geometry, which is a broader concept. We show how geometry emerges in quantum as an intrinsic consequence of unitary evolution, composing a frame work compatible with quantum metric and independent of specific details or approximations, suggesting quantum geometry may have widespread applicability. Indeed, we exemplify geometric observables, such as oscillation, dephasing, in spin and band scenarios. Anomalies, supported by both analytic and numerical solutions, underscore the advantages of adopting a geometric perspective, potentially yielding distinguishable experimental signatures.