Correspondence Principle, Ergodicity, and Finite-Time Dynamics

The quantum-classical correspondence stands out as one of the most intriguing challenges in the realm of quantum mechanics. Within the context of quantum chaos, the conventional approach, which relies on spectral statistics to infer distinct classical dynamical properties, fails in typical scenarios, such as slow relaxation processes, dynamical localization, or other deviations from ergodicity. To address this difficulty, we propose a novel approach, the essence of which lies in associating a quantum energy shell with classical finite-time trajectories. Our results extend the knowledge of quantum-classical correspondence, offer explanations for previously conflicting findings, and also bear significance for applications in quantum metrology, sensing, and computation when the determination of the ergodicity degree of the quantum states is in need.