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Physical Review Letters

Correspondence Principle, Ergodicity, and Finite-Time Dynamics

Zhen-Qi Chen, Rui-Hua Ni, Yalei Song, Liang Huang, Jiao Wang, Giulio Casati

Published Mar 31, 2025• 4 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.134.130402

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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.
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Conventional wisdom currently has it that chaotic behavior is either strongly suppressed or absent in quantum models. Indeed, some researchers have concluded that these considerations serve to undermine the correspondence principle, thereby raising serious doubts about the adequacy of quantum mechanics as a fundamental physical theory. Thus, the quantum chaos question is a prime subject for philosophical analysis. The most significant reasons given for the absence or suppression of chaotic behavior in quantum models are the linearity of Schrödinger's equation and the unitarity of the time-evolution described by that equation. Both are shown in this essay to be irrelevant by demonstrating that the crucial feature for chaos is the nonseparability of the Hamiltonian. That demonstration indicates that quantum chaos is likely to be exhibited in models of open quantum systems. A measure for probing such models for chaotic behavior is developed, and then used to show that quantum mechanics has chaotic models for systems having a continuous energy spectrum. The prospects of this result for vindicating the essence of the correspondence principle are then briefly examined.
The recovery of classical chaotic dynamics from quantum systems has long been a subject of interest. Furthermore, recent work indicates that quantum chaos may well be significant in quantum information processing. In this paper we discuss the quantum to classical crossover of a superconducting quantum inference device (SQUID) ring. Such devices comprise of thick superconducting loop enclosing a Josephson weak link. These devices are currently strong candidates for many applications in quantum technologies. The weak link brings with it a non-linearity such that semi-classical models of this system can exhibit chaotic dynamics. For many similar systems an application of the correspondence principle together with the inclusion of environmental degrees of freedom through a quantum trajectories approach can be used to effectively recover classical dynamics. Here we show (i) that the standard expression of the correspondence principle is incompatible with the ring Hamiltonian and we present a more pragmatic and general expression which finds application here and (ii) that practical limitations to circuit parameters of the SQUID ring prevent arbitrarily accurate recovery of classical chaotic dynamics.
The famous Nils Bohr's quantum-classical correspondence principle states that the classical mechanics is a limiting case of the more general quantum mechanics. This implies that ``under certain conditions" quantum laws of motion become equivalent to classical laws. One of the conditions is fairly obvious: the corresponding classical action should be very large as compared with the Planck's constant. But this condition is not sufficient. Suppression of effects of quantum interference ("decoherence") is the phenomenon of primary importance for understanding of the Bohr's principle. Being, in essence, of quite general nature, this problem takes on special significance in the case of non-linear classically chaotic quantum systems. Whereas the rapid decay of phase correlations is an underlying feature of the classical dynamical chaos, the "quantum chaos" by itself is not capable of destroying the quantum phase coherence. Strictly speaking, any initially pure quantum state remains pure during an arbitrary long unitary evolution. Basically, formation of incoherent mixed states (decoherence) results only from the process of preparation of a mixed initial state or interaction with a noisy environment. Just the decoherence brigdes the quantum and classical worlds. Peculiarities of the time behavior of Peres fidelity, state purity, Shanon and von Neumann entropies are analyzed in detail. We demonstrate the ways the decoherence shows up in periodically driven systems that can be associated with Ramsey-type interferomentry experiments with ion traps. Finally, decoherence in ballistic electron quantum transport caused by interaction with a disordered environment is considered.