Abstract
The research on quantum chaos finds its roots in the study of the spectrum of complex nuclei in the 1950s and the pioneering experiments in microwave billiards in the 1970s. Since then, a large number of new results was produced. Nevertheless, the work on the subject is, even at present, a superposition of several approaches expressed in different mathematical formalisms and weakly linked to each other. The purpose of this paper is to supply a unified framework for describing quantum chaos using the quantum ergodic hierarchy. Using the factorization property of this framework, we characterize the dynamical aspects of quantum chaos by obtaining the Ehrenfest time. We also outline a generalization of the quantum mixing level of the kicked rotator in the context of the impulsive differential equations.
Highlights
Unstable systems were studied in detail in many different areas of classical and quantum physics
Another important approach to quantum chaos was proposed by Michael Berry, who identifies a chaotic quantum system as a quantum system with a chaotic classical limit [11]; he originally called the study of this kind of quantum systems chaology
In this paper we propose a unified framework for describing quantum chaos using the quantum ergodic hierarchy, previously developed by two of us in [12,13]
Summary
Unstable systems were studied in detail in many different areas of classical and quantum physics. The universal statistical properties of the energy spectrum are given by the Random Matrix Theory approach, which characterizes the stationary aspects of quantum chaos in the energy domain [8,9,10] Another important approach to quantum chaos was proposed by Michael Berry, who identifies a chaotic quantum system as a quantum system with a chaotic classical limit [11]; he originally called the study of this kind of quantum systems chaology. On the basis of this result, we will present a formalism developed to treat quantum systems whose classical limits belong to one of the levels of the classical ergodic hierarchy This will allow us to identify some conditions that a quantum system must satisfy to lead to chaotic behavior in its classical limit.
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