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arXiv:quant-ph/9707039

Intrinsic chaoticity in stable classical systems and quantum fluctuations

S. De Martino, Silvio De Siena, Fabrizio Illuminati

Published Jul 21, 1997• 0 citations
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Abstract

The paper has been withdrawn
DOI
10.48550/arxiv.quant-ph/9707039

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We investigate the classical and quantum properties of ultracold sodium atoms in a one-dimensional optical lattice and a three-dimensional harmonic trap. The energy versus crystal momentum dispersion relation for the lowest energy band of the optical lattice, together with the harmonic potential, define an effective Hamiltonian that we use to calculate classical atom paths. When one of the symmetry axes of the harmonic trap is aligned with the optical lattice, the atoms follow stable trajectories. But tilting this symmetry axis away from the optical lattice direction creates an unusual type of mixed stable-chaotic classical dynamics, which originates from the intrinsically quantum-mechanical nature of energy bands. In this regime, the density of quantized energy levels for the system exhibits periodic fluctuations associated with both stable and unstable periodic classical orbits. One of the unstable orbits also produces well-defined scar patterns in a subset of eigenfunctions. Wave functions with distinct spatial forms are identified and related directly to particular parts of the classical phase space using a Wigner function analysis.
Quantum-Classical convergence describes the inevitable merging of quantum and classical systems, often observed in complex systems exhibiting chaotic behavior. This paper explores a novel mathematical framework for understanding this convergence, proposing a hybrid approach that integrates concepts from quantum mechanics (superposition, entanglement) and classical chaos theory. We introduce a model where classical elements are intrinsically influenced by quantum fluctuations, aiming to illuminate the dynamic interplay between these two fundamental scales. The paper details the core mechanism, outlining the integration of quantum and chaotic elements into a unified system. The resulting model offers a new perspective on the emergence of patterns and stability within complex dynamical systems, particularly in the context of chaotic systems. This work provides a foundation for advanced theoretical investigations into the fundamental principles underlying the convergence of these seemingly disparate realms.
The possibility is discussed of inferring or simulating some aspects of quantum dynamics by adding classical statistical fluctuations to classical mechanics. We introduce a general principle of mechanical stability and derive a necessary condition for classical chaotic fluctuations to affect confined dynamical systems, on any scale, ranging from microscopic to macroscopic domains. As a consequence we obtain, both for microscopic and macroscopic aggregates, dimensional relations defining the minimum unit of action of individual constituents, yielding in all cases Planck action constant.