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International Journal of Quantum Chemistry

Quantum analogue of the Kolmogorov–Arnold–Moser transition in different quantum anharmonic oscillators

Pratim Kumar Chattaraj, Biswajit Maiti, S. Sengupta

Published Jan 1, 2004• 7 citations
Publisher

Abstract

Abstract The results of numerical computations are presented for the Bohmian trajectories of the family of different one‐ and two‐dimensional anharmonic oscillators, which exhibit regular or chaotic motion in both classical and quantum domains, depending on the values of the parameters appearing in the respective Hamiltonians. Quantum signatures of the Kolmogorov–Arnold–Moser (KAM) transition from the regular to chaotic classical dynamics of these oscillators are studied using a quantum theory of motion (QTM) as developed by de Broglie and Bohm. A phase space distance function between two initially close Bohmian trajectories, the associated Kolmogorov–Sinai–Lyapunov (KSL) entropy, the phase space volume, the autocorrelation function, the associated power spectrum, and the nearest‐neighbor spacing distribution, clearly differentiate the quantum analogues of the corresponding regular and chaotic motions in the classical domain. These quantum anharmonic oscillators are known to be useful in several diverse branches of science. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

DOI
10.1002/qua.20161

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Abstract The results of numerical computations are presented for the Bohmian trajectories of the family of different one‐ and two‐dimensional anharmonic oscillators, which exhibit regular or chaotic motion in both classical and quantum domains, depending on the values of the parameters appearing in the respective Hamiltonians. Quantum signatures of the Kolmogorov–Arnold–Moser (KAM) transition from the regular to chaotic classical dynamics of these oscillators are studied using a quantum theory of motion (QTM) as developed by de Broglie and Bohm. A phase space distance function between two initially close Bohmian trajectories, the associated Kolmogorov–Sinai–Lyapunov (KSL) entropy, the phase space volume, the autocorrelation function, the associated power spectrum, and the nearest‐neighbor spacing distribution, clearly differentiate the quantum analogues of the corresponding regular and chaotic motions in the classical domain. These quantum anharmonic oscillators are known to be useful in several diverse branches of science. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004
Quantum domain behaviour of classically chaotic systems is studied using the quantum theory of motion in teh sense of clasical interpretationof quantum mechanics as developed by de Broglie and Bohm. Dynamics of quantum Henon- Heiles oscillatorm barbanis oxcillatior and CTW oscillator are ananysed with the help of quantum Lypunov exponent and Kolmogorov-Sinai entropy defined in terms of the distance between two initially close Bohmian trajectories. Standard diagnostics of quantum chaos like auticorrelation function and the assocated power spectrum, nearest neighbour spacing distribution phase space volume, spectral rigidity, etc. support these results. Quantum theory of motion provides an alternative route for understanding quantum chaos. Nonlinear dynamics of integrable systems in quantum domain is also properly taken care of within this framework.
Abstract The quantum domain behavior of classical nonintegrable systems is well‐understood by the implementation of quantum fluid dynamics and quantum theory of motion. These approaches properly explain the quantum analogs of the classical Kolmogorov–Arnold–Moser type transitions from regular to chaotic domain in different anharmonic oscillators. Field‐induced tunneling and chaotic ionization in Rydberg atoms are also analyzed with the help of these theories. Quantum fluid density functional theory may be used to understand different time‐dependent processes like ion‐atom/molecule collisions, atom‐field interactions, and so forth. Regioselectivity as well as confined atomic/molecular systems and their reactivity dynamics have also been explained. © 2013 Wiley Periodicals, Inc.
The quantum-classical correspondence is studied for a periodically driven quartic oscillator exhibiting integrable and chaotic dynamics, by studying the Bohmian trajectory of the corresponding "classical" Schrödinger equation. Phase plots and the Kolmogorov-Sinai entropy are computed and compared with the classical trajectory as well as the Bohmian trajectory obtained from the time dependent Schrödinger equation. Bohmian mechanics at the classical limit appears to mimick the behavior of a dissipative dynamical system.
This paper presents a number of numerical investigations of orbits in the de Broglie–Bohm version of quantum mechanics. We first clarify how the notion of chaos should be implemented in the case of Bohmian orbits. Then, we investigate the Bohmian orbits in three different characteristic quantum systems: (a) superposition of three stationary states in the Hamiltonian of two uncoupled harmonic oscillators with incommensurable frequencies, (b) wave packets in a Hénon–Heiles-type Hamiltonian and (c) a modified two-slit experiment. In these examples, we identify regular or chaotic orbits and also orbits exhibiting a temporarily regular and then chaotic behaviour. Then, we focus on a numerical investigation of the Bohm–Vigier (Bohm and Vigier 1954 Phys. Rev. 26 208) theory, that an arbitrary initial particle distribution P should asymptotically tend to |ψ| 2 , by considering the role of chaotic mixing in causing irregularity of Madelung's flow, a necessary condition for P to tend to |ψ| 2 . We find that the degree of chaos of a particular system correlates with the speed of convergence of P to |ψ| 2 . In the case of wave-packet dynamics, our numerical data show that the time of convergence scales exponentially with the inverse of the effective perturbation from the harmonic oscillator Hamiltonian. The latter result can be viewed as a quantum analogue of Nekhoroshev's (Nekhoroshev 1977 Russ. Math. Surveys 32 1) theorem of exponential stability in classical nonlinear Hamiltonian dynamics.
The classical Kolmogorov-Arnold-Moser (KAM) theorem provides the underlying mechanism for the stability of the solar system under some small chaotic perturbations. Despite many previous efforts, any quantum version of the KAM theorem remains elusive. In this work, we provide a quantum KAM theorem in the context of the anisotropic Dicke model, which is the most important quantum optics model. It describes a single mode of photons coupled to NNN qubits with both a rotating wave (RW) term and a counter-RW (CRW) term. As the ratio of the CRW over the RW term increases from zero to one, the systems evolves from quantum integrable to quantum chaotic. We establish a quantum KAM theorem to characterize such an evolution quantitatively by both large-NNN expansion and random matrix theory and find agreement from the two complementary approaches. Connections and differences between the Dicke models and Sachdev-Ye-Kitaev (SYK) or hybrid SYK models are examined. A possible quantum KAM theorem in terms of other quantum chaos criteria such as the quantum Lyapunov exponent is also discussed.