Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:chao-dyn/9404001 · Physical Review Letters

Semiquantal dynamics of fluctuations: Ostensible quantum chaos

Arjendu K. Pattanayak, William C. Schieve

Published May 2, 1994• 61 citations
arXivHTMLPDF

Abstract

The time-dependent variational principle using generalized Gaussian trial functions yields a finite dimensional approximation to the full quantum dynamics and is used in many disciplines. It is shown how these ``semiquantum'' dynamics may be derived via the Ehrenfest theorem and recast as an extended classical gradient system with the fluctuation variables coupled to the average variables. An extended potential is constructed for a one-dimensional system. The semiquantal behavior is shown to be chaotic even though the system has regular classical behavior and the quantum behavior had been assumed regular.

DOI
10.1103/physrevlett.72.2855

Related papers

Chaos in time-dependent variational approximations to quantum dynamics

Fred Cooper, John F. Dawson, Salman Habib +1 more

Dynamical chaos has recently been shown to exist in the Gaussian approximation in quantum mechanics and in the self-consistent mean field approach to studying the dynamics of quantum fields. In this study, we first note that any variational approximation to the dynamics of a quantum system based on the Dirac action principle leads to a classical Hamiltonian dynamics for the variational parameters. Since this Hamiltonian is generically nonlinear and nonintegrable, the dynamics thus generated can be chaotic, in distinction to the exact quantum evolution. We then restrict our attention to a system of two biquadratically coupled quantum oscillators and study two variational schemes, the leading order large-$N$ (four canonical variables) and Hartree (six canonical variables) approximations. The chaos seen in the approximate dynamics is an artifact of the approximations: this is demonstrated by the fact that its onset occurs on the same characteristic time scale as the breakdown of the approximations when compared to numerical solutions of the time-dependent Schr\"odinger equation.

arXiv:quant-ph/9610013•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Feb 1, 1998•46 citations

Semiquantum Chaos

Jun Ma, Renkuan Yuan

In this paper it is shown by numerical calculations that chaos exists in a quantum-classical coupled system and behavior of the system depends on the initial condition of the system. It is also found that relatively regular or irregular behavior of one subsystem corresponds to similar behavior of the other. So one way arise naturally in which quantum information about such a coupled system can be obtained only by observing the different kinds of motions of the classical subsystem.

Journal of the Physical Society of Japan•Aug 15, 1997•3 citations

Quantum fluctuations in classical chaos

Wei-Min Zhang, Jian‐Min Yuan, D.H. Feng +2 more

The effects of quantum fluctuation on the classical dynamical structure are investigated by using the semiquantal dynamics derived from the effective action. A pulsed-spin system is studied as an example, which shows that global chaos is suppressed by quantum fluctuation. The quantum-classical correspondence is explicitly examined for low-spin cases.

Physical Review A•Sep 1, 1990•11 citations

Quantum fluctuation dynamics in momentum spread of cold atoms

WANGZhong-jie, WangQin-Mou, HuangWan-Xia

Dynamics of the quantum system that consists of a cold atom in a modulated standing wave of light is analyzed using the time-dependent variational principle formulation based on squeezed coherent states. A group of ordinary differential equations describing evolution of two pairs of canonically conjugate variables (q(t), p(t); G(t), Π(t)) are derived where G(t) and Π(t) describe the quantum fluctuations of the system. It has been shown that a transition from the regular motion to the chaotic motion in G(t), Π(t) phase space. Quantum system seems to be capable to show the classical-like chaotic structure.

Chinese journal of atomic and molecular physics•Jan 1, 2003•0 citations

Quantal Cumulant Mechanics as Extended Ehrenfest Theorem

Yasuteru Shigeta

Since Schrodinger proposed wave mechanics for quantum phenomena in 1926 [1-4], referred as Schrodinger equation named after his name, this equation has been applied to atom-mol‐ ecules, condensed matter, particle, and elementary particle physics and succeeded to repro‐ duce various experiments. Although the Schrodinger equation is in principle the differential equation and difficult to solve, by introducing trial wave functions it is reduced to matrix equations on the basis of the variational principle. The accuracy of the approximate Schro‐ dinger equation depends strongly on the quality of the trial wave function. He also derived the time-dependent Schrodinger equation by imposing the time-energy correspondence. This extension opened to describe time-dependent phenomena within quantum mechanics. However there exist a few exactly solvable systems so that the methodology to solve Schro‐ dinger equation approximately is extensively explored, yet.

InTech eBooks•Apr 3, 2013•2 citations

Quantum Chaos in a Rydberg Atom System

Yochai Werman

A recent proposal by Hallam et al. suggested using the chaotic properties of the semiclassical equations of motion, obtained by the time dependent variational principle (TDVP), as a characterization of quantum chaos. In this paper, we calculate the Lyapunov spectrum of the semiclassical theory approximating the quantum dynamics of a strongly interacting Rydberg atom array, which lead to periodic motion. In addition, we calculate the effect of quantum fluctuations around this approximation, and obtain the escape rate from the periodic orbit. We compare this rate to the rate extracted from the exact solution of the quantum theory, and find an order of magnitude discrepancy. We conclude that in this case, chaos in the TDVP equations does not correpond to phsyical properties of the system. Our result complement those of Ho et al. regarding the escape rate from the semiclassical periodic orbit.

arXiv:2001.06110•Jan 16, 2020•2 citations
The time-dependent variational principle using generalized Gaussian trial functions yields a finite dimensional approximation to the full quantum dynamics and is used in many disciplines. It is shown how these ``semiquantum'' dynamics may be derived via the Ehrenfest theorem and recast as an extended classical gradient system with the fluctuation variables coupled to the average variables. An extended potential is constructed for a one-dimensional system. The semiquantal behavior is shown to be chaotic even though the system has regular classical behavior and the quantum behavior had been assumed regular.
Dynamical chaos has recently been shown to exist in the Gaussian approximation in quantum mechanics and in the self-consistent mean field approach to studying the dynamics of quantum fields. In this study, we first note that any variational approximation to the dynamics of a quantum system based on the Dirac action principle leads to a classical Hamiltonian dynamics for the variational parameters. Since this Hamiltonian is generically nonlinear and nonintegrable, the dynamics thus generated can be chaotic, in distinction to the exact quantum evolution. We then restrict our attention to a system of two biquadratically coupled quantum oscillators and study two variational schemes, the leading order large-NNN (four canonical variables) and Hartree (six canonical variables) approximations. The chaos seen in the approximate dynamics is an artifact of the approximations: this is demonstrated by the fact that its onset occurs on the same characteristic time scale as the breakdown of the approximations when compared to numerical solutions of the time-dependent Schr\"odinger equation.
Dynamics of the quantum system that consists of a cold atom in a modulated standing wave of light is analyzed using the time-dependent variational principle formulation based on squeezed coherent states. A group of ordinary differential equations describing evolution of two pairs of canonically conjugate variables (q(t), p(t); G(t), Π(t)) are derived where G(t) and Π(t) describe the quantum fluctuations of the system. It has been shown that a transition from the regular motion to the chaotic motion in G(t), Π(t) phase space. Quantum system seems to be capable to show the classical-like chaotic structure.
Since Schrodinger proposed wave mechanics for quantum phenomena in 1926 [1-4], referred as Schrodinger equation named after his name, this equation has been applied to atom-mol‐ ecules, condensed matter, particle, and elementary particle physics and succeeded to repro‐ duce various experiments. Although the Schrodinger equation is in principle the differential equation and difficult to solve, by introducing trial wave functions it is reduced to matrix equations on the basis of the variational principle. The accuracy of the approximate Schro‐ dinger equation depends strongly on the quality of the trial wave function. He also derived the time-dependent Schrodinger equation by imposing the time-energy correspondence. This extension opened to describe time-dependent phenomena within quantum mechanics. However there exist a few exactly solvable systems so that the methodology to solve Schro‐ dinger equation approximately is extensively explored, yet.
A recent proposal by Hallam et al. suggested using the chaotic properties of the semiclassical equations of motion, obtained by the time dependent variational principle (TDVP), as a characterization of quantum chaos. In this paper, we calculate the Lyapunov spectrum of the semiclassical theory approximating the quantum dynamics of a strongly interacting Rydberg atom array, which lead to periodic motion. In addition, we calculate the effect of quantum fluctuations around this approximation, and obtain the escape rate from the periodic orbit. We compare this rate to the rate extracted from the exact solution of the quantum theory, and find an order of magnitude discrepancy. We conclude that in this case, chaos in the TDVP equations does not correpond to phsyical properties of the system. Our result complement those of Ho et al. regarding the escape rate from the semiclassical periodic orbit.