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

Physical Review A

Limit of classical chaos in quantum systems

Tomio Yamakoshi Petrosky, William C. Schieve

Published Jun 1, 1985• 12 citations
Publisher

Abstract

The nonlocal effect of quantum mechanics upon the classical chaos around the separatrix in a Hamiltonian system is investigated by extending the definition of the Melnikov function in the semiclassical approximation. It is shown that the quantum correction of the Melnikov function is related to the quantum fluctuation of the energy on the stable and unstable manifolds. This correction is a constant shift of the center of oscillation of the classical Melnikov function from zero. Because of this shift, the effect of quantum mechanics suppresses the classical chaos around the separatrix. Physical estimates are made of the magnitude of the quantum effect for a double-well oscillator system. As examples, we treat the case of the electron for the molecular scale and the proton for the nuclear scale, and also comment on the ammonia molecules ${\mathrm{NH}}_{3}$, ${\mathrm{ND}}_{3}$, and ${\mathrm{NT}}_{3}$.

DOI
10.1103/physreva.31.3907

Related papers

Phase Space Structures in Chaotic Quantum Dynamics.

B. S. Helmkamp

I compare the quantum and classical dynamics of a particle moving in a cosine potential while subject to a time-dependent force. I concentrate here on the behavior of an initially well-localized wave packet at times before the classically chaotic motion is fully developed. I find that the quantum and classical dynamics are indistinguishable well beyond the Ehrenfest time where the wave packet delocalizes. The quantum and classical descriptions first differ precisely when the classical probability density is folded in the vicinity of a hyperbolic fixed point. At this point, the wave function acquires a nodal structure which I show to be the result of a simple beating phenomenon between paths in the semiclassical propagator. When the interaction of the classical manifold with the hyperbolic fixed point leads to escape from the remnant separatrix, representing the onset of classical chaos, the interference associated with the tendril leads to a long-lived accumulation of quantum amplitude on top of the cosine barrier. This effect also has a semiclassical interpretation, meaning that it arises from the interference between classical paths, but one must expand to second order in Planck's constant to describe the behavior correctly. Both the nodal structure and this pinning of amplitude on the barrier are dynamic mechanisms for the quantum inhibition of mixing. I then couple the system to a bath of harmonic oscillators in order to study the effect of the environment on these mechanisms. Although an oscillator bath environment brings dissipation as well as noise to the problem, the noise effect dominates for the high temperature, weak coupling regime I study. When there is sufficient noise to render the interfering classical paths indistinguishable, I find that the quantum interference gets erased. This dephasing occurs at very early times, long before there is appreciable dissipation of energy to the environment. Consequently, one can argue that the presence of an environment, even if its effect would be negligible in a nonchaotic setting, allows for the possibility of quantum mixing.

Civil War Book Review•Jan 1, 1995•0 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

Fragility of classical Hamiltonian period doubling to quantum fluctuations

Reyhaneh Khasseh, Angelo Russomanno, Rosario Fazio

We add quantum fluctuations to a classical period-doubling Hamiltonian time crystal, replacing the $N$ classical interacting angular momenta with quantum spins of size $l$. The full permutation symmetry of the Hamiltonian allows a mapping to a bosonic model and the application of exact diagonalization for a quite large system size. In the thermodynamic limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$ the model is described by a system of Gross-Pitaevskii equations whose classical-chaos properties closely mirror the finite-$N$ quantum chaos. For $N\ensuremath{\rightarrow}\ensuremath{\infty}$, and $l$ finite, Rabi oscillations mark the absence of persistent period doubling, which is recovered for $l\ensuremath{\rightarrow}\ensuremath{\infty}$ with Rabi-oscillation frequency tending exponentially to 0. For the chosen initial conditions, we can represent this model in terms of Pauli matrices and apply the discrete truncated Wigner approximation. For finite $l$ this approximation reproduces no Rabi oscillations but correctly predicts the absence of period doubling. Our results show the instability of time-translation symmetry breaking in this classical system even to the smallest quantum fluctuations, because of tunneling effects.

arXiv:2108.11408•Physical review. B./Physical review. B•Oct 21, 2021•3 citations

The semiclassical regime of the chaotic quantum-classical transition

Benjamin D. Greenbaum, Salman Habib, Kosuke Shizume +1 more

An analysis of the semiclassical regime of the quantum-classical transition is given for open, bounded, one-dimensional chaotic dynamical systems. Environmental fluctuations-characteristic of all realistic dynamical systems-suppress the development of a fine structure in classical phase space and damp nonlocal contributions to the semiclassical Wigner function, which would otherwise invalidate the approximation. This dual regularization of the singular nature of the semiclassical limit is demonstrated by a numerical investigation of the chaotic Duffing oscillator.

arXiv:quant-ph/0401174•Chaos An Interdisciplinary Journal of Nonlinear Science•Jul 20, 2005•9 citations

Applications of the Decoherence Formalism

Todd A. Brun

In this work the decoherence formalism of quantum mechanics is explored and applied to a number of interesting problems in quantum physics. The boundary between quantum and classical physics is examined, and demonstration made that quantum histories corresponding to classical equations of motion become more probable for a broad class of models, including linear and nonlinear models of Brownian motion. The link between noise, dissipation, and decoherence is studied. This work is then applied to systems which classically exhibit dissipative chaotic dynamics. A theory is explicated for treating these systems, and the ideas are applied to a particular model of the forced, damped Duffing oscillator, which is chaotic for certain parameter values. Differences between classical and quantum chaos are examined, particularly differences arising in the structure of fractal strange attractors, and the conceptual difficulties in framing standard notions of chaos in a quantum system. A brief discussion of previous work on quantum chaos is included, and the differences between Hamiltonian and dissipative chaos pointed out; a somewhat different interpretation of quantum chaos from the standard one is suggested. A class of histories for quantum systems, in phase space rather than configuration space, is studied. Different ways of representing projections in phase space are discussed, and expressions for the probability of phase space histories are derived; conditions for such histories to decohere are also estimated in the semiclassical limit.

•Mar 16, 2009•4 citations

Nonmonotonicity in the Quantum-Classical Transition: Chaos Induced by Quantum Effects

Arie Kapulkin, Arjendu K. Pattanayak

The classical-quantum transition for chaotic systems is understood to be accompanied by the suppression of chaotic effects as the relative variant Planck's over 2pi is increased. We show evidence to the contrary in the behavior of the quantum trajectory dynamics of a dissipative quantum chaotic system, the double-well Duffing oscillator. The classical limit in the case considered has regular behavior, but as the effective variant Planck's over 2pi is increased we see chaotic behavior. This chaos then disappears deeper into the quantum regime, which means that the quantum-classical transition in this case is nonmonotonic in variant Planck's over 2pi.

arXiv:quant-ph/0702174•Physical Review Letters•Aug 11, 2008•27 citations
The nonlocal effect of quantum mechanics upon the classical chaos around the separatrix in a Hamiltonian system is investigated by extending the definition of the Melnikov function in the semiclassical approximation. It is shown that the quantum correction of the Melnikov function is related to the quantum fluctuation of the energy on the stable and unstable manifolds. This correction is a constant shift of the center of oscillation of the classical Melnikov function from zero. Because of this shift, the effect of quantum mechanics suppresses the classical chaos around the separatrix. Physical estimates are made of the magnitude of the quantum effect for a double-well oscillator system. As examples, we treat the case of the electron for the molecular scale and the proton for the nuclear scale, and also comment on the ammonia molecules NH3{\mathrm{NH}}_{3}NH3​, ND3{\mathrm{ND}}_{3}ND3​, and NT3{\mathrm{NT}}_{3}NT3​.
I compare the quantum and classical dynamics of a particle moving in a cosine potential while subject to a time-dependent force. I concentrate here on the behavior of an initially well-localized wave packet at times before the classically chaotic motion is fully developed. I find that the quantum and classical dynamics are indistinguishable well beyond the Ehrenfest time where the wave packet delocalizes. The quantum and classical descriptions first differ precisely when the classical probability density is folded in the vicinity of a hyperbolic fixed point. At this point, the wave function acquires a nodal structure which I show to be the result of a simple beating phenomenon between paths in the semiclassical propagator. When the interaction of the classical manifold with the hyperbolic fixed point leads to escape from the remnant separatrix, representing the onset of classical chaos, the interference associated with the tendril leads to a long-lived accumulation of quantum amplitude on top of the cosine barrier. This effect also has a semiclassical interpretation, meaning that it arises from the interference between classical paths, but one must expand to second order in Planck's constant to describe the behavior correctly. Both the nodal structure and this pinning of amplitude on the barrier are dynamic mechanisms for the quantum inhibition of mixing. I then couple the system to a bath of harmonic oscillators in order to study the effect of the environment on these mechanisms. Although an oscillator bath environment brings dissipation as well as noise to the problem, the noise effect dominates for the high temperature, weak coupling regime I study. When there is sufficient noise to render the interfering classical paths indistinguishable, I find that the quantum interference gets erased. This dephasing occurs at very early times, long before there is appreciable dissipation of energy to the environment. Consequently, one can argue that the presence of an environment, even if its effect would be negligible in a nonchaotic setting, allows for the possibility of quantum mixing.
We add quantum fluctuations to a classical period-doubling Hamiltonian time crystal, replacing the NNN classical interacting angular momenta with quantum spins of size lll. The full permutation symmetry of the Hamiltonian allows a mapping to a bosonic model and the application of exact diagonalization for a quite large system size. In the thermodynamic limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$ the model is described by a system of Gross-Pitaevskii equations whose classical-chaos properties closely mirror the finite-NNN quantum chaos. For $N\ensuremath{\rightarrow}\ensuremath{\infty}$, and lll finite, Rabi oscillations mark the absence of persistent period doubling, which is recovered for $l\ensuremath{\rightarrow}\ensuremath{\infty}$ with Rabi-oscillation frequency tending exponentially to 0. For the chosen initial conditions, we can represent this model in terms of Pauli matrices and apply the discrete truncated Wigner approximation. For finite lll this approximation reproduces no Rabi oscillations but correctly predicts the absence of period doubling. Our results show the instability of time-translation symmetry breaking in this classical system even to the smallest quantum fluctuations, because of tunneling effects.
An analysis of the semiclassical regime of the quantum-classical transition is given for open, bounded, one-dimensional chaotic dynamical systems. Environmental fluctuations-characteristic of all realistic dynamical systems-suppress the development of a fine structure in classical phase space and damp nonlocal contributions to the semiclassical Wigner function, which would otherwise invalidate the approximation. This dual regularization of the singular nature of the semiclassical limit is demonstrated by a numerical investigation of the chaotic Duffing oscillator.
In this work the decoherence formalism of quantum mechanics is explored and applied to a number of interesting problems in quantum physics. The boundary between quantum and classical physics is examined, and demonstration made that quantum histories corresponding to classical equations of motion become more probable for a broad class of models, including linear and nonlinear models of Brownian motion. The link between noise, dissipation, and decoherence is studied. This work is then applied to systems which classically exhibit dissipative chaotic dynamics. A theory is explicated for treating these systems, and the ideas are applied to a particular model of the forced, damped Duffing oscillator, which is chaotic for certain parameter values. Differences between classical and quantum chaos are examined, particularly differences arising in the structure of fractal strange attractors, and the conceptual difficulties in framing standard notions of chaos in a quantum system. A brief discussion of previous work on quantum chaos is included, and the differences between Hamiltonian and dissipative chaos pointed out; a somewhat different interpretation of quantum chaos from the standard one is suggested. A class of histories for quantum systems, in phase space rather than configuration space, is studied. Different ways of representing projections in phase space are discussed, and expressions for the probability of phase space histories are derived; conditions for such histories to decohere are also estimated in the semiclassical limit.
The classical-quantum transition for chaotic systems is understood to be accompanied by the suppression of chaotic effects as the relative variant Planck's over 2pi is increased. We show evidence to the contrary in the behavior of the quantum trajectory dynamics of a dissipative quantum chaotic system, the double-well Duffing oscillator. The classical limit in the case considered has regular behavior, but as the effective variant Planck's over 2pi is increased we see chaotic behavior. This chaos then disappears deeper into the quantum regime, which means that the quantum-classical transition in this case is nonmonotonic in variant Planck's over 2pi.