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:1007.2771 · Laser Physics Letters

Universal behavior in quantum chaotic dynamics

Hongwei Xiong, Biao Wu

Published Mar 1, 2011• 13 citations
arXivHTMLPDF

Abstract

We discover numerically that a moving wave packet in a chaotic billiard will always evolve into a quantum state, whose density probability distribution is exponential. This exponential distribution is found to be universal for quantum chaotic systems with rigorous proof. In contrast, for the corresponding classical system, the distribution is Gaussian. We find that the quantum exponential distribution can smoothly change to the classical Gaussian distribution with coarse graining. This universal dynamical behavior can be observed experimentally with Bose-Einstein condensates.

DOI
10.1002/lapl.201010144

Related papers

Quantum dispersion and its exponential growth of a wave packet in chaotic systems

Junqing Li, Liu Fang, Xing Yong-Zhong +2 more

The quantum correspondence of one particular signature of classical chaos--the exponential instability of motion--can be characterized by the initial exponential growth rate of the spreading of the propagating quantum wave packet. The growth rate is approximately twice the classical maximum Lyapunov exponent of the system. In the regular case, the dispersion of the wave packet is only due to the usual quantum effect that should vanish in the classical limit. In contrast, in the chaotic case, the evolution behavior of the wave packet is due to the dynamical effect associated with the nonlinearity and persists as long as the spatial extension of the initial wave packet is kept finite.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Apr 11, 2002•2 citations

Exponential Divergence and Long Time Relaxation in Chaotic Quantum Dynamics

Arjendu K. Pattanayak, Paul W. Brumer

Phase-space representations of the dynamics of the quantal and classical cat map are used to explore quantum-classical correspondence in a $K$ system: As $\ensuremath{\Elzxh}\ensuremath{\rightarrow}0$, the classical chaotic behavior is shown to emerge smoothly and exactly. The quantum dynamics near the classical limit displays both exponential separation of adjacent distributions and long time relaxation, two characteristic features of classical chaotic motion.

arXiv:chao-dyn/9605014•Physical Review Letters•Jul 1, 1996•21 citations

Quantum chaos and the double-slit experiment

Giulio Casati, Tomaž Prosen

We report on the numerical simulation of the double-slit experiment, where the initial wave packet is bounded inside a billiard domain with perfectly reflecting walls. If the shape of the billiard is such that the classical ray dynamics is regular, we obtain interference fringes whose visibility can be controlled by changing the parameters of the initial state. However, if we modify the shape of the billiard thus rendering classical (ray) dynamics fully chaotic, the interference fringes disappear and the intensity on the screen becomes the (classical) sum of intensities for the two corresponding one-slit experiments. Thus we show a clear and fundamental example in which transition to chaotic motion in a deterministic classical system, in absence of any external noise, leads to a profound modification in the quantum behavior.

arXiv:nlin/0403038•Physical Review A•Sep 29, 2005•55 citations

Quantum Chaos and Randomness

Nobuhiko Saitô, Hironori Makino

In contrast to the classical Hamiltonian mechanics whose chaotic behaviors are now well established, quantum mechanics is still obscure about the origin of its random behavior. It is shown in this paper that in case of the stadium billiard, the fact that the nodal lines are smooth but are randomly oscillating, as first described by Heller, and MacDonald and Kaufman, can be interpreted as due to the irregular pseudochaotic behavior of the phase. To do this, we employ the path integral representation of propagator and the reinterpreted Bohm's formulation of quantum mechanics. Thus we reach the conclusion that although the chaotic behavior in quantum mechanics is suppressed by the Einstein quantization condition, it is observed in the properly defined phase of the propagator, or of the wave function, as a pseudochaos. In other words quantum mechanics has the counterpart of classical chaos in the phase of the propagator, and consequently, the wave function is also, pseudochaotic (random) with respect to q , but it does not exhibit exponential instability. The essential feature of quantum chaos manifests itself in the vanishings of the average of the wave function and the two-point correlation of the wave function at high energies. They are closely related to ergodicity and quantum measurement, and in particular new light will be shed on the measurement theory. The basic assumptions in the random matrix theory are discussed, confirming the universality of level spacing distribution.

Journal of the Physical Society of Japan•Jul 15, 2004•2 citations

Chaotic dynamics in billiards using Bohm’s quantum mechanics

O. F. de Alcantara Bonfim, J. Florencio, F. C. Sá Barreto

The dynamics of a particle in square and circular billiards is studied within the framework of Bohm's quantum mechanics. While conventional quantum mechanics predicts that the system shows no indication of chaotic behavior for these geometries from either the eigenvalue spectra distribution or the structure of the eigenfunctions, we find that in Bohm's quantum mechanics these systems exhibit both regular and chaotic behavior, depending on the form of the initial wave packet and on the particle's initial position.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Sep 1, 1998•27 citations

Quantum chaos, dynamical stability and decoherence

Giulio Casati, Tomaž Prosen

We discuss the stability of quantum motion under system's perturbations in the light of the corresponding classical behavior. In particular we focus our attention on the so called "fidelity" or Loschmidt echo, its relation with the decay of correlations, and discuss the quantum-classical correspondence. We then report on the numerical simulation of the double-slit experiment, where the initial wave-packet is bounded inside a billiard domain with perfectly reflecting walls. If the shape of the billiard is such that the classical ray dynamics is regular, we obtain interference fringes whose visibility can be controlled by changing the parameters of the initial state. However, if we modify the shape of the billiard thus rendering classical (ray) dynamics fully chaotic, the interference fringes disappear and the intensity on the screen becomes the (classical) sum of intensities for the two corresponding one-slit experiments. Thus we show a clear and fundamental example in which transition to chaotic motion in a deterministic classical system, in absence of any external noise, leads to a profound modification in the quantum behavior.

Brazilian Journal of Physics•Jun 1, 2005•2 citations
We discover numerically that a moving wave packet in a chaotic billiard will always evolve into a quantum state, whose density probability distribution is exponential. This exponential distribution is found to be universal for quantum chaotic systems with rigorous proof. In contrast, for the corresponding classical system, the distribution is Gaussian. We find that the quantum exponential distribution can smoothly change to the classical Gaussian distribution with coarse graining. This universal dynamical behavior can be observed experimentally with Bose-Einstein condensates.
The quantum correspondence of one particular signature of classical chaos--the exponential instability of motion--can be characterized by the initial exponential growth rate of the spreading of the propagating quantum wave packet. The growth rate is approximately twice the classical maximum Lyapunov exponent of the system. In the regular case, the dispersion of the wave packet is only due to the usual quantum effect that should vanish in the classical limit. In contrast, in the chaotic case, the evolution behavior of the wave packet is due to the dynamical effect associated with the nonlinearity and persists as long as the spatial extension of the initial wave packet is kept finite.
Phase-space representations of the dynamics of the quantal and classical cat map are used to explore quantum-classical correspondence in a KKK system: As $\ensuremath{\Elzxh}\ensuremath{\rightarrow}0$, the classical chaotic behavior is shown to emerge smoothly and exactly. The quantum dynamics near the classical limit displays both exponential separation of adjacent distributions and long time relaxation, two characteristic features of classical chaotic motion.
We report on the numerical simulation of the double-slit experiment, where the initial wave packet is bounded inside a billiard domain with perfectly reflecting walls. If the shape of the billiard is such that the classical ray dynamics is regular, we obtain interference fringes whose visibility can be controlled by changing the parameters of the initial state. However, if we modify the shape of the billiard thus rendering classical (ray) dynamics fully chaotic, the interference fringes disappear and the intensity on the screen becomes the (classical) sum of intensities for the two corresponding one-slit experiments. Thus we show a clear and fundamental example in which transition to chaotic motion in a deterministic classical system, in absence of any external noise, leads to a profound modification in the quantum behavior.
In contrast to the classical Hamiltonian mechanics whose chaotic behaviors are now well established, quantum mechanics is still obscure about the origin of its random behavior. It is shown in this paper that in case of the stadium billiard, the fact that the nodal lines are smooth but are randomly oscillating, as first described by Heller, and MacDonald and Kaufman, can be interpreted as due to the irregular pseudochaotic behavior of the phase. To do this, we employ the path integral representation of propagator and the reinterpreted Bohm's formulation of quantum mechanics. Thus we reach the conclusion that although the chaotic behavior in quantum mechanics is suppressed by the Einstein quantization condition, it is observed in the properly defined phase of the propagator, or of the wave function, as a pseudochaos. In other words quantum mechanics has the counterpart of classical chaos in the phase of the propagator, and consequently, the wave function is also, pseudochaotic (random) with respect to q , but it does not exhibit exponential instability. The essential feature of quantum chaos manifests itself in the vanishings of the average of the wave function and the two-point correlation of the wave function at high energies. They are closely related to ergodicity and quantum measurement, and in particular new light will be shed on the measurement theory. The basic assumptions in the random matrix theory are discussed, confirming the universality of level spacing distribution.
The dynamics of a particle in square and circular billiards is studied within the framework of Bohm's quantum mechanics. While conventional quantum mechanics predicts that the system shows no indication of chaotic behavior for these geometries from either the eigenvalue spectra distribution or the structure of the eigenfunctions, we find that in Bohm's quantum mechanics these systems exhibit both regular and chaotic behavior, depending on the form of the initial wave packet and on the particle's initial position.
We discuss the stability of quantum motion under system's perturbations in the light of the corresponding classical behavior. In particular we focus our attention on the so called "fidelity" or Loschmidt echo, its relation with the decay of correlations, and discuss the quantum-classical correspondence. We then report on the numerical simulation of the double-slit experiment, where the initial wave-packet is bounded inside a billiard domain with perfectly reflecting walls. If the shape of the billiard is such that the classical ray dynamics is regular, we obtain interference fringes whose visibility can be controlled by changing the parameters of the initial state. However, if we modify the shape of the billiard thus rendering classical (ray) dynamics fully chaotic, the interference fringes disappear and the intensity on the screen becomes the (classical) sum of intensities for the two corresponding one-slit experiments. Thus we show a clear and fundamental example in which transition to chaotic motion in a deterministic classical system, in absence of any external noise, leads to a profound modification in the quantum behavior.