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

Random perturbation of systematic degeneracies and their Fourier transforms and periodic orbits

W. D. Heiss, Markus Müller

Published Oct 1, 1993• 6 citations
Publisher

Abstract

The effect of random perturbation is investigated for a Hamiltonian with systematic degeneracies. It turns out that the influence of the degeneracies remains significant under perturbation. The model simulates quantized versions of classically chaotic systems in that the Fourier transform of the level density shows the characteristic lines which would be associated with the periods of periodic orbits if the system had a classical analogy. The decaying magnitude of the recurring lines is related to the spread of the probability distribution under the influence of perturbation of the degenerate levels.

DOI
10.1103/physreva.48.2558

Related papers

Random perturbation of systematic degeneracies and quantum chaos

W. D. Heiss, John C. H. Chiang

The effect of a random perturbation is investigated for a Hamiltonian with systematic degeneracies. It turns out that the influence of the degeneracies remains significant under a perturbation. In particular, the eigenvectors manifest specific patterns when the perturbation produces all characteristics of quantum chaos for the spectrum.

Physical Review A•Apr 1, 1993•7 citations

Scar and antiscar quantum effects in open chaotic systems

Lev Kaplan

We predict and numerically observe strong periodic orbit effects in the properties of weakly open quantum systems with a chaotic classical limit. Antiscars lead to a large number of exponentially narrow isolated resonances when the single-channel (or tunneling) opening is located on a short unstable orbit of the closed system; the probability to remain in the system at long times is thus exponentially enhanced over the random matrix theory prediction. The distribution of resonance widths and the probability to remain are quantitatively given in terms of only the stability matrix of the orbit on which the opening is placed. The long-time remaining probability density is nontrivially distributed over the available phase space; it can be enhanced or suppressed near orbits other than the one on which the lead is located, depending on the periods and classical actions of these other orbits. These effects of the short periodic orbits on quantum decay rates have no classical counterpart, and first appear on times scales much larger than the Heisenberg time of the system. All the predictions are quantitatively compared with numerical data.

arXiv:chao-dyn/9809013•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•May 1, 1999•22 citations

Introduction to classical chaos and quantum chaos.

Faure, Frédéric

This article gives an introduction to chaotic systems in classical mechanics and their quantum mechanical counterparts.The author assumes very few prerequisites and starts with a basic introduction to the mathematical formulation of classical and quantum mechanics and discusses the relation between classical and quantum systems by means of the mathematical procedure of quantization. Then, the article proceeds with discussing the notion of ``chaos'' in classical mechanics and particularly focuses on the concepts of hyperbolicity, ergodicity and mixing. In the last section, the quantum mechanical properties of classically chaotic systems are discussed. Here, the focus lies in particular on quantum ergodicity and the role of Egorov's theorem. Finally, the distribution of eigenvalues, according to Weyl's law, {\it M.C. Gutzwiller}'s trace formula [``Periodic orbits and classical quantization conditions'', J. Math. Phys., 12, No. 3, 343--358 (1971; \url{doi:10.1063/1.1665596})] as well as the random matrix conjecture of {\it O. Bohigas} et al. [``Characterization of chaotic quantum spectra and universality of level fluctuation laws'', Phys. Rev. Lett. 52, No. 1, 1--4 (1984; \url{doi:10.1103/PhysRevLett.52.1})] is also discussed.A particularly nice feature throughout the whole text is that it includes both mathematically rigorous definitions which can be illustrated for toy models such as the cat map, as well as phyiscally relevant sytems such as billiards. For the latter it is then outlined, where the difficulties in a mathematically rigorous description lie.

•Jan 1, 2014•0 citations

Scars in quantum chaotic wavefunctions

Lev Kaplan

We review recent progress in attaining a quantitative understanding of the scarring phenomenon, the non-random behaviour of quantum wavefunctions near unstable periodic orbits of a classically chaotic system. The wavepacket dynamics framework leads to predictions about statistical long-time and stationary properties of quantum systems with chaotic classical analogues. Many long-time quantum properties can be quantitatively understood using only short-time classical dynamics information; these include wavefunction intensity distributions, intensity correlations in phase space and correlations between wavefunctions, and distributions of decay rates and conductance peaks in weakly open systems. Strong deviations from random matrix theory are predicted and observed in the presence of short unstable periodic orbits.

arXiv:chao-dyn/9810013•Nonlinearity•Jan 1, 1999•112 citations

ADIABATIC APPROXIMATION FOR DEGENERATE OR ALMOST DEGENERATE STATES

G. F. Drukarev

The behavior of a system under the action of a timedependent perturbation is considered in the adiabatic approximation, degeneracy being assumed in the absence of the perturbation. The state of the system is characterized by the density matrix in space of eigenfunctions belonging to a given degenerate level. A unitary operator is set up that relates the density matrix before the perturbation is turned on to that after the perturbation is turned off. The case when there is no exact degeneracy in the absence of a perturbation but rather a set of closely spaced levels (almost degenerate states) is also considered. As an illustration of application of the theory, the behavior of spin 1 and 1/2 particles in a variable magnetic field is examined. (auth)

•Nov 1, 1963•0 citations

Spectral oscillations, periodic orbits, and scaling

S A Fulling

The eigenvalue density of a quantum-mechanical system exhibits oscillations, determined by the closed orbits of the corresponding classical system; this relationship is simple and strong for waves in billiards or on manifolds, but becomes slightly muddy for a Schrödinger equation with a potential, where the orbits depend on the energy.In special cases the simplicity has been restored by rescaling the size of the orbit, and perhaps a coupling constant at the same time.We point out that the goal can be achieved for any system of this class simply by rescaling the overall coupling constant.In each of these situations we inspect critically the relation between the oscillation frequency and the period of the orbit; in many cases it is observed that a characteristic length of the orbit is a better indicator.When these matters are properly understood, the periodic-orbit theory for generic quantum systems recovers the clarity and simplicity that it always had for the wave equation in a cavity.Finally, we comment on the alleged 'paradox' that semiclassical periodic-orbit theory is more effective in calculating low energy levels than high ones.

arXiv:quant-ph/0012070•Journal of Physics A Mathematical and General•Apr 26, 2002•4 citations
The effect of random perturbation is investigated for a Hamiltonian with systematic degeneracies. It turns out that the influence of the degeneracies remains significant under perturbation. The model simulates quantized versions of classically chaotic systems in that the Fourier transform of the level density shows the characteristic lines which would be associated with the periods of periodic orbits if the system had a classical analogy. The decaying magnitude of the recurring lines is related to the spread of the probability distribution under the influence of perturbation of the degenerate levels.
We predict and numerically observe strong periodic orbit effects in the properties of weakly open quantum systems with a chaotic classical limit. Antiscars lead to a large number of exponentially narrow isolated resonances when the single-channel (or tunneling) opening is located on a short unstable orbit of the closed system; the probability to remain in the system at long times is thus exponentially enhanced over the random matrix theory prediction. The distribution of resonance widths and the probability to remain are quantitatively given in terms of only the stability matrix of the orbit on which the opening is placed. The long-time remaining probability density is nontrivially distributed over the available phase space; it can be enhanced or suppressed near orbits other than the one on which the lead is located, depending on the periods and classical actions of these other orbits. These effects of the short periodic orbits on quantum decay rates have no classical counterpart, and first appear on times scales much larger than the Heisenberg time of the system. All the predictions are quantitatively compared with numerical data.
This article gives an introduction to chaotic systems in classical mechanics and their quantum mechanical counterparts.The author assumes very few prerequisites and starts with a basic introduction to the mathematical formulation of classical and quantum mechanics and discusses the relation between classical and quantum systems by means of the mathematical procedure of quantization. Then, the article proceeds with discussing the notion of ``chaos'' in classical mechanics and particularly focuses on the concepts of hyperbolicity, ergodicity and mixing. In the last section, the quantum mechanical properties of classically chaotic systems are discussed. Here, the focus lies in particular on quantum ergodicity and the role of Egorov's theorem. Finally, the distribution of eigenvalues, according to Weyl's law, {\it M.C. Gutzwiller}'s trace formula [``Periodic orbits and classical quantization conditions'', J. Math. Phys., 12, No. 3, 343--358 (1971; \url{doi:10.1063/1.1665596})] as well as the random matrix conjecture of {\it O. Bohigas} et al. [``Characterization of chaotic quantum spectra and universality of level fluctuation laws'', Phys. Rev. Lett. 52, No. 1, 1--4 (1984; \url{doi:10.1103/PhysRevLett.52.1})] is also discussed.A particularly nice feature throughout the whole text is that it includes both mathematically rigorous definitions which can be illustrated for toy models such as the cat map, as well as phyiscally relevant sytems such as billiards. For the latter it is then outlined, where the difficulties in a mathematically rigorous description lie.
We review recent progress in attaining a quantitative understanding of the scarring phenomenon, the non-random behaviour of quantum wavefunctions near unstable periodic orbits of a classically chaotic system. The wavepacket dynamics framework leads to predictions about statistical long-time and stationary properties of quantum systems with chaotic classical analogues. Many long-time quantum properties can be quantitatively understood using only short-time classical dynamics information; these include wavefunction intensity distributions, intensity correlations in phase space and correlations between wavefunctions, and distributions of decay rates and conductance peaks in weakly open systems. Strong deviations from random matrix theory are predicted and observed in the presence of short unstable periodic orbits.
The behavior of a system under the action of a timedependent perturbation is considered in the adiabatic approximation, degeneracy being assumed in the absence of the perturbation. The state of the system is characterized by the density matrix in space of eigenfunctions belonging to a given degenerate level. A unitary operator is set up that relates the density matrix before the perturbation is turned on to that after the perturbation is turned off. The case when there is no exact degeneracy in the absence of a perturbation but rather a set of closely spaced levels (almost degenerate states) is also considered. As an illustration of application of the theory, the behavior of spin 1 and 1/2 particles in a variable magnetic field is examined. (auth)
The eigenvalue density of a quantum-mechanical system exhibits oscillations, determined by the closed orbits of the corresponding classical system; this relationship is simple and strong for waves in billiards or on manifolds, but becomes slightly muddy for a Schrödinger equation with a potential, where the orbits depend on the energy.In special cases the simplicity has been restored by rescaling the size of the orbit, and perhaps a coupling constant at the same time.We point out that the goal can be achieved for any system of this class simply by rescaling the overall coupling constant.In each of these situations we inspect critically the relation between the oscillation frequency and the period of the orbit; in many cases it is observed that a characteristic length of the orbit is a better indicator.When these matters are properly understood, the periodic-orbit theory for generic quantum systems recovers the clarity and simplicity that it always had for the wave equation in a cavity.Finally, we comment on the alleged 'paradox' that semiclassical periodic-orbit theory is more effective in calculating low energy levels than high ones.