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:nlin/0112015 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Measuring the Lyapunov exponent using quantum mechanics

Fernando Cucchietti, Caio Lewenkopf, Eduardo R. Mucciolo, Horacio M. Pastawski, Raúl O. Vallejos

Published Apr 1, 2002• 102 citations
arXivHTMLPDF

Abstract

We study the time evolution of two wave packets prepared at the same initial state, but evolving under slightly different Hamiltonians. For chaotic systems, we determine the circumstances that lead to an exponential decay with time of the wave packet overlap function. We show that for sufficiently weak perturbations, the exponential decay follows a Fermi golden rule, while by making the difference between the two Hamiltonians larger, the characteristic exponential decay time becomes the Lyapunov exponent of the classical system. We illustrate our theoretical findings by investigating numerically the overlap decay function of a two-dimensional dynamical system.

DOI
10.1103/physreve.65.046209

Related papers

Golden rule decay versus Lyapunov decay of the quantum Loschmidt echo

Philippe Jacquod, P. G. Silvestrov, C. W. J. Beenakker

The overlap of two wave packets evolving in time with slightly different Hamiltonians decays exponentially approximate to e(-gammat), for perturbation strengths U greater than the level spacing Delta. We present numerical evidence for a dynamical system that the decay rate gamma is given by the smallest of the Lyapunov exponent lambda of the classical chaotic dynamics and the level broadening U(2)/Delta that follows from the golden rule of quantum mechanics. This implies the range of validity U > the square root of [lambdaDelta] for the perturbation-strength independent decay rate discovered by Jalabert and Pastawski [Phys. Rev. Lett. 86, 2490 (2001)].

arXiv:nlin/0107044•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Oct 15, 2001•252 citations

Hypersensitivity to perturbations of quantum-chaotic wave-packet dynamics

P. G. Silvestrov, J. Tworzydło, C. W. J. Beenakker

We reexamine the problem of the ``Loschmidt echo,'' that measures the sensitivity to perturbation of quantum-chaotic dynamics. The overlap squared $M(t)$ of two wave packets evolving under slightly different Hamiltonian is shown to have the double-exponential initial decay $\ensuremath{\propto}\mathrm{exp}(\ensuremath{-}\mathrm{constant}\ifmmode\times\else\texttimes\fi{}{e}^{2{\ensuremath{\lambda}}_{0}t})$ in the main part of the phase space. The coefficient ${\ensuremath{\lambda}}_{0}$ is the self-averaging Lyapunov exponent. The average decay $\overline{M}\ensuremath{\propto}{e}^{\ensuremath{-}{\ensuremath{\lambda}}_{1}t}$ is single exponential with a different coefficient ${\ensuremath{\lambda}}_{1}.$ The volume of phase space that contributes to $\overline{M}$ vanishes in the classical limit $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Elzxh}}0$ for times less than the Ehrenfest time ${\ensuremath{\tau}}_{E}=\frac{1}{2}{\ensuremath{\lambda}}_{0}^{\ensuremath{-}1}|\mathrm{ln}\ensuremath{\Elzxh}|.$ It is only after the Ehrenfest time that the average decay is representative for a typical initial condition.

arXiv:nlin/0207002•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Feb 26, 2003•47 citations

Decoherence as decay of the Loschmidt echo in a Lorentz gas

Fernando Cucchietti, Horacio M. Pastawski, Diego A. Wisniacki

Classical chaotic dynamics is characterized by exponential sensitivity to initial conditions. Quantum mechanics, however, does not show this feature. We consider instead the sensitivity of quantum evolution to perturbations in the Hamiltonian. This is observed as an attenuation of the Loschmidt echo M(t), i.e., the amount of the original state (wave packet of width sigma) which is recovered after a time reversed evolution, in the presence of a classically weak perturbation. By considering a Lorentz gas of size L, which for large L is a model for an unbounded classically chaotic system, we find numerical evidence that, if the perturbation is within a certain range, M(t) decays exponentially with a rate 1/tau(phi) determined by the Lyapunov exponent lambda of the corresponding classical dynamics. This exponential decay extends much beyond the Eherenfest time t(E) and saturates at a time t(s) approximately equal to lambda(-1)ln[N], where N approximately (L/sigma)(2) is the effective dimensionality of the Hilbert space. Since tau(phi) quantifies the increasing uncontrollability of the quantum phase (decoherence) its characterization and control has fundamental interest.

arXiv:cond-mat/0102135•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Apr 8, 2002•62 citations

Sensitivity to perturbations in a quantum chaotic billiard

Diego A. Wisniacki, E. Vergini, Horacio M. Pastawski +1 more

The Loschmidt echo (LE) measures the ability of a system to return to the initial state after a forward quantum evolution followed by a backward perturbed one. It has been conjectured that the echo of a classically chaotic system decays exponentially, with a decay rate given by the minimum between the width Gamma of the local density of states and the Lyapunov exponent. As the perturbation strength is increased one obtains a crossover between both regimes. These predictions are based on situations where the Fermi golden rule (FGR) is valid. By considering a paradigmatic fully chaotic system, the Bunimovich stadium billiard, with a perturbation in a regime for which the FGR manifestly does not work, we find a crossover from Gamma to Lyapunov decay. We find that, challenging the analytic interpretation, these conjectures are valid even beyond the expected range.

arXiv:nlin/0111051•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•May 17, 2002•35 citations

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

Measuring the distance between quantum many-body wave functions

Xiao Chen, Tianci Zhou, Cenke Xu

Abstract We study the distance of two wave functions under chaotic time evolution. The two initial states differ only by a local perturbation. To be entitled ‘chaos’ the distance should have a rapid growth afterwards. Instead of focusing on the entire wave function, we measure the distance d 2 ( t ) by investigating the difference of two reduced density matrices of the subsystem A that is spatially separated from the local perturbation. This distance d 2 ( t ) grows with time and eventually saturates to a small constant. We interpret the distance growth in terms of operator scrambling picture, which relates d 2 ( t ) to the square of commutator (out-of-time-order correlator) and shows that both these quantities measure the area of the operator wave front in subsystem A . Among various one-dimensional spin- models, we numerically show that the models with non-local power-law interaction can have an exponentially growing regime in d 2 ( t ) when the local perturbation and subsystem A are well separated. This so-called Lyapunov regime is absent in the spin- chain with local interaction only. After sufficiently long time evolution, d 2 ( t ) relaxes to a small constant, which decays exponentially as we increase the system size and is consistent with eigenstate thermalization hypothesis. Based on these results, we demonstrate that d 2 ( t ) is a useful quantity to characterize both quantum chaos and quantum thermalization in many-body wave functions.

arXiv:1712.06054•Journal of Statistical Mechanics Theory and Experiment•Jul 1, 2018•13 citations
We study the time evolution of two wave packets prepared at the same initial state, but evolving under slightly different Hamiltonians. For chaotic systems, we determine the circumstances that lead to an exponential decay with time of the wave packet overlap function. We show that for sufficiently weak perturbations, the exponential decay follows a Fermi golden rule, while by making the difference between the two Hamiltonians larger, the characteristic exponential decay time becomes the Lyapunov exponent of the classical system. We illustrate our theoretical findings by investigating numerically the overlap decay function of a two-dimensional dynamical system.
The overlap of two wave packets evolving in time with slightly different Hamiltonians decays exponentially approximate to e(-gammat), for perturbation strengths U greater than the level spacing Delta. We present numerical evidence for a dynamical system that the decay rate gamma is given by the smallest of the Lyapunov exponent lambda of the classical chaotic dynamics and the level broadening U(2)/Delta that follows from the golden rule of quantum mechanics. This implies the range of validity U > the square root of [lambdaDelta] for the perturbation-strength independent decay rate discovered by Jalabert and Pastawski [Phys. Rev. Lett. 86, 2490 (2001)].
We reexamine the problem of the ``Loschmidt echo,'' that measures the sensitivity to perturbation of quantum-chaotic dynamics. The overlap squared M(t)M(t)M(t) of two wave packets evolving under slightly different Hamiltonian is shown to have the double-exponential initial decay $\ensuremath{\propto}\mathrm{exp}(\ensuremath{-}\mathrm{constant}\ifmmode\times\else\texttimes\fi{}{e}^{2{\ensuremath{\lambda}}_{0}t})$ in the main part of the phase space. The coefficient ${\ensuremath{\lambda}}_{0}$ is the self-averaging Lyapunov exponent. The average decay $\overline{M}\ensuremath{\propto}{e}^{\ensuremath{-}{\ensuremath{\lambda}}_{1}t}$ is single exponential with a different coefficient ${\ensuremath{\lambda}}_{1}.$ The volume of phase space that contributes to M‾\overline{M}M vanishes in the classical limit $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Elzxh}}0$ for times less than the Ehrenfest time ${\ensuremath{\tau}}_{E}=\frac{1}{2}{\ensuremath{\lambda}}_{0}^{\ensuremath{-}1}|\mathrm{ln}\ensuremath{\Elzxh}|.$ It is only after the Ehrenfest time that the average decay is representative for a typical initial condition.
Classical chaotic dynamics is characterized by exponential sensitivity to initial conditions. Quantum mechanics, however, does not show this feature. We consider instead the sensitivity of quantum evolution to perturbations in the Hamiltonian. This is observed as an attenuation of the Loschmidt echo M(t), i.e., the amount of the original state (wave packet of width sigma) which is recovered after a time reversed evolution, in the presence of a classically weak perturbation. By considering a Lorentz gas of size L, which for large L is a model for an unbounded classically chaotic system, we find numerical evidence that, if the perturbation is within a certain range, M(t) decays exponentially with a rate 1/tau(phi) determined by the Lyapunov exponent lambda of the corresponding classical dynamics. This exponential decay extends much beyond the Eherenfest time t(E) and saturates at a time t(s) approximately equal to lambda(-1)ln[N], where N approximately (L/sigma)(2) is the effective dimensionality of the Hilbert space. Since tau(phi) quantifies the increasing uncontrollability of the quantum phase (decoherence) its characterization and control has fundamental interest.
The Loschmidt echo (LE) measures the ability of a system to return to the initial state after a forward quantum evolution followed by a backward perturbed one. It has been conjectured that the echo of a classically chaotic system decays exponentially, with a decay rate given by the minimum between the width Gamma of the local density of states and the Lyapunov exponent. As the perturbation strength is increased one obtains a crossover between both regimes. These predictions are based on situations where the Fermi golden rule (FGR) is valid. By considering a paradigmatic fully chaotic system, the Bunimovich stadium billiard, with a perturbation in a regime for which the FGR manifestly does not work, we find a crossover from Gamma to Lyapunov decay. We find that, challenging the analytic interpretation, these conjectures are valid even beyond the expected range.
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.
Abstract We study the distance of two wave functions under chaotic time evolution. The two initial states differ only by a local perturbation. To be entitled ‘chaos’ the distance should have a rapid growth afterwards. Instead of focusing on the entire wave function, we measure the distance d 2 ( t ) by investigating the difference of two reduced density matrices of the subsystem A that is spatially separated from the local perturbation. This distance d 2 ( t ) grows with time and eventually saturates to a small constant. We interpret the distance growth in terms of operator scrambling picture, which relates d 2 ( t ) to the square of commutator (out-of-time-order correlator) and shows that both these quantities measure the area of the operator wave front in subsystem A . Among various one-dimensional spin- models, we numerically show that the models with non-local power-law interaction can have an exponentially growing regime in d 2 ( t ) when the local perturbation and subsystem A are well separated. This so-called Lyapunov regime is absent in the spin- chain with local interaction only. After sufficiently long time evolution, d 2 ( t ) relaxes to a small constant, which decays exponentially as we increase the system size and is consistent with eigenstate thermalization hypothesis. Based on these results, we demonstrate that d 2 ( t ) is a useful quantity to characterize both quantum chaos and quantum thermalization in many-body wave functions.