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:quant-ph/0410146 · Physical Review A

Decoherence and the quantum-classical limit in the presence of chaos

Fabricio Toscano, R. L. de Matos Filho, L. Davidovich

Published Jan 31, 2005• 24 citations
arXivHTMLPDF

Abstract

We investigate how decoherence affects the short-time separation between quantum and classical dynamics for classically chaotic systems, within the framework of a specific model. For a wide range of parameters, the distance between the corresponding phase-space distributions depends on a single parameter $\ensuremath{\chi}$ that relates an effective Planck constant ${\ensuremath{\hbar}}_{\mathrm{eff}}$, the Lyapunov coefficient, and the diffusion constant. This distance peaks at a time that depends logarithmically on ${\ensuremath{\hbar}}_{\mathrm{eff}}$, in agreement with previous estimations of the separation time for Hamiltonian systems. However, for $\ensuremath{\chi}\ensuremath{\lesssim}1$, the separation remains small, going down with ${\ensuremath{\hbar}}_{\mathrm{eff}}^{2}$, so the concept of separation time loses its meaning.

DOI
10.1103/physreva.71.010101

Related papers

Decoherence and the Quantum-Classical Transition in Phase Space

L. Davidovich

We investigate how decoherence affects the short‐time separation between quantum and classical dynamics for classically chaotic systems, within the framework of a specific model, which corresponds to an ion trapped in a harmonic trap and submitted to non‐resonant laser pulses. For a wide range of parameters, the distance between the classical and quantum (Wigner function) phase‐space distributions depends on a single parameter χ that relates an effective Planck constant ℏeff (given by the square of the Lamb‐Dicke parameter), the Lyapunov coeffficient, and the diffusion constant. This distance peaks at a time that depends logarithmically on ℏeff, in agreement with previous estimations of the separation time for Hamiltonian systems. Due to the logarithmic dependence, this time may be very small, even for macroscopis sytems, which would seem to imply that quantum behavior should become manifest after a short time scale. We show however that, for χ1, the separation remains small, going down with ℏeff2. In this limit, the concept of separation time does not make sense anymore, and the classical behavior of the system is recovered.

AIP conference proceedings•Jan 1, 2005•1 citation

Decoherence and the quantum-classical limit in phase space (Invited Paper)

L. Davidovich, Fabricio Toscano, R. L. de Matos Filho

We discuss the emergence of the classical description from quantum mechanics for chaotic systems. We consider a specific model, which corresponds to an ion trapped in a harmonic potential and submitted to a sequence of laser pulses: the kicked harmonic oscillator, under conditions that lead to chaotic behavior for the classical system. We show that noise plays an essential role in the transition from quantum to classical behavior. This transition is described in terms of the separation between the classical phase space distribution and the Wigner function corresponding to the quantum system, obtained by integrating over the whole phase space the magnitude of the difference between the two distributions. It is shown that, in the semiclassical limit, this separation is governed by a single parameter, which depends on the diffusion coefficient, the Lamb-Dicke parameter, and the kick strength. The Lamb-Dicke parameter plays the role of a dimensioneless Planck constant. As this parameter goes to zero, we show that both distributions remain close together for all times.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•May 23, 2005•0 citations

Decoherence, Chaos, Quantum-Classical Correspondence, and the Algorithmic Arrow of Time

Wojciech Hubert Zurek

The environment – external or internal degrees of freedom coupled to the system – can, in effect, monitor some of its observables. As a result, the eigenstates of these observables decohere and behave like classical states: Continuous destruction of superpositions leads to environment-induced superselection (einselection). Here I investigate it in the context of quantum chaos (i.e., quantum dynamics of systems which are classically chaotic). I show that the evolution of a chaotic macroscopic (but, ultimately, quantum) system is not just difficult to predict (requiring an accuracy exponentially increasing with time) but quickly ceases to be deterministic in principle as a result of the Heisenberg indeterminacy (which limits the resolution available in the initial conditions). This happens after a time t ℏ which is only logarithmic in the Planck constant. A definitely macroscopic (if somewhat outrageous) example is afforded by various components of the solar system which are chaotic, with the Lyapunov timescales ranging from a bit more than a month (Hyperion) to millions of years (planetary system as a whole). On the timescale t ℏ the initial minimum uncertainty wavepackets corresponding to celestial bodies would be smeared over distances of the order of radii of their orbits into “Schrödinger cat – like” states, and the concept of a trajectory would cease to apply. In reality, such paradoxical states are eliminated by decoherence which helps restore quantum-classical correspondence. The price for the recovery of classicality is the loss of predictability: In the classical limit (associated with effective decoherence, and not just with the smallness of ℏ) the rate of increase of the von Neumann entropy of the decohering system is independent of the strength of the coupling to the environment, and equal to the sum of the positive Lyapunov exponents. Algorithmic aspects of entropy production are briefly explored to illustrate the effect of decoherence from the point of view of the observer. We show that “decoherence strikes twice”, introducing unpredictability into the system and extracting quantum coherence from the observers memory, where it enters as a price for the classicality of his records.

arXiv:quant-ph/9802054•Physica Scripta•Jan 1, 1998•150 citations

Decoherence in a classically chaotic quantum system: Entropy production and quantum-classical correspondence

Diana Monteoliva, Juan Pablo Paz

We study the decoherence process for an open quantum system that is classically chaotic (a quartic double well with harmonic driving coupled to a sea of harmonic oscillators). We carefully analyze the time dependence of the rate of entropy production showing that it has two relevant regimes: For short times it is proportional to the diffusion coefficient (fixed by the system-environment coupling strength); for longer times (but before equilibration) it is fixed by dynamical properties of the system (and is related to the Lyapunov exponent). The nature of the transition time between both regimes is investigated and the issue of quantum to classical correspondence is addressed. Finally, the impact of the interaction with the environment on coherent tunneling is analyzed.

arXiv:quant-ph/0106090•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Oct 30, 2001•28 citations

Quantum-to-classical transition in a system with mixed classical dynamics

Fabricio Toscano, Diego A. Wisniacki

We study how decoherence rules the quantum-classical transition of the kicked harmonic oscillator. The system presents classical dynamics that ranges from regular to strong chaotic behavior depending on the amplitude of the kicks. We show that for regular and mixed classical dynamics, and in the presence of noise, the distance between the classical and quantum phase space distributions is proportional to a single parameter chi identical to K Planck's (eff)(2)/4D(3/2) , which relates the effective Planck constant, Planck's (eff), to the kicking strength, K, and the diffusion constant, D. This relation between classical and quantum distributions is valid when chi<1 , a case that is always attainable in the semiclassical regime, independent of the value of the strength of noise given by D. Our results extend a recent study performed in the chaotic regime.

arXiv:quant-ph/0606024•Physical Review E•Nov 27, 2006•6 citations

The threshold for quantum-classical correspondence is $D \sim \hbar^{\frac43}$

Felipe Hernández, Daniel Ranard, C. Jess Riedel

In chaotic quantum systems, an initially localized quantum state can deviate strongly from the corresponding classical phase-space distribution after the Ehrenfest time $t_{\mathrm{E}} \sim \log(\hbar^{-1})$, even in the limit $\hbar \to 0$. Decoherence by the environment is often invoked to explain the persistence of the quantum-classical correspondence at longer timescales. Recent rigorous results for Lindblad dynamics with phase-space diffusion strength $D$ show that quantum and classical evolutions remain close for times that are exponentially longer than the Ehrenfest time whenever $D \gg \hbar^{\frac43}$, in units set by the classical Hamiltonian. At the same time, some heuristic arguments have suggested the weaker condition $D \gg \hbar^{2}$ always suffices. Here we construct an explicit Lindbladian that demonstrates that the scaling $D \sim \hbar^{\frac43}$ is indeed the threshold for quantum-classical correspondence beyond the Ehrenfest time. Our example uses a smooth time-dependent Hamiltonian and linear Lindblad operators generating homogeneous isotropic diffusion. It exhibits an $\hbar$-independent quantum-classical discrepancy at the Ehrenfest time whenever $D \ll \hbar^{\frac43}$, even for $\hbar$-independent "macroscopic" smooth observables.

quant-pharXiv:2512.17623•Dec 22, 2025•0 citations
We investigate how decoherence affects the short-time separation between quantum and classical dynamics for classically chaotic systems, within the framework of a specific model. For a wide range of parameters, the distance between the corresponding phase-space distributions depends on a single parameter \ensuremathχ\ensuremath{\chi}\ensuremathχ that relates an effective Planck constant \ensuremathℏeff{\ensuremath{\hbar}}_{\mathrm{eff}}\ensuremathℏeff​, the Lyapunov coefficient, and the diffusion constant. This distance peaks at a time that depends logarithmically on \ensuremathℏeff{\ensuremath{\hbar}}_{\mathrm{eff}}\ensuremathℏeff​, in agreement with previous estimations of the separation time for Hamiltonian systems. However, for \ensuremathχ\ensuremath≲1\ensuremath{\chi}\ensuremath{\lesssim}1\ensuremathχ\ensuremath≲1, the separation remains small, going down with \ensuremathℏeff2{\ensuremath{\hbar}}_{\mathrm{eff}}^{2}\ensuremathℏeff2​, so the concept of separation time loses its meaning.
We investigate how decoherence affects the short‐time separation between quantum and classical dynamics for classically chaotic systems, within the framework of a specific model, which corresponds to an ion trapped in a harmonic trap and submitted to non‐resonant laser pulses. For a wide range of parameters, the distance between the classical and quantum (Wigner function) phase‐space distributions depends on a single parameter χ that relates an effective Planck constant ℏeff (given by the square of the Lamb‐Dicke parameter), the Lyapunov coeffficient, and the diffusion constant. This distance peaks at a time that depends logarithmically on ℏeff, in agreement with previous estimations of the separation time for Hamiltonian systems. Due to the logarithmic dependence, this time may be very small, even for macroscopis sytems, which would seem to imply that quantum behavior should become manifest after a short time scale. We show however that, for χ1, the separation remains small, going down with ℏeff2. In this limit, the concept of separation time does not make sense anymore, and the classical behavior of the system is recovered.
We discuss the emergence of the classical description from quantum mechanics for chaotic systems. We consider a specific model, which corresponds to an ion trapped in a harmonic potential and submitted to a sequence of laser pulses: the kicked harmonic oscillator, under conditions that lead to chaotic behavior for the classical system. We show that noise plays an essential role in the transition from quantum to classical behavior. This transition is described in terms of the separation between the classical phase space distribution and the Wigner function corresponding to the quantum system, obtained by integrating over the whole phase space the magnitude of the difference between the two distributions. It is shown that, in the semiclassical limit, this separation is governed by a single parameter, which depends on the diffusion coefficient, the Lamb-Dicke parameter, and the kick strength. The Lamb-Dicke parameter plays the role of a dimensioneless Planck constant. As this parameter goes to zero, we show that both distributions remain close together for all times.
The environment – external or internal degrees of freedom coupled to the system – can, in effect, monitor some of its observables. As a result, the eigenstates of these observables decohere and behave like classical states: Continuous destruction of superpositions leads to environment-induced superselection (einselection). Here I investigate it in the context of quantum chaos (i.e., quantum dynamics of systems which are classically chaotic). I show that the evolution of a chaotic macroscopic (but, ultimately, quantum) system is not just difficult to predict (requiring an accuracy exponentially increasing with time) but quickly ceases to be deterministic in principle as a result of the Heisenberg indeterminacy (which limits the resolution available in the initial conditions). This happens after a time t ℏ which is only logarithmic in the Planck constant. A definitely macroscopic (if somewhat outrageous) example is afforded by various components of the solar system which are chaotic, with the Lyapunov timescales ranging from a bit more than a month (Hyperion) to millions of years (planetary system as a whole). On the timescale t ℏ the initial minimum uncertainty wavepackets corresponding to celestial bodies would be smeared over distances of the order of radii of their orbits into “Schrödinger cat – like” states, and the concept of a trajectory would cease to apply. In reality, such paradoxical states are eliminated by decoherence which helps restore quantum-classical correspondence. The price for the recovery of classicality is the loss of predictability: In the classical limit (associated with effective decoherence, and not just with the smallness of ℏ) the rate of increase of the von Neumann entropy of the decohering system is independent of the strength of the coupling to the environment, and equal to the sum of the positive Lyapunov exponents. Algorithmic aspects of entropy production are briefly explored to illustrate the effect of decoherence from the point of view of the observer. We show that “decoherence strikes twice”, introducing unpredictability into the system and extracting quantum coherence from the observers memory, where it enters as a price for the classicality of his records.
We study the decoherence process for an open quantum system that is classically chaotic (a quartic double well with harmonic driving coupled to a sea of harmonic oscillators). We carefully analyze the time dependence of the rate of entropy production showing that it has two relevant regimes: For short times it is proportional to the diffusion coefficient (fixed by the system-environment coupling strength); for longer times (but before equilibration) it is fixed by dynamical properties of the system (and is related to the Lyapunov exponent). The nature of the transition time between both regimes is investigated and the issue of quantum to classical correspondence is addressed. Finally, the impact of the interaction with the environment on coherent tunneling is analyzed.
We study how decoherence rules the quantum-classical transition of the kicked harmonic oscillator. The system presents classical dynamics that ranges from regular to strong chaotic behavior depending on the amplitude of the kicks. We show that for regular and mixed classical dynamics, and in the presence of noise, the distance between the classical and quantum phase space distributions is proportional to a single parameter chi identical to K Planck's (eff)(2)/4D(3/2) , which relates the effective Planck constant, Planck's (eff), to the kicking strength, K, and the diffusion constant, D. This relation between classical and quantum distributions is valid when chi<1 , a case that is always attainable in the semiclassical regime, independent of the value of the strength of noise given by D. Our results extend a recent study performed in the chaotic regime.
The threshold for quantum-classical correspondence is D∼ℏ43D \sim \hbar^{\frac43}D∼ℏ34​
In chaotic quantum systems, an initially localized quantum state can deviate strongly from the corresponding classical phase-space distribution after the Ehrenfest time tE∼log⁡(ℏ−1)t_{\mathrm{E}} \sim \log(\hbar^{-1})tE​∼log(ℏ−1), even in the limit ℏ→0\hbar \to 0ℏ→0. Decoherence by the environment is often invoked to explain the persistence of the quantum-classical correspondence at longer timescales. Recent rigorous results for Lindblad dynamics with phase-space diffusion strength DDD show that quantum and classical evolutions remain close for times that are exponentially longer than the Ehrenfest time whenever D≫ℏ43D \gg \hbar^{\frac43}D≫ℏ34​, in units set by the classical Hamiltonian. At the same time, some heuristic arguments have suggested the weaker condition D≫ℏ2D \gg \hbar^{2}D≫ℏ2 always suffices. Here we construct an explicit Lindbladian that demonstrates that the scaling D∼ℏ43D \sim \hbar^{\frac43}D∼ℏ34​ is indeed the threshold for quantum-classical correspondence beyond the Ehrenfest time. Our example uses a smooth time-dependent Hamiltonian and linear Lindblad operators generating homogeneous isotropic diffusion. It exhibits an ℏ\hbarℏ-independent quantum-classical discrepancy at the Ehrenfest time whenever D≪ℏ43D \ll \hbar^{\frac43}D≪ℏ34​, even for ℏ\hbarℏ-independent "macroscopic" smooth observables.