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:chao-dyn/9807009 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Fingerprints of classical instability in open quantum dynamics

Paul Miller, Sarben Sarkar

Published Oct 1, 1998• 33 citations
arXivHTMLPDF

Abstract

The dynamics near a hyperbolic fixed point in phase space is modeled by an inverted harmonic oscillator. We investigate the effect of the classical instability on the open quantum dynamics of the oscillator, introduced through the interaction with a thermal bath, using both the survival probability function and the rate of von Neumann entropy increase, for large times. In this parameter range we prove, using influence functional techniques, that the survival probability function decreases exponentially at a rate ${\ensuremath{\kappa}}^{\ensuremath{'}}$ depending not only on the measure of instability in the model but also on the strength of interaction with the environment. We also show that ${\ensuremath{\kappa}}^{\ensuremath{'}}$ determines the rate of the von Neumann entropy increase and that this result is independent of the temperature of the environment. This generalizes earlier results that are valid in the limit of vanishing dissipation. The validity of inferring similar rates of survival probability decrease and entropy increase for quantum chaotic systems is also discussed.

DOI
10.1103/physreve.58.4217

Related papers

Quantum system under periodic perturbation: Effect of environment

Masahiro Hotta, I. Joichi, Sh. Matsumoto +1 more

In many physical situations the behavior of a quantum system is affected by interaction with a larger environment. We develop, using the method of an influence functional, how to deduce the density matrix of the quantum system incorporating the effect of environment. After introducing the characterization of the environment by spectral weight, we first devise schemes to approximate the spectral weight, and then a perturbation method in field theory models, in order to approximately describe the environment. All of these approximate models may be classified as extended Ohmic models of dissipation whose differences are in the high frequency part. The quantum system we deal with in the present work is a general class of harmonic oscillators with an arbitrary time-dependent frequency. The late time behavior of the system is well described by an approximation that employs a localized friction in the dissipative part of the correlation function appearing in the influence functional. The density matrix of the quantum system is then determined in terms of a single classical solution obtained with the time-dependent frequency. With this one can compute the entropy, the energy distribution function, and other physical quantities of the system in a closed form. A specific application is made to the case of a periodically varying frequency. This dynamical system has a remarkable property when the environmental interaction is switched off: The effect of the parametric resonance gives rise to an exponential growth of the populated number in higher excitation levels, or particle production in field theory models. The effect of the environment is investigated for this dynamical system and it is demonstrated that there exists a critical strength of the friction for the parametric effect. In the model of a periodically oscillating field coupled to a system quantum field, it is verified that the parametric effect occurs in medium, with a somewhat diminished rate, if the relaxation time scale of the system field towards thermalization given by the friction term is larger than the time scale of the coherent parametric amplification. The effect persists until the back reaction against the periodic oscillation stops particle production. The resulting energy distribution of produced particles described by a universal function deviates from the thermal one, having an average energy that exponentially increases with time. The dynamical system driven by the parametric oscillator thus maintains and does not lose its quantum nature even in thermal bath. In the present work analytic formulas of how physical quantities behave at late times both in the high and in the low temperature regions are given, along with the results of a numerical computation displaying time evolution.

arXiv:hep-ph/9608374•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Apr 15, 1997•12 citations

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

Bound on the rate of entropy change in open quantum systems

Pharnam Bakhshinezhad, A. T. Rezakhani

We study the temporal rate of variations of the von Neumann entropy in an open quantum system which interacts with a bath. We show that for almost all initial states of the bath and the system, the time-average of the rate of entropy change is bounded by a function which depends on various properties of the system and environment, and is mostly relatively small. This result holds true under fairly general conditions in almost any arbitrary quantum system.

arXiv:1808.08305•Aug 24, 2018•1 citation

Survival of coherence for open quantum systems in thermal baths

Filippo Giraldi, Francesco Petruccione

The loss of coherence in a general open quantum system interacting with a bosonic environment is analyzed. The reservoir is initially in a thermal state. The reduced dynamics is described by a non-Markovian time-local master equation. We consider spectral densities that are sub- or super-Ohmic at low frequencies and arbitrarily shaped at high frequencies. In the super-Ohmic regime, for noninteger frequency powers larger than 2, long time survival of coherence appears. In the latter regime, at vanishing temperature, the asymptotic amount of surviving coherence is stabilized to its initial value, up to a phase factor, by properly increasing the bandwidth and decreasing the low-frequency profile of the spectral density. For noninteger positive frequency powers less than 2, stretched exponential-like decoherence is found over long times. The relaxations to the asymptotic configurations become arbitrarily slow by approaching the frequency power 2 of the super-Ohmic regime. The same dependence on temperature, spectral density, and scale frequency appears for purity and concurrence of two qubits and coherence of a qubit.

Physical Review A•Oct 2, 2013•14 citations

Quantum chaos: a decoherent definition

Zurek, W H, Paz, J P

We show that the rate of increase of von Neumann entropy computed from the reduced density matrix of an open quantum system is an excellent indicator of the dynamical behavior of its classical hamiltonian counterpart. In decohering quantum analogs of systems which exhibit classical hamiltonian chaos entropy production rate quickly tends to a constant which is given by the sum of the positive Lyapunov exponents, and falls off only as the system approaches equilibrium. By contrast, integrable systems tend to have entropy production rate which decreases as t^{-1} well before equilibrium is attained. Thus, behavior of quantum systems in contact with the environment can be used as a test to determine the nature of their hamiltonian evolution.

•Jan 1, 1995•94 citations

Entropy of a quantum oscillator in a structured environment

Jishad Kumar

Entropy of a quantum oscillator in the presence of a quantum environment (or heat bath) is studied here. The spectral density, which comprises the environmental influences, here corresponds to a quasi-monochromatic thermal harmonic noise. This further implies that the environment is a structured one compared to the usually employed unstructured strictly ohmic environment. The entropy is found to be vanishing as the temperature approaches its absolute zero value, which thereby guarantees the third law of thermodynamics. At very high temperatures the classical behavior is observed. Similar to the usually employed ohmic-Drude bath model, the coefficient of the leading temperature dependent term in the entropy (at low temperatures) is proportional to the damping strength but not on other parameters of the spectral density characterising the bath.

Physica Scripta•Aug 8, 2014•0 citations
The dynamics near a hyperbolic fixed point in phase space is modeled by an inverted harmonic oscillator. We investigate the effect of the classical instability on the open quantum dynamics of the oscillator, introduced through the interaction with a thermal bath, using both the survival probability function and the rate of von Neumann entropy increase, for large times. In this parameter range we prove, using influence functional techniques, that the survival probability function decreases exponentially at a rate \ensuremathκ\ensuremath′{\ensuremath{\kappa}}^{\ensuremath{'}}\ensuremathκ\ensuremath′ depending not only on the measure of instability in the model but also on the strength of interaction with the environment. We also show that \ensuremathκ\ensuremath′{\ensuremath{\kappa}}^{\ensuremath{'}}\ensuremathκ\ensuremath′ determines the rate of the von Neumann entropy increase and that this result is independent of the temperature of the environment. This generalizes earlier results that are valid in the limit of vanishing dissipation. The validity of inferring similar rates of survival probability decrease and entropy increase for quantum chaotic systems is also discussed.
In many physical situations the behavior of a quantum system is affected by interaction with a larger environment. We develop, using the method of an influence functional, how to deduce the density matrix of the quantum system incorporating the effect of environment. After introducing the characterization of the environment by spectral weight, we first devise schemes to approximate the spectral weight, and then a perturbation method in field theory models, in order to approximately describe the environment. All of these approximate models may be classified as extended Ohmic models of dissipation whose differences are in the high frequency part. The quantum system we deal with in the present work is a general class of harmonic oscillators with an arbitrary time-dependent frequency. The late time behavior of the system is well described by an approximation that employs a localized friction in the dissipative part of the correlation function appearing in the influence functional. The density matrix of the quantum system is then determined in terms of a single classical solution obtained with the time-dependent frequency. With this one can compute the entropy, the energy distribution function, and other physical quantities of the system in a closed form. A specific application is made to the case of a periodically varying frequency. This dynamical system has a remarkable property when the environmental interaction is switched off: The effect of the parametric resonance gives rise to an exponential growth of the populated number in higher excitation levels, or particle production in field theory models. The effect of the environment is investigated for this dynamical system and it is demonstrated that there exists a critical strength of the friction for the parametric effect. In the model of a periodically oscillating field coupled to a system quantum field, it is verified that the parametric effect occurs in medium, with a somewhat diminished rate, if the relaxation time scale of the system field towards thermalization given by the friction term is larger than the time scale of the coherent parametric amplification. The effect persists until the back reaction against the periodic oscillation stops particle production. The resulting energy distribution of produced particles described by a universal function deviates from the thermal one, having an average energy that exponentially increases with time. The dynamical system driven by the parametric oscillator thus maintains and does not lose its quantum nature even in thermal bath. In the present work analytic formulas of how physical quantities behave at late times both in the high and in the low temperature regions are given, along with the results of a numerical computation displaying time evolution.
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.
The loss of coherence in a general open quantum system interacting with a bosonic environment is analyzed. The reservoir is initially in a thermal state. The reduced dynamics is described by a non-Markovian time-local master equation. We consider spectral densities that are sub- or super-Ohmic at low frequencies and arbitrarily shaped at high frequencies. In the super-Ohmic regime, for noninteger frequency powers larger than 2, long time survival of coherence appears. In the latter regime, at vanishing temperature, the asymptotic amount of surviving coherence is stabilized to its initial value, up to a phase factor, by properly increasing the bandwidth and decreasing the low-frequency profile of the spectral density. For noninteger positive frequency powers less than 2, stretched exponential-like decoherence is found over long times. The relaxations to the asymptotic configurations become arbitrarily slow by approaching the frequency power 2 of the super-Ohmic regime. The same dependence on temperature, spectral density, and scale frequency appears for purity and concurrence of two qubits and coherence of a qubit.
We show that the rate of increase of von Neumann entropy computed from the reduced density matrix of an open quantum system is an excellent indicator of the dynamical behavior of its classical hamiltonian counterpart. In decohering quantum analogs of systems which exhibit classical hamiltonian chaos entropy production rate quickly tends to a constant which is given by the sum of the positive Lyapunov exponents, and falls off only as the system approaches equilibrium. By contrast, integrable systems tend to have entropy production rate which decreases as t^{-1} well before equilibrium is attained. Thus, behavior of quantum systems in contact with the environment can be used as a test to determine the nature of their hamiltonian evolution.
Entropy of a quantum oscillator in the presence of a quantum environment (or heat bath) is studied here. The spectral density, which comprises the environmental influences, here corresponds to a quasi-monochromatic thermal harmonic noise. This further implies that the environment is a structured one compared to the usually employed unstructured strictly ohmic environment. The entropy is found to be vanishing as the temperature approaches its absolute zero value, which thereby guarantees the third law of thermodynamics. At very high temperatures the classical behavior is observed. Similar to the usually employed ohmic-Drude bath model, the coefficient of the leading temperature dependent term in the entropy (at low temperatures) is proportional to the damping strength but not on other parameters of the spectral density characterising the bath.