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/9812019 · Physica Scripta

The 1/N-expansion, Quantum-Classical Correspondence and Nonclassical States Generation in Dissipative Higher-Order Anharmonic Oscillators

K. N. Alekseev, Jan Peřina

Published Jan 1, 2000• 2 citations
arXivHTMLPDF

Abstract

We develop a method for the determination of the dynamics of dissipative quantum systems in the limit of large number of quanta N , based on the 1/ N -expansion of Heidmann et al. (1985 Opt. Commun. 54 189) and the quantum-classical correspondence. Using this method, we find analytically the dynamics of nonclassical states generation in the higher-order anharmonic dissipative oscillators for an arbitrary temperature of a reservoir. We show that the quantum correction to the classical motion increases with time quadratically up to some maximal value, which is dependent on the degree of nonlinearity and a damping constant, and then it decreases. Similarities and differences with the corresponding behavior of the quantum corrections to the classical motion in the Hamiltonian chaotic systems are discussed. We also compare our results obtained for some limiting cases with the results obtained by using other semiclassical tools and discuss the conditions for validity of our approach.

DOI
10.1238/physica.regular.061a00007

Related papers

Quantum-classical correspondence and nonclassical state generation in dissipative quantum optical systems

K. N. Alekseev, N. V. Alekseeva, Jan Peřina

We develop a semiclassical method to determine the nonlinear dynamics of dissipative quantum optical systems in the limit of large number of photons N; it is based on the 1/N-expansion and the quantum-classical correspondence. The method is used to tackle two problems: the study of the dynamics of nonclassical state generation in higher order anharmonic dissipative oscillators and the establishment of the difference between the quantum and classical dynamics of the second-harmonic generation in a self-pulsing regime. In addressing the first problem, we obtain an explicit time dependence of the squeezing and the Fano factor for an arbitrary degree of anharmonism in the short-time approximation. For the second problem, we analytically find a characteristic time scale at which the quantum dynamics differs insignificantly from the classical one.

arXiv:chao-dyn/9912013•Journal of Experimental and Theoretical Physics•Apr 1, 2000•1 citation

The influence of dissipation on the quantum-classical correspondence: Stability of stochastic trajectories

Maksym Kryvohuz, Jianshu Cao

The quantum-classical correspondence in the presence of dissipation is studied. The semiclassical expression for the linear response function of an anharmonic system is expressed in a series containing classical stability matrix elements, which can diverge due to the chaotic behavior of stochastic trajectories. The presence of dissipation in most cases removes the divergence of higher-order correction terms, thus suppressing quantum effects and making the system more classical. The regime of system-bath coupling, which makes quantum dynamics completely classical, is obtained in terms of friction, temperature, and anharmonicity. Special cases when bath coupling may lead to enhancement of quantum effects are discussed.

The Journal of Chemical Physics•Jun 18, 2009•18 citations

Dissipation and Quantum-Classical Correspondence in Nonlinear Nonintegrable Systems

Kazuo Hida

The effect of dissipation on the recovery of the classical chaotic dynamics is discussed in the semiclassical regime of the quantum nonintegrable systems. In the classically chaotic Hamiltonian systems, the semiclassical approach breaks down within rather short time scale due to the interference of the wave function. It is shown that the dissipation suppresses the interference and confines the quantum fluctuation around the semiclassical trajectory which is described by the semiclassical Langevin equation. We also show that the semiclassical condition obtained by Schmid is equivalent to the condition for the suppression of interference. The discussion is valid regardless whether the classical motion is chaotic or not and therefore dissipation helps the recovery of the classical dynamics in the chaotic systems.

Journal of the Physical Society of Japan•Jan 15, 1988•1 citation

Quantum-Classical Correspondence in a Quartic Oscillator -- Corrections to Frequency and Bounds for Periodic Motion

Biswas, Mandas, Deb Shankar Ray

We take a qualitative comparative look at quantum and classical quartic anharmonic oscillators. It has been shown that the behavior of the quantum anharmonic oscillator mimics that of the classical anharmonic oscillators with the structurally same Hamiltonian in the coherent state basis of the harmonic oscillators. The associated equation of motion allows us to use Lindstet-Poincare perturbation method to compute the classical frequency of the oscillation order by order, by taking care of its dependence on amplitude and the quantum corrections. We also derive a bound for periodicity of such oscillations in both the classical and quantum cases.

arXiv:2410.09722•Oct 13, 2024•0 citations

Analytic solution for the dissipative anharmonic quantum oscillator and semiclassical analysis

Franz X. Kärtner, Axel Schenzle

The analytic solution of the time evolution of a dissipative nonlinear quantum oscillator in the generalized s-ordered representation is derived. The analytical solution allows for comparsion of the various representations with the classical and semiclassical dynamics. It is shown by the eigenvalue analysis of the evolution operators that in the case of a small nonlinearity a semiclassical analysis is an excellent approximation to the exact quantum dynamics. This is further demonstrated by using the Green's function of the anharmonic oscillator to solve the initial-value problem for a superposition of coherent states.

Physical Review A•Aug 1, 1993•14 citations

Time evolution of correlation functions for classical and quantum anharmonic oscillators

Luís M. A. Bettencourt, CHRISTOF WETTERICH

The time evolution of the correlation functions of an ensemble of anharmonic N-component oscillators with O(N) symmetry is described by a flow equation, exact up to corrections of order $1/N^2$. We find effective irreversibility. Nevertheless, analytical and numerical investigation reveals that the system does not reach thermal equilibrium for large times, even when $N\to \infty$. Depending on the initial distribution, the dynamics is asymptotically stable or it exhibits growing modes which break the conditions for the validity of the 1/N expansion for large time. We investigate both classical and quantum systems, the latter being the limit of an O(N) symmetric scalar quantum field theory in zero spatial dimensions.

arXiv:hep-ph/9805360•May 18, 1998•4 citations
We develop a method for the determination of the dynamics of dissipative quantum systems in the limit of large number of quanta N , based on the 1/ N -expansion of Heidmann et al. (1985 Opt. Commun. 54 189) and the quantum-classical correspondence. Using this method, we find analytically the dynamics of nonclassical states generation in the higher-order anharmonic dissipative oscillators for an arbitrary temperature of a reservoir. We show that the quantum correction to the classical motion increases with time quadratically up to some maximal value, which is dependent on the degree of nonlinearity and a damping constant, and then it decreases. Similarities and differences with the corresponding behavior of the quantum corrections to the classical motion in the Hamiltonian chaotic systems are discussed. We also compare our results obtained for some limiting cases with the results obtained by using other semiclassical tools and discuss the conditions for validity of our approach.
We develop a semiclassical method to determine the nonlinear dynamics of dissipative quantum optical systems in the limit of large number of photons N; it is based on the 1/N-expansion and the quantum-classical correspondence. The method is used to tackle two problems: the study of the dynamics of nonclassical state generation in higher order anharmonic dissipative oscillators and the establishment of the difference between the quantum and classical dynamics of the second-harmonic generation in a self-pulsing regime. In addressing the first problem, we obtain an explicit time dependence of the squeezing and the Fano factor for an arbitrary degree of anharmonism in the short-time approximation. For the second problem, we analytically find a characteristic time scale at which the quantum dynamics differs insignificantly from the classical one.
The quantum-classical correspondence in the presence of dissipation is studied. The semiclassical expression for the linear response function of an anharmonic system is expressed in a series containing classical stability matrix elements, which can diverge due to the chaotic behavior of stochastic trajectories. The presence of dissipation in most cases removes the divergence of higher-order correction terms, thus suppressing quantum effects and making the system more classical. The regime of system-bath coupling, which makes quantum dynamics completely classical, is obtained in terms of friction, temperature, and anharmonicity. Special cases when bath coupling may lead to enhancement of quantum effects are discussed.
The effect of dissipation on the recovery of the classical chaotic dynamics is discussed in the semiclassical regime of the quantum nonintegrable systems. In the classically chaotic Hamiltonian systems, the semiclassical approach breaks down within rather short time scale due to the interference of the wave function. It is shown that the dissipation suppresses the interference and confines the quantum fluctuation around the semiclassical trajectory which is described by the semiclassical Langevin equation. We also show that the semiclassical condition obtained by Schmid is equivalent to the condition for the suppression of interference. The discussion is valid regardless whether the classical motion is chaotic or not and therefore dissipation helps the recovery of the classical dynamics in the chaotic systems.
We take a qualitative comparative look at quantum and classical quartic anharmonic oscillators. It has been shown that the behavior of the quantum anharmonic oscillator mimics that of the classical anharmonic oscillators with the structurally same Hamiltonian in the coherent state basis of the harmonic oscillators. The associated equation of motion allows us to use Lindstet-Poincare perturbation method to compute the classical frequency of the oscillation order by order, by taking care of its dependence on amplitude and the quantum corrections. We also derive a bound for periodicity of such oscillations in both the classical and quantum cases.
The analytic solution of the time evolution of a dissipative nonlinear quantum oscillator in the generalized s-ordered representation is derived. The analytical solution allows for comparsion of the various representations with the classical and semiclassical dynamics. It is shown by the eigenvalue analysis of the evolution operators that in the case of a small nonlinearity a semiclassical analysis is an excellent approximation to the exact quantum dynamics. This is further demonstrated by using the Green's function of the anharmonic oscillator to solve the initial-value problem for a superposition of coherent states.
The time evolution of the correlation functions of an ensemble of anharmonic N-component oscillators with O(N) symmetry is described by a flow equation, exact up to corrections of order 1/N21/N^21/N2. We find effective irreversibility. Nevertheless, analytical and numerical investigation reveals that the system does not reach thermal equilibrium for large times, even when N→∞N\to \inftyN→∞. Depending on the initial distribution, the dynamics is asymptotically stable or it exhibits growing modes which break the conditions for the validity of the 1/N expansion for large time. We investigate both classical and quantum systems, the latter being the limit of an O(N) symmetric scalar quantum field theory in zero spatial dimensions.