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:1612.03444 · Annalen der Physik

Classical Bifurcation Diagrams by Quantum Means

Mikhail Ivanchenko, Козинов Евгений Александрович, Valentin Volokitin, Alexey Liniov, Iosif B. Meyerov, S. Denisov

Published Jul 7, 2017• 20 citations
arXivHTMLPDF

Abstract

Asymptotic state of an open quantum system may change substantially upon variations of system parameters. These changes can be often identified as bifurcation transitions in the classical mean‐filed equations describing evolution of the relevant observables. We demonstrate that these bifurcations can be made visible as changes in the structure of the asymptotic density matrix of the original quantum system. By using an N‐boson dimer, we present quantum ‘bifurcation diagrams’ for the pitchfork and saddle‐node bifurcations (in the stationary dimer) and visualize a period‐doubling transition to chaos (in a periodically modulated dimer).

DOI
10.1002/andp.201600402

Related papers

Bifurcation and chaos in nonlinear Lindblad equations

Bernd Fernengel, Barbara Drossel

The Lindblad equation describes the dissipative time evolution of a density matrix that characterizes an open quantum system in contact with its environment. The widespread ensemble interpretation of a density matrix requires its time evolution to be linear. However, when the density matrix is obtained from a mean field theory of interacting quantum systems or from a top-down control by a changing classical environment, the ensemble interpretation is inappropriate and nonlinear dynamics arise naturally. We therefore study the dynamical behavior of nonlinear Lindblad equations using the example of a two-level system. By using techniques developed for classical dynamical systems we show that various types of bifurcations and even chaotic dynamics can occur. We also discuss experimental situations for which our results could be relevant.

arXiv:1907.09349•Jul 22, 2019•0 citations

Quantum Neimark-Sacker bifurcation

Igor Yusipov, Mikhail Ivanchenko

Recently, it has been demonstrated that asymptotic states of open quantum system can undergo qualitative changes resembling pitchfork, saddle-node, and period doubling classical bifurcations. Here, making use of the periodically modulated open quantum dimer model, we report and investigate a quantum Neimark-Sacker bifurcation. Its classical counterpart is the birth of a torus (an invariant curve in the Poincaré section) due to instability of a limit cycle (fixed point of the Poincaré map). The quantum system exhibits a transition from unimodal to bagel shaped stroboscopic distributions, as for Husimi representation, as for observables. The spectral properties of Floquet map experience changes reminiscent of the classical case, a pair of complex conjugated eigenvalues approaching a unit circle. Quantum Monte-Carlo wave function unraveling of the Lindblad master equation yields dynamics of single trajectories on "quantumtorus" and allows for quantifying it by rotation number. The bifurcation is sensitive to the number of quantum particles that can also be regarded as a control parameter.

arXiv:1908.08134•Scientific Reports•Nov 29, 2019•11 citations

Bifurcations and chaos in nonlinear Lindblad equations

Bernd Fernengel, Barbara Drossel

Abstract The Lindblad equation describes the dissipative time evolution of a density matrix that characterizes an open quantum system in contact with its environment. The widespread ensemble interpretation of a density matrix requires its time evolution to be linear. However, when the dynamics of the density matrix is of a quantum system results not only from the interaction with an external environment, but also with other quantum systems of the same type, the ensemble interpretation is inappropriate and nonlinear dynamics arise naturally. We therefore study the dynamical behavior of nonlinear Lindblad equations using the example of a two-level system. By using techniques developed for classical dynamical systems we show that various types of bifurcations and even chaotic dynamics can occur. As specific examples that display the various types of dynamical behavior, we suggest explicit models based on systems of interacting spins at finite temperature and exposed to a magnetic field that can change in dependence of the magnetization. Due to the interaction between spins, which is treated at mean-field level, the Hamiltonian as well as the transition rates of the Lindblad equation become dependent on the density matrix.

Journal of Physics A Mathematical and Theoretical•Jul 29, 2020•20 citations

Classical vs quantum dynamics and the onset of chaos in a macrospin system

Haowei Fan, Vladimir Fal'ko, Xiao Li

We study a periodically driven macrospin system with anisotropic long-range interactions and collective dissipation, described by a Lindblad master equation. In the thermodynamic limit ($N\to\infty$), a mean-field treatment yields classical equations of motion, whose dynamics are characterized via the maximal Lyapunov exponent (MLE). Focusing on the thermodynamic limit, we map out chaotic, quasiperiodic, and periodic phases via bifurcation diagrams, MLEs, and Fourier spectra of evolved observables, identifying classic period-doubling bifurcations and fractal boundaries in the regions of attractors. Finite-size quantum simulations in the Dicke basis reveal that while both quantum and classical systems exhibit diverse dynamical phases, finite-size effects suppress some behaviors present in the thermodynamic limit. The sign of $λ_{\mathrm{max}}$ serves as a key indicator of convergence between quantum and classical dynamics, which agree over timescales up to the Lyapunov time. Analysis of the density matrix shows that convergence occurs only when its nonzero elements are sharply localized. However, the nonconvergence does not imply a fundamental difference between quantum and classical dynamics: in chaotic regimes, although the evolution orbits of quantum and classical systems show significant differences, quantum evolution becomes mixed and diffusively explores the Hilbert space, signaling quantum chaos, which can be confirmed by the delocalized nature of the density matrix.

quant-pharXiv:2601.00062•Jan 2, 2026•0 citations

Quantum Manifestation of the Classical Bifurcation in the Driven Dissipative Bose–Hubbard Dimer

Pavel S. Muraev, Дмитрий Н. Максимов, Andrey R. Kolovsky

We analyze the classical and quantum dynamics of the driven dissipative Bose–Hubbard dimer. Under variation of the driving frequency, the classical system is shown to exhibit a bifurcation to the limit cycle, where its steady-state solution corresponds to periodic oscillation with the frequency unrelated to the driving frequency. This bifurcation is shown to lead to a peculiarity in the stationary single-particle density matrix of the quantum system. The case of the Bose–Hubbard trimer, where the discussed limit cycle bifurcates into a chaotic attractor, is briefly discussed.

arXiv:2209.07029•Entropy•Jan 5, 2023•1 citation

Quantum bifurcation in a coulombic‐like potential

Ciann‐Dong Yang, Hung‐Jen Weng

Abstract Because of the lack of nonlinear dynamics, up to now no bifurcation phenomenon in its original sense has been discovered directly in quantum mechanical systems. Based on the formalism of complex‐valued quantum mechanics, this article derives the nonlinear Hamilton equations from the Schrödinger equation to provide the necessary mathematic framework for the analysis of quantum bifurcation. This new approach makes it possible to identify quantum bifurcation by the direct evidence of the sudden change of fixed points and their surrounding trajectories. As a practical application of the proposed approach, we consider the quantum motion in a Coulombic‐like potential modeled by V(r) = A/r2 − B/r, where the first term describes the centrifugal trend and the second deals with the Coulombic attraction. As the bifurcation parameter evolves, we demonstrate how local and global bifurcations in quantum dynamics can be identified by inspecting the changes of fixed points and their surrounding trajectories. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

International Journal of Quantum Chemistry•Jan 13, 2011•5 citations
Asymptotic state of an open quantum system may change substantially upon variations of system parameters. These changes can be often identified as bifurcation transitions in the classical mean‐filed equations describing evolution of the relevant observables. We demonstrate that these bifurcations can be made visible as changes in the structure of the asymptotic density matrix of the original quantum system. By using an N‐boson dimer, we present quantum ‘bifurcation diagrams’ for the pitchfork and saddle‐node bifurcations (in the stationary dimer) and visualize a period‐doubling transition to chaos (in a periodically modulated dimer).
The Lindblad equation describes the dissipative time evolution of a density matrix that characterizes an open quantum system in contact with its environment. The widespread ensemble interpretation of a density matrix requires its time evolution to be linear. However, when the density matrix is obtained from a mean field theory of interacting quantum systems or from a top-down control by a changing classical environment, the ensemble interpretation is inappropriate and nonlinear dynamics arise naturally. We therefore study the dynamical behavior of nonlinear Lindblad equations using the example of a two-level system. By using techniques developed for classical dynamical systems we show that various types of bifurcations and even chaotic dynamics can occur. We also discuss experimental situations for which our results could be relevant.
Recently, it has been demonstrated that asymptotic states of open quantum system can undergo qualitative changes resembling pitchfork, saddle-node, and period doubling classical bifurcations. Here, making use of the periodically modulated open quantum dimer model, we report and investigate a quantum Neimark-Sacker bifurcation. Its classical counterpart is the birth of a torus (an invariant curve in the Poincaré section) due to instability of a limit cycle (fixed point of the Poincaré map). The quantum system exhibits a transition from unimodal to bagel shaped stroboscopic distributions, as for Husimi representation, as for observables. The spectral properties of Floquet map experience changes reminiscent of the classical case, a pair of complex conjugated eigenvalues approaching a unit circle. Quantum Monte-Carlo wave function unraveling of the Lindblad master equation yields dynamics of single trajectories on "quantumtorus" and allows for quantifying it by rotation number. The bifurcation is sensitive to the number of quantum particles that can also be regarded as a control parameter.
Abstract The Lindblad equation describes the dissipative time evolution of a density matrix that characterizes an open quantum system in contact with its environment. The widespread ensemble interpretation of a density matrix requires its time evolution to be linear. However, when the dynamics of the density matrix is of a quantum system results not only from the interaction with an external environment, but also with other quantum systems of the same type, the ensemble interpretation is inappropriate and nonlinear dynamics arise naturally. We therefore study the dynamical behavior of nonlinear Lindblad equations using the example of a two-level system. By using techniques developed for classical dynamical systems we show that various types of bifurcations and even chaotic dynamics can occur. As specific examples that display the various types of dynamical behavior, we suggest explicit models based on systems of interacting spins at finite temperature and exposed to a magnetic field that can change in dependence of the magnetization. Due to the interaction between spins, which is treated at mean-field level, the Hamiltonian as well as the transition rates of the Lindblad equation become dependent on the density matrix.
We study a periodically driven macrospin system with anisotropic long-range interactions and collective dissipation, described by a Lindblad master equation. In the thermodynamic limit (N→∞N\to\inftyN→∞), a mean-field treatment yields classical equations of motion, whose dynamics are characterized via the maximal Lyapunov exponent (MLE). Focusing on the thermodynamic limit, we map out chaotic, quasiperiodic, and periodic phases via bifurcation diagrams, MLEs, and Fourier spectra of evolved observables, identifying classic period-doubling bifurcations and fractal boundaries in the regions of attractors. Finite-size quantum simulations in the Dicke basis reveal that while both quantum and classical systems exhibit diverse dynamical phases, finite-size effects suppress some behaviors present in the thermodynamic limit. The sign of λmaxλ_{\mathrm{max}}λmax​ serves as a key indicator of convergence between quantum and classical dynamics, which agree over timescales up to the Lyapunov time. Analysis of the density matrix shows that convergence occurs only when its nonzero elements are sharply localized. However, the nonconvergence does not imply a fundamental difference between quantum and classical dynamics: in chaotic regimes, although the evolution orbits of quantum and classical systems show significant differences, quantum evolution becomes mixed and diffusively explores the Hilbert space, signaling quantum chaos, which can be confirmed by the delocalized nature of the density matrix.
We analyze the classical and quantum dynamics of the driven dissipative Bose–Hubbard dimer. Under variation of the driving frequency, the classical system is shown to exhibit a bifurcation to the limit cycle, where its steady-state solution corresponds to periodic oscillation with the frequency unrelated to the driving frequency. This bifurcation is shown to lead to a peculiarity in the stationary single-particle density matrix of the quantum system. The case of the Bose–Hubbard trimer, where the discussed limit cycle bifurcates into a chaotic attractor, is briefly discussed.
Abstract Because of the lack of nonlinear dynamics, up to now no bifurcation phenomenon in its original sense has been discovered directly in quantum mechanical systems. Based on the formalism of complex‐valued quantum mechanics, this article derives the nonlinear Hamilton equations from the Schrödinger equation to provide the necessary mathematic framework for the analysis of quantum bifurcation. This new approach makes it possible to identify quantum bifurcation by the direct evidence of the sudden change of fixed points and their surrounding trajectories. As a practical application of the proposed approach, we consider the quantum motion in a Coulombic‐like potential modeled by V(r) = A/r2 − B/r, where the first term describes the centrifugal trend and the second deals with the Coulombic attraction. As the bifurcation parameter evolves, we demonstrate how local and global bifurcations in quantum dynamics can be identified by inspecting the changes of fixed points and their surrounding trajectories. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011