The double pendulum serves as the paradigmatic model for the transition from integrability to chaos, yet its potential as a rigorous bridge between classical nonlinear dynamics and quantum open systems remains underutilized. By treating the system’s phase space as a topologically tiled manifold under $2\pi$ periodicity, we establish a structural isomorphism between classical Lyapunov instability and quantum decoherence rates. Leveraging recent advances in open quantum system dynamics (Hernández et al., 2025) and topological critical phases (Yu et al., 2026), we demonstrate that classical “flip” events correspond to topological winding number transitions that drive quantum state delocalization. Numerical simulations reveal that the fractal geometry of stability islands quantitatively predicts the resilience of quantum states to environmental coupling, suggesting a novel “decoherence-free” encoding scheme based on Kolmogorov-Arnold-Moser (KAM) tori. These findings unify the deterministic geometry of chaos with the probabilistic nature of quantum mechanics, resolving the Ehrenfest time paradox through a topological gauge argument.
The double pendulum serves as the paradigmatic model for the transition from integrability to chaos, yet its potential as a rigorous bridge between classical nonlinear dynamics and quantum open systems remains underutilized. By treating the system’s phase space as a topologically tiled manifold under 2π periodicity, we establish a structural isomorphism between classical Lyapunov instability and quantum decoherence rates. Leveraging recent advances in open quantum system dynamics (Hernández et al., 2025) and topological critical phases (Yu et al., 2026), we demonstrate that classical “flip” events correspond to topological winding number transitions that drive quantum state delocalization. Numerical simulations reveal that the fractal geometry of stability islands quantitatively predicts the resilience of quantum states to environmental coupling, suggesting a novel “decoherence-free” encoding scheme based on Kolmogorov-Arnold-Moser (KAM) tori. These findings unify the deterministic geometry of chaos with the probabilistic nature of quantum mechanics, resolving the Ehrenfest time paradox through a topological gauge argument.
In this work the decoherence formalism of quantum mechanics is explored and applied to a number of interesting problems in quantum physics. The boundary between quantum and classical physics is examined, and demonstration made that quantum histories corresponding to classical equations of motion become more probable for a broad class of models, including linear and nonlinear models of Brownian motion. The link between noise, dissipation, and decoherence is studied. This work is then applied to systems which classically exhibit dissipative chaotic dynamics. A theory is explicated for treating these systems, and the ideas are applied to a particular model of the forced, damped Duffing oscillator, which is chaotic for certain parameter values. Differences between classical and quantum chaos are examined, particularly differences arising in the structure of fractal strange attractors, and the conceptual difficulties in framing standard notions of chaos in a quantum system. A brief discussion of previous work on quantum chaos is included, and the differences between Hamiltonian and dissipative chaos pointed out; a somewhat different interpretation of quantum chaos from the standard one is suggested. A class of histories for quantum systems, in phase space rather than configuration space, is studied. Different ways of representing projections in phase space are discussed, and expressions for the probability of phase space histories are derived; conditions for such histories to decohere are also estimated in the semiclassical limit.
It is widely recognized that entanglement generation and dynamical chaos are intimately related in semiclassical models via the process of decoherence. In this paper, we propose a unifying framework which directly connects the bipartite and multipartite entanglement growth to the quantifiers of classical and quantum chaos. In the semiclassical regime, the dynamics of the von Neumann entanglement entropy, the spin squeezing, the quantum Fisher information, and the out-of-time-order square commutator are governed by the divergence of nearby phase-space trajectories via the local Lyapunov spectrum, as suggested by previous conjectures in the literature. General analytical predictions are confirmed by detailed numerical calculations for two paradigmatic models, relevant in atomic and optical experiments, which exhibit a regular-to-chaotic transition: the quantum kicked top and the Dicke model.
Description/Abstract: Every physics student learns two separate stories. In quantum mechanics, systems exist in superposition until measured, then collapse into definite states through decoherence. In nonlinear dynamics, systems undergo structured transitions — period-doubling cascades governed by Feigenbaum's universal constants. These two stories have never been connected. This paper connects them. Using the Universal Cascade Law diagnostic — which predicts that any bounded system with nonlinear coupling necessarily produces fractal geometry organized by the Feigenbaum constants — we examine quantum decoherence across three layers: the linear Lindblad master equation (no cascade, as predicted by the absence of nonlinearity), classical nonlinear oscillators (full Feigenbaum cascade confirmed with δ₂ = 5.09 converging toward 4.669 across five independent systems), and the full quantum density matrix of a driven-dissipative Kerr oscillator solved in Hilbert space. The result is more profound than simple confirmation. The Feigenbaum cascade exists in the classical limit of quantum hardware but vanishes in the full quantum dynamics. The quantum density matrix holds all branches simultaneously as a mixture, encoding the cascade in its phase-space topology — bimodal Wigner functions, variance spikes, purity drops — while reporting smooth averages. Decoherence is the process by which the Feigenbaum cascade comes into existence. The cascade does not live in the quantum regime. It does not simply live in the classical regime. It lives in the transition between them. Decoherence is the birth of structure. Additional results include: correction of the published Rössler second bifurcation point from c₂ ≈ 3.53 to c₂ = 3.836; confirmation that coupling topology (quadratic vs cubic nonlinearity) selects the Feigenbaum family member (δ ≈ 4.67 vs δ ≈ 5.84); discovery that detection thresholds themselves follow Feigenbaum scaling (the observability scaling property); quantification of the quantum tunneling shift (Δγ = 0.733) between quantum and semiclassical transition points; and identification of the cascade "whisper" in quantum fluctuation structure at classical bifurcation points. Keywords: quantum decoherence, Feigenbaum constant, period-doubling cascade, Universal Cascade Law, Kerr oscillator, Duffing oscillator, Rössler system, Wigner function, bifurcation, nonlinear dynamics, quantum-classical transition, Lindblad master equation, universality, coupling topology, wave function collapse, measurement problem, fractal geometry
The reconciliation of unitary quantum mechanics with the smooth geometry of General Relativity remains a central challenge in quantum gravity, epitomized by the Firewall Paradox. In this work, we investigate the emergent geometry of the black hole information manifold using a metric-preserving machine learning framework. We simulate a 1D quantum black hole using the Sachdev-Ye-Kitaev (SYK) model and reconstruct its bulk geometry via an autoencoder constrained by the Quantum Bures Metric. Our analysis reveals a strong negative correlation between the thermodynamic entropy and the box-counting dimension of the emergent manifold, with finite size scaling consistent with perfect anti-correlation in the thermodynamic limit. Crucially, control experiments reveal that this correspondence is chaos-dependent: integrable free fermion systems show dramatically weaker correlations, while spatially local Heisenberg chains exhibit intermediate behavior. This establishes chaos as a key requirement for strong entropy-topology correspondence, distinguishing maximally chaotic from integrable thermal quantum states. We observe no topological phase transitions across the thermal crossover, supporting the hypothesis that high-entropy chaotic thermalization drives geometric smoothing, consistent with the ER=EPR conjecture.
Chaos in classical systems can be characterized by Lyapunov exponents that measure the exponential divergence of nearby trajectories, but directly extending this framework to quantum mechanics has been a persistent challenge. The wavelike nature of quantum states and the non-commutative geometry of quantum phase space obstruct a straightforward generalization of classical chaos theory. Here we develop a rigorous approach to quantum chaos by leveraging quantum optimal transport theory, which provides the missing geometric foundation for measuring distances between extended quantum distributions. We define quantum Lyapunov exponents that naturally avoid divergences encountered when naïvely generalizing classical exponents, and show that in the semiclassical limit they recover the classical global expansion rate, and hence the usual maximal Lyapunov exponent when they coincide. Our framework provides a tight connection between the divergence of classical trajectories and semiclassical phase space evolution, and additionally clarifies the role of out-of-time-order correlators as diagnostics of quantum chaos. These results establish quantum optimal transport as a unifying mathematical foundation for quantum chaos theory, providing new tools to characterize dynamical behavior across the full range of quantum dynamics from simple few-body models to complex many-body systems.
Recently, the phenomenon of quantum-classical correspondence breakdown was uncovered in optomechanics, where in the classical regime the system exhibits chaos but in the corresponding quantum regime the motion is regular - there appears to be no signature of classical chaos whatsoever in the corresponding quantum system, generating a paradox. We find that transient chaos, besides being a physically meaningful phenomenon by itself, provides a resolution. Using the method of quantum state diffusion to simulate the system dynamics subject to continuous homodyne detection, we uncover transient chaos associated with quantum trajectories. The transient behavior is consistent with chaos in the classical limit, while the long term evolution of the quantum system is regular. Transient chaos thus serves as a bridge for the quantum-classical transition (QCT). Strikingly, as the system transitions from the quantum to the classical regime, the average chaotic transient lifetime increases dramatically (faster than the Ehrenfest time characterizing the QCT for isolated quantum systems). We develop a physical theory to explain the scaling law.