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arXiv:2610.28415

Chaos and scrambling in anisotropic Dicke and Rabi models

Long-Jie Li, Li-Li Gao, Zi-Cheng Liu, Zi-Min Li, Murray T. Batchelor

quant-phnlin.CDSubmitted Oct 20, 2026• Announced Oct 21, 2026• 0 citations
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Abstract

Identifying the quantum signatures of classical chaos remains a central question in quantum physics, and quantum scrambling provides a new perspective on this correspondence. We study two distinct limits of the same model to examine how the correspondence between chaos and scrambling changes across different regimes. The thermodynamic-limit anisotropic Dicke model exhibits a close correspondence between chaos and scrambling, whereas this correspondence breaks down in the large-frequency-ratio limit of the anisotropic Rabi model (ARM). The contrasting chaotic behavior in the two limits originates from the different numbers of classical degrees of freedom, whereas their similar scrambling behavior stems from eigenstate delocalization. This implies that, in the large-frequency-ratio limit, eigenstate delocalization alone is insufficient to qualify the ARM as a classically chaotic model. This finding suggests that identifying chaos in quantum systems requires jointly considering the classical degrees of freedom and eigenstate delocalization.

Comments
12 pages, 7 figures

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Identifying the quantum signatures of classical chaos remains a central question in quantum physics, and quantum scrambling provides a new perspective on this correspondence. We study two distinct limits of the same model to examine how the correspondence between chaos and scrambling changes across different regimes. The thermodynamic-limit anisotropic Dicke model exhibits a close correspondence between chaos and scrambling, whereas this correspondence breaks down in the large-frequency-ratio limit of the anisotropic Rabi model (ARM). The contrasting chaotic behavior in the two limits originates from the different numbers of classical degrees of freedom, whereas their similar scrambling behavior stems from eigenstate delocalization. This implies that, in the large-frequency-ratio limit, eigenstate delocalization alone is insufficient to qualify the ARM as a classically chaotic model. This finding suggests that identifying chaos in quantum systems requires jointly considering the classical degrees of freedom and eigenstate delocalization.
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.
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We present a detailed analysis of the connection between chaos and the onset of thermalization in the spin-boson Dicke model. This system has a well-defined classical limit with two degrees of freedom, and it presents both regular and chaotic regions. Our studies of the eigenstate expectation values and the distributions of the off-diagonal elements of the number of photons and the number of excited atoms validate the diagonal and off-diagonal eigenstate thermalization hypothesis (ETH) in the chaotic region, thus ensuring thermalization. The validity of the ETH reflects the chaotic structure of the eigenstates, which we corroborate using the von Neumann entanglement entropy and the Shannon entropy. Our results for the Shannon entropy also make evident the advantages of the so-called "efficient basis" over the widespread employed Fock basis when investigating the unbounded spectrum of the Dicke model. The efficient basis gives us access to a larger number of converged states than what can be reached with the Fock basis.
Dissipative quantum chaos plays a central role in the characterization and control of information scrambling, nonunitary evolution, and thermalization, but it still lacks a precise definition. The Grobe-Haake-Sommers conjecture, which links Ginibre level repulsion to classical chaotic dynamics, was recently shown to fail [Phys. Rev. Lett. 133, 240404 (2024)PRLTAO0031-900710.1103/PhysRevLett.133.240404]. We properly restore the quantum-classical correspondence through a dynamical approach based on the von Neumann entropy (VNE) and out-of-time-order correlators (OTOCs), which reveal signatures of chaos beyond spectral statistics. Focusing on the open anisotropic Dicke model, we identify two distinct regimes: transient chaos, marked by rapid early-time growth of VNE and OTOCs followed by low saturation values, and steady-state chaos, characterized by high long-time values. We introduce a random matrix toy model and show that Ginibre spectral statistics signals short-time chaos rather than steady-state chaos. Our results establish VNE dynamics and OTOCs as reliable diagnostics of dissipative quantum chaos across different timescales.
Quantum chaos remains a challenging concept because classical signatures of chaos do not directly transfer to quantum systems. This thesis investigates which indicators of chaos are shared between classical and quantum mechanics and how they can be used to characterise non-integrable many-body systems. We examine ergodicity and thermalisation by means of the Eigenstate Thermalisation Hypothesis (ETH) as signatures of non-integrability, and study the Out-of-Time-Ordered Correlator (OTOC) as a probe of information scrambling and as a measure of sensitivity to initial conditions. We briefly discuss the appearance of chaotic wavefunctions and observations such as quantum scars and quantum ergodicity in individual eigenmodes of the Bunimovich stadium potential. Using the OTOC, we discuss extensions of the Lyapunov exponent to quantum systems and apply these ideas to the Sachdev–Ye–Kitaev (SYK) model. In the large-N limit, and using the AdS/CFT correspondence, we find that the model might potentially be the dual of a black hole and that it is maximally chaotic. We conclude by discussing additional proposed signatures of quantum chaos and prospects for experimental realisations of the SYK model.