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

TL;DR

This study analyzes a periodically driven dissipative macrospin with all-to-all anisotropic interactions, comparing the classical mean-field dynamics in the thermodynamic limit $N \to \infty$ to finite-$N$ quantum dynamics governed by a Lindblad equation. By deriving nonautonomous mean-field equations on the unit sphere and using a QR-based Lyapunov analysis, the authors map chaotic, quasiperiodic, and periodic regimes, including period-doubling routes to chaos and fractal attractor boundaries. Finite-$N$ quantum simulations in the Dicke basis reveal that quantum-classical correspondence depends on the sign of the maximal Lyapunov exponent $\lambda_{\max}$: negative $\lambda_{\max}$ yields convergence up to the Lyapunov time $t_L \sim 1/\lambda_{\max}$, while positive $\lambda_{\max}$ leads to quantum chaos with diffusion across the Hilbert space and density-matrix delocalization, suppressing nontrivial classical periodic attractors due to tunneling. Overall, the work clarifies how chaos, finite-size effects, and dissipation shape the quantum-classical connection in Floquet-driven, long-range interacting systems and provides diagnostics via MLE and density-matrix structure that are relevant for experimental platforms like cavity QED and trapped ions.

Abstract

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.

Figures (17)

• Figure 1: (a) A schematic diagram of the spin ring with periodic single-particle driving, all-to-all interactions, and dissipation. (b) An example of a bifurcation process from periodic behavior to chaos in the system.
• Figure 2: Schematic overview of this work. The macrospin Hamiltonian [Eq. \ref{['collectiveHamiltonian']}] branches into the classical thermodynamic limit ($N\to\infty$; Sec. \ref{['sec_bifurctaion']}-\ref{['sec_finiteNandinfiniteN']}) and the quantum description via the Lindblad master equation ($1 \ll N < 200$; Sec. \ref{['sec_finiteNandinfiniteN']}-\ref{['Sec_densitymatrix']}). Three dynamical regimes are identified according to the maximal Lyapunov exponent $\lambda_{\max}$: stable ($\lambda_{\max}<0$), quasiperiodic ($0<\lambda_{\max}\ll 1$), and chaotic ($\lambda_{\max}>0$). Key characteristics include the degree of quantum noise, agreement between quantum and classical trajectories, and localization of the density matrix.
• Figure 3: Two representative examples of the evolution of $m^z(t)$ in finite-size quantum systems with system sizes ranging from 100 to 200, shown in different colors. The corresponding dissipation strength $\kappa$ and driving strength $\Gamma$ are indicated at the top of each panel. In addition, the parameter combinations corresponding to the two plots are also indicated in Fig. \ref{['MLE_PD']} using the same symbols as in the titles. In both panels, the initial condition is set to be an $x$-polarized state. Black dashed lines represent the corresponding classical orbits in the thermodynamic limit.
• Figure 4: (Classical) The Phase diagrams of the MLE for different choices of the interaction strength as a function of the driving strength $\Gamma$ and the dissipation strength $\kappa$. The initial condition is uniformly set to be $\mathbf{m}(0)=(1,0,0)$. We use red to indicate the chaotic phase where the MLE is positive, and white to denote regions where the MLE is zero. Blue is used to mark the stable phase with negative MLE values, allowing for a clear visual distinction between regions with positive and negative MLE. The interaction strength is indicated above each panel. Those parameter combinations to be analyzed later are marked by colored stars in (b).
• Figure 5: (Classical) (a) Bifurcation diagram of $m^x(t)$ for driving strength $\Gamma=8.427$. Black dots correspond to dynamics starting from the $x$-polarized initial state, highlighting state-specific behavior. Red dots represent results from $500$ uniformly sampled initial states over the sphere, capturing the global dynamical structure. (b) MLE as a function of $\kappa$, computed for the $x$-polarized initial state. (c, d) Zoomed-in views of selected sub-regions of panels (a) and (b), respectively. Blue-shaded areas mark stable windows, identified by both stable periodic orbits and negative MLE values.

Theorems & Definitions (3)

• Definition 1
• Theorem 1: Block-diagonal structure of $H$
• Theorem 2: Dynamical closure of each Young sector