Quantum Signatures of Strange Attractors

TL;DR

The paper investigates how classical strange attractors in dissipative chaos manifest in quantum mechanics by employing the Caldirola–Kanai framework to embed dissipation into a time-dependent Hamiltonian for the driven Duffing oscillator. Phase-space visualization via the Husimi distribution reveals a first quantum strange attractor, showing localization onto the classical invariant set with quantum smoothing due to the uncertainty principle. By analyzing four dynamical regimes—harmonic, non-chaotic dissipative, conservative chaotic, and chaotic dissipative—the work demonstrates the quantum-classical correspondence and the unique role of dissipation in shaping quantum chaotic states. The out-of-time-ordered correlator (OTOC) further validates semiclassical behavior, as stronger dissipation enhances the exponential growth that aligns with the classical Lyapunov exponent and extends the Ehrenfest time, illustrating a concrete quantum-to-classical transition in open chaotic systems.

Abstract

In classical mechanics, driven systems with dissipation often exhibit complex, fractal dynamics known as strange attractors. This paper addresses the fundamental question of how such structures manifest in the quantum realm. We investigate the quantum Duffing oscillator, a paradigmatic chaotic system, using the Caldirola-Kanai (CK) framework, where dissipation is integrated directly into a time-dependent Hamiltonian. By employing the Husimi distribution to represent the quantum state in phase space, we present the first visualization of a quantum strange attractor within this model. Our simulations demonstrate how an initially simple Gaussian wave packet is stretched, folded, and sculpted by the interplay of chaotic dynamics and energy loss, causing it to localize onto a structure that beautifully mirrors the classical attractor. This quantum "photograph" is inherently smoothed, blurring the infinitely fine fractal details of its classical counterpart as a direct consequence of the uncertainty principle. We supplement this analysis by examining the out-of-time-ordered correlator (OTOC), which shows that stronger dissipation clarifies the exponential growth associated with the classical Lyapunov exponent, thereby confirming the model's semiclassical behavior. This work offers a compelling geometric perspective on open chaotic quantum systems and sheds new light on the quantum-classical transition.

Figures (8)

• Figure 1: Initial Husimi distribution (\ref{['eq:init']}) and the distribution of the initial classical trajectories used in the simulations. In our numerical examples, we use the parameters $x_0=+1$, $p_0=-1.5$, $\sigma=0.5$ in units of $\hbar=1$.
• Figure 2: The harmonic dissipative Duffing equation $\alpha=+1$, $\beta=0$, $\delta=0.1$, $\gamma=2.5$, $\omega=2$. Snapshot of the classical and quantum evolution after $13.37$ cycles $T_{cy}$ of the external forcing.
• Figure 3: The non-chaotic dissipative Duffing equation $\alpha=+1$, $\beta=0.25$, $\delta=0.1$, $\gamma=2.5$, $\omega=2$. Snapshot of the classical and quantum evolution after $13.37$ cycles $T_{cy}$ of the external forcing.
• Figure 4: Frequency response curve of the Duffing Equation at parameters of Fig. \ref{['fig3:main']}. At frequency $\omega=2.0$, we have two stable and one unstable solution corresponding to two stable and one unstable periodic orbits in the phase space.
• Figure 5: The conservative Duffing Equation $\alpha=-1.0$, $\beta=0.25$, $\delta=0.0$, $\gamma=2.5$, $\omega=2.0$. Snapshot of the classical and quantum evolution after $13.37$ cycles $T_{cy}$ of the external forcing.