Transient and steady-state chaos in dissipative quantum systems

TL;DR

The paper addresses how chaos is defined in open quantum systems and argues that spectral statistics alone are insufficient. It introduces a dynamical framework based on entanglement entropy $\mathcal{S}_{VN}$ and out-of-time-order correlators (FOTOC) to diagnose chaos across timescales, applying it to the open anisotropic Dicke model with photon loss (dissipation rate $\kappa$). The main results show two distinct chaos regimes: transient chaos with rapid early-time growth of $\mathcal{S}_{VN}$ and FOTOC scrambling but low long-time saturation, and steady-state chaos with large long-time $\mathcal{S}_{VN}$; the steady-state entropy correlates with the classical Lyapunov exponent $\Lambda_{ss}$, while Ginibre spectral statistics reflect only short-time chaos. A random-matrix toy model demonstrates that Ginibre statistics signal transient chaos and can be decoupled from steady-state chaos by spectral-structure engineering, thereby restoring the quantum-classical correspondence in dissipative dynamics.

Abstract

Dissipative quantum chaos plays a central role in the characterization and control of information scrambling, non-unitary 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)]. We properly restore the quantum-classical correspondence through a dynamical approach based on entanglement entropy 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 entanglement 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 entanglement dynamics and OTOCs as reliable diagnostics of dissipative quantum chaos across different timescales.

Figures (8)

• Figure 1: Schematic illustration. A dynamical framework restores the quantum-classical correspondence by identifying three regimes: (I) steady-state chaos, featuring a chaotic attractor in the classical dynamics (left), and linear growth followed by large saturation of the von Neumann entropy (middle panel); (II) transient chaos, characterized by a regular attractor, and rapid short-time VNE growth followed by decay at long times; and (III) regular dynamics, showing slow VNE growth and low saturation. Regimes (I) and (II) exhibit Ginibre Liouvillian spectral statistics (right panels), while (III) corresponds to 2d-Poisson statistics.
• Figure 2: Signature of steady-state chaos on the $\lambda_--\lambda_+$ plane. (a), (c) Averaged long-time Lyapunov exponent $\Lambda_{\rm ss}$ and (b), (d) total von Neumann entropy of the spin and photon subsystems $\mathcal{S}_{\rm ss}^{\rm VN}$ in the (a)-(b) absence of dissipation, $\kappa= 0$, and (c)-(d) presence of dissipation, $\kappa= 1$.
• Figure 3: Transient chaos vs steady-state chaos in the Dicke limit $\lambda_-=\lambda_+=\lambda$. (a) Finite-time ensemble averaged Lyapunov exponent $\Lambda_t$, (b) total VNE $\mathcal{S}^{\rm VN}(t)$, and (c) standard deviation $\Delta S_z(t)\approx \sqrt{1-\mathcal{F}_z(t)}/\delta\phi$ obtained from FOTOC $\mathcal{F}_z(t)$ of $z$ component of spin for $\delta\phi \ll 1/S$, for atom-photon coupling strengths $\lambda=1.2, 2.0$ and $\kappa = 0, 1$. In (b), the time axis is linear for $t\le 0.5$ and logarithmic for $t>0.5$. (d) The linear growth rate of VNE, $\mathcal{S}^{\rm VN}_{\rm slope}$, and the time-averaged Lyapunov exponent, $\overline{\Lambda}$, within a time interval $t\in[0, 0.5]$ as a function of the coupling strength $\lambda$ for $\kappa = 1$.
• Figure 4: Transient and steady-state chaos for the random matrix toy model. (a) Time evolution of the von Neumann entropy $\mathcal{S}_{\rm VN}$, averaged over an ensemble of random matrices, for the Liouvillian in Eq. (\ref{['Liouvillian']}) with $\mu = 0$ (solid red line), resulting in 2d Poisson level statistics, with $\mu = 1$, $\chi=0$ (dashed violet line), resulting in Ginibre spectral statistics, and the projected case with $\mu = 1$, $\chi=1$ (solid blue line), also resulting in Ginibre spectral statistics. (b) Evolution of $\mathcal{S}_{\rm VN}$ for different values of the deformation parameter $\chi$ and fixed $\mu = 1$. The horizontal dashed line in (a)-(b) indicates the maximal entropy of the subsystem, $\mathcal{S}^{\rm max}_{\rm VN} = \frac{1}{2}\ln(N)$. We use $N=49$ and $\gamma = 1$. The random numbers are drawn from a Gaussian distribution with zero mean and unit variance.
• Figure S5: The classical dynamics of the open anisotropic Dicke model. (a) The classical phase-diagram on the $\lambda_--\lambda_+$ plane. (b) The spin dynamics over the Bloch sphere at different values of $\lambda_{+}$ for $\lambda_{-}=2.0$. All energies (time) are measured by $\omega (1/\omega)$. We set $\omega=1.0,\omega_0=1.0$ and dissipation strength $\kappa = 1$.