Correspondence principle, dissipation, and Ginibre ensemble

TL;DR

This work challenges the dissipative quantum chaos conjecture by testing the quantum–classical correspondence in the prototypical periodically kicked top with dissipation across a broad semiclassical regime. It analyzes the complex spectrum of the dissipative Floquet operator, utilizes complex spectral ratios to diagnose spectral statistics, and contrasts these with classical chaos indicators such as chaos fractions and Lyapunov (and Hausdorff) dimensions. The main result is that Ginibre spectral correlations are not a robust or universal diagnostic of dissipative quantum chaos, with breakdowns arising even in parameter regions previously thought to exhibit clear quantum–classical alignment. The findings prompt a shift toward quantum fingerprints tied to classical chaotic structures and motivate developing diagnostics beyond spectral correlations for dissipative quantum chaos.

Abstract

The correspondence between quantum and classical behavior has been essential since the advent of quantum mechanics. This principle serves as a cornerstone for understanding quantum chaos, which has garnered increased attention due to its strong impact in various theoretical and experimental fields. When dissipation is considered, quantum chaos takes concepts from isolated quantum chaos to link classical chaotic motion with spectral correlations of Ginibre ensembles. This correspondence was first identified in periodically kicked systems with damping, but it has been shown to break down in dissipative atom-photon systems [Phys. Rev. Lett. 133, 240404 (2024)]. In this contribution, we revisit the original kicked model and perform a systematic exploration across a broad parameter space, reaching a genuine semiclassical limit. Our results demonstrate that the correspondence principle, as defined through this spectral connection, fails even in this prototypical system. These findings provide conclusive evidence that Ginibre spectral correlations are neither a robust nor a universal diagnostic of dissipative quantum chaos.

Figures (10)

• Figure 1: System size analysis in the kicked top with dissipation. [(a1)-(a3)] Average $\langle r \rangle$ and [(b1)-(b3)] average $-\langle \cos\theta \rangle$ from the complex spectral ratio in Eq. \ref{['eq:ComplexRatio']} as a function of the kick strength $k_{1}$. Each column identifies a different case: $k_{0}=0$, $k_{0}=10$, and $k_{0}=20$. [(c1)-(c3)] Average $\langle r \rangle$ and [(d1)-(d3)] average $-\langle \cos\theta \rangle$ as a function of the parameter $k_{0}$. Each column identifies a different case: $k_{1}=10^{-3}$, $k_{1}=1.5$, and $k_{1}=8$. In all panels, we show the last averages for different system sizes $j=10,30,60,80$. The horizontal blue (red) dashed line represents the limit of a quantum system described by 2D Poisson (GinUE) statistics. System parameters: $p=2$ and $\Gamma=0.1$.
• Figure 2: Quantum and classical analysis of the kicked top with dissipation. [(a1)-(a3)] Average $\langle r \rangle$ and [(b1)-(b3)] average $-\langle \cos\theta \rangle$ from the complex spectral ratio in Eq. \ref{['eq:ComplexRatio']} as a function of the kick and dissipation strengths ($k_{1}$,$\Gamma$). Panels (c1)-(c3) and (d1)-(d3) show the last averages for specific dissipation strengths, $\Gamma=0.1,0.2,0.3,0.4$. Each column identifies a different case: $k_{0}=0$, $k_{0}=10$, and $k_{0}=20$. In panels (a1)-(a3) and (b1)-(b3), the gray stripes represent the limits ($k_{1} \to 0$ and $\Gamma \to 0$) where the 2D Poisson and GinUE statistics do not hold. In panels (a1)-(a3) and (b1)-(b3), the horizontal blue (red) dashed line represents the limit of a quantum system described by 2D Poisson (GinUE) statistics. In panels (a1)-(d3), we use a system size $j=80$ that provides 12961 eigenvalues with positive parity. [(e1)-(e3)] Fraction of chaotic initial conditions in Eq. \ref{['eq:ChaoticAttractorFraction']} and [(f1)-(f3)] average Lyapunov dimension in Eq. \ref{['eq:LyapunovDimension']} as a function of the kick and dissipation strengths ($k_{1},\Gamma$). Panels (g1)-(g3) and (h1)-(h3) show the last quantities for specific dissipation strengths, $\Gamma=0.1,0.2,0.3,0.4$. Each column identifies a different case: $k_{0}=0$, $k_{0}=10$, and $k_{0}=20$. In panels (g1)-(g3), the horizontal blue (red) dashed line represents the limit of a classical system with simple (chaotic) attractors, while in panels (f1)-(f3), corresponds to the Lyapunov dimension of a point (surface). The average in phase space takes 1245 initial conditions evolved until $10^3$ periods. We use $p=2$ in all panels.
• Figure S1: Classical and quantum signatures of chaos for the kicked top in absence of dissipation. [(a1)-(a3)] Chaos fraction in Eq. \ref{['eq:ChaosFraction']} and [(b1)-(b3)] normalized spectral ratio in Eq. \ref{['eq:SpectralRatio']} as a function of the kick strength $k_{1}$. Each column represents a different case: [(a1)-(b1)] $k_{0}=0$, [(a2)-(b2)] $k_{0}=10$, and [(a3)-(b3)] $k_{0}=20$. In panels (a1)-(a3), we evolve a set of 1250 initial conditions until $10^{3}$ periods for each kick strength. The horizontal blue (red) dashed line represents the limit for a regular (chaotic) classical system. In panels (b1)-(b3), we chose a system size $j=4096$ and average the spectral ratios from the two parity sectors of the Floquet operator for each kick strength. The horizontal blue (red) dashed line represents the limit for a regular (chaotic) quantum system. System parameters: $p=2$ and $\Gamma=0$.
• Figure S2: Complex eigenvalues of the dissipative Floquet operator of the kicked top. The first three columns identify a weak kick $k_{1}=10^{-3}$: [(a1)-(a4)] $k_{0}=0$, [(b1)-(b4)] $k_{0}=10$, and [(c1)-(c4)] $k_{0}=20$. The last three columns identify a strong kick $k_{1}=8$: [(d1)-(d4)] $k_{0}=0$, [(e1)-(e4)] $k_{0}=10$, and [(f1)-(f4)] $k_{0}=20$. In contrast, each row identifies a different dissipation strength: [(a1)-(f1)] $\Gamma=0.1$, [(a2)-(f2)] $\Gamma=0.2$, [(a3)-(f3)] $\Gamma=0.3$, and [(a4)-(f4)] $\Gamma=0.4$. System parameters: $p=2$ and $j=10$.
• Figure S3: [(a1)-(a3)] Fraction of eigenvalues of the dissipative Floquet operator, which are below the machine precision, as a function the kick strength $k_{1}$. Fraction of eigenvalues below the machine precision as a function of the dissipation strength $\Gamma$ for [(b1)-(b3)] weak kick $k_{1}=10^{-3}$ and [(c1)-(c3)] strong kick $k_{1}=8$. Each column represents a different case: [(a1)-(c1)] $k_{0}=0$, [(a2)-(c2)] $k_{0}=10$, and [(a3)-(c3)] $k_{0}=20$. In panels (a1)-(a3), we chose different dissipation strengths $\Gamma=0.1,0.2,0.3,0.4$ and the system size $j=80$. In panels (b1)-(b3) and (c1)-(c3), we chose different system sizes $j=10,30,60,80$ and the vertical red dashed line represents the dissipation threshold at $\Gamma=0.4$. We use $p=2$ in all panels.