Universality of spectral fluctuations in open quantum chaotic systems

TL;DR

This work extends random matrix theory to non-Hermitian, non-unitary ensembles derived from matrix-element symmetries (symm-GinE, GinE, self-dual-GinE) and demonstrates that their spectral fluctuations are universal across dissipative quantum chaotic systems, corresponding to OE/UE/SE classes in the conservative limit. The authors develop a general method to unfold two-dimensional spectra with nonuniform density by introducing a metric-based unfolding on the complex plane and analyze both short-range (spacing distributions, spacing ratios) and long-range (number variance) statistics; they verify universality through a dissipative kicked-top model and a dissipative Floquet framework. Notably, they obtain analytic expressions for number variance in GinE and reveal a simple relation for self-dual-GinE, Sigma^2_sdG(n) = Sigma^2_G(n/√2). Overall, the results establish robust universality of spectral correlations in open quantum chaotic systems and provide a practical toolkit for studying 2D spectral statistics in dissipative contexts.

Abstract

Quantum chaotic systems with one-dimensional spectra follow spectral correlations of orthogonal (OE), unitary (UE), or symplectic ensembles (SE) of random matrices depending on their invariance under time reversal and rotation. In this letter, we study the non-Hermitian and non-unitary ensembles based on the symmetry of matrix elements, viz. ensemble of complex symmetric, complex asymmetric (Ginibre), and self-dual matrices of complex quaternions. The eigenvalues for these ensembles lie in the two-dimensional plane. We show that the fluctuation statistics of these ensembles are universal and quantum chaotic systems belonging to OE, UE, and SE in the presence of a dissipative environment show similar spectral fluctuations. The short-range correlations are studied using spacing ratio and spacing distribution. For long-range correlations, unfolding at a non-local scale is crucial. We describe a generic method to unfold the two-dimensional spectra with non-uniform density and evaluate correlations using number variance. We find that both short-range and long-range correlations are universal. We verify our results with the quantum kicked top in a dissipative environment that can be tuned to exhibit symmetries of OE, UE, and SE in its conservative limit.

Figures (6)

• Figure 1: The spectral density for all three ensembles are shown in (a). The scatter plots of eigenvalues for symm-GinE, GinE and self-dual-GinE are shown in (b), (c) and (d) respectively.
• Figure 2: Nearest neighbor spacing distribution for symm-GinE, GinE and self-dual-GinE on log-log scale and its agreement with the dissipative kicked top represented by $F_\text{sG}, F_\text{G}$ and $F_\text{sdG}$ respectively. Same spacing distributions on linear scale are shown in the inset.
• Figure 3: Distribution of the spacing ratio of both types for random matrix ensembles represented by $M_\text{sG}, M_\text{G}$ and $M_\text{sdG}$. The spacing ratios for dissipative kicked tops in Eq. (\ref{['eq-F-dissp']}) show an excellent agreement with that of corresponding random matrix ensembles.
• Figure 4: Number variance for three ensembles viz., symm-GinE ($M_\text{sG}$), GinE ($M_\text{G}$) and self-dual-GinE ($M_\text{sdG}$). Number variance for GinE follow Eq. (\ref{['nmcr-sigmasq-disc']}) and that of self-dual-GinE shows excellent agreement with Eq. (\ref{['eq-nvar-self-dual-GinE']}).
• Figure 5: (a) The spectral density for the eigenvalues of the Floquet operators given by Eq. (\ref{['eq-F-dissp']}). The scatter plot of eigenvalues for $F_\text{sG}, F_\text{G}$ and $F_\text{sdG}$ are shown in (b), (c) and (d) respectively.