Beyond Ginibre statistics in open Floquet chaotic systems with localized leaks

TL;DR

This work shows that Ginibre statistics do not universally describe open, chaotic quantum systems with spatially localized leaks. By studying the leaky quantum standard map and comparing its complex resonance spectrum to the Ginibre ensemble and the truncated circular orthogonal ensemble (TCOE), the authors demonstrate that long-lived resonances obey TCOE-like density and short-range correlations, while short-lived resonances and certain leakage regimes deviate from both benchmarks. As leakage grows, the density of states of the truncated ensemble tends toward the Ginibre circular law, yet its local spectral correlations persistently differ from Ginibre, indicating a distinct universal class for truncated non-Hermitian matrices. These results have implications for interpreting experiments in optical and microwave billiards and underscore the nuanced limits of Ginibre universality in open quantum chaotic systems.

Abstract

We show that the spectral properties of driven quantum systems with a classically chaotic counterpart and spatially localized openness, such as optical or microwave billiards with leaks, deviate from predictions of Ginibre ensembles. Our analysis focuses on the leaky quantum standard map (QSM) of the kicked rotor. We compare its complex resonance spectrum with both Ginibre and truncated circular orthogonal ensembles (TCOEs). We find that the long-lived resonances follow TCOE statistics, reproducing the density of states and level spacing correlations, but depart from Ginibre predictions. Short-lived resonances, however, do not show a clear correspondence with either random-matrix ensemble. We also demonstrate that increasing the leak size takes the density of states of the TCOE toward the Ginibre limit, yet their spectral correlations remain distinct.

Figures (7)

• Figure 1: Sketch of the distributions of the eigenvalues over the complex plane for the (a) leaky quantum standard map, (b) truncated circular orthogonal ensemble, and (c) Ginibre unitary ensemble.
• Figure 2: Classical-quantum correspondence of the closed standard map for (a),(c) mixed phase-space regime at $K=1.5$ and (b),(d) fully chaotic regime at $K=10$. (a)-(b) Classical phase-space landscapes and (c)-(d) Husimi Q-functions of selected eigenstates.
• Figure 3: Open (a)-(b) classical and (c)-(d) quantum standard map with leakage; $\Delta q=0.2$, $K=10$, and, for the quantum case, $N=10^4$. (a) Map of the classical dwell time $\tau$ for a slit (shaded area) centered at $\bar{q}_L = 0.2$. Dark blue indicates trajectories that escape after $\tau=1$ iteration; blue for $\tau=2$; green for $\tau=3$; and white corresponds to long-lived trajectories ($\tau > 3$). (b) Classical dwell time averaged over the phase space, $\langle\tau\rangle$, as a function of the slit position $\bar{q}_L$. (c) Husimi Q-function averaged over the 20 longest-lived eigenstates. The probability density follows the classical dwell-time landscape: it is suppressed in regions where the classical trajectories escape rapidly and enhanced in regions associated with long classical dwell times. (d) Quantum dwell time averaged over all eigenstates, $\langle T\rangle$, as a function of the slit position: its behavior mirrors the dependence of the classical dwell time on $\bar{q}_L$.
• Figure 4: (a) Averaged density of states $R_1 (\lambda)$ of the complex eigenvalues $\lambda$ of the leaky QSM (circles), TCOE (solid curves), and GinUE (horizontal line at $1/\pi$) plotted as a function of the modulus $|\lambda|$. The leak is centered at $\bar{q}_L = 0.2$ and three slit sizes are shown: $\Delta q = 0.2$ (blue circles and red line), $\Delta q = 0.1$ (green circles and light-green line), and $\Delta q = 0.01$ (purple circles and orange line). The agreement between the leaky QSM and the TCOE progressively extends to the region of low $|\lambda|$ as $\Delta q$ decreases. The inset highlights the region of long-lived resonances, where there is excellent agreement between QSM and TCOE for all slit sizes. (b)-(d): Rescaled two-dimensional density of states of the leaky QSM for increasing slit size from (b) to (d). The white hole around the origin, clearly visible in (d), corresponds to the excluded short-lived resonances, as described in the text. For each value of $\Delta q$, multiple ensemble realizations are considered to ensure a total of at least $10^5$ eigenvalues. The QSM ensemble is generated with $9.99 < K < 10.01$.
• Figure 5: Short-range spectral correlations for the leaky QSM with slit size and position $\Delta q = \bar{q}_L = 0.2$ (blue), compared with the TCOE (red) and the GinUE (black). (a) Nearest-neighbor eigenvalue spacing distributions for all three cases. (b)-(c) Distributions of ratios of consecutive eigenvalue spacings for (b) the leaky QSM and (c) the TCOE. The mean values $\langle r \rangle$ and $\langle \cos \theta \rangle$ are indicated. (d) Radius distribution of the ratio of consecutive spacings showing close agreement among the three ensembles. (e) Angle distribution of the ratio of consecutive spacings, demonstrating strong correspondence between the leaky QSM and the TCOE, and their deviation from the GinUE. Multiple ensemble realizations are used to ensure a total of at least $10^5$ eigenvalues. The QSM ensemble is generated with $9.99 < K < 10.01$ and $N = 10^4$.