Local Wigner-Mass Maps and Integrated Negativity as Measures of nonclassicality in Quantum Chaotic Billiards

TL;DR

The paper addresses how nonclassical phase-space structure emerges in wave-chaotic systems by introducing local Wigner-mass maps and the integrated negativity as a compact diagnostic. It combines a group-theoretic Wigner–Weyl framework with a density-operator perspective to show that off-diagonal coherence between hybridizing modes generates oscillatory, sign-changing features in the Wigner function, with negativity peaking near avoided crossings. The authors define global negativity $ left( $) and local masses $P(r)$ and $N(r)$, linking them via $P(r)-N(r)=| abla ext{psi}(r)|^2$ and $ left(N= int N(r) d^2r ight)$, and demonstrate that negativity concentrates at low momentum $k o0$ and near the cavity center, while mixed marginals remain nonnegative due to Wiener–Khinchin. These results, validated in oval and quadrupole billiards, reveal a universal interference mechanism controlled by relative modal weights and off-diagonal coherence, with practical implications for mode engineering and coherence control in wave-chaotic platforms such as optical, acoustic, and microwave resonators.

Abstract

The Wigner function is a phase space quasi-probability distribution whose negative regions provide a direct, local signature of nonclassicality. To identify where phase-sensitive structure concentrates, we introduce local positive- and negative Wigner-mass maps and adopt the integrated Wigner negativity as a compact scalar measure of nonclassical phase space structure. A decomposition of the density operator reveals that off-diagonal coherences between hybridizing components generate oscillatory, sign-alternating patterns, with the negative contribution maximized when component weights are comparable. Non-Gaussian chaotic eigenmodes exhibit a baseline negativity that is further amplified by such hybridization. We validate these diagnostics across two billiard geometries and argue that the framework is transferable to other wave-chaotic platforms, where it can aid mode engineering and coherence control.

Figures (6)

• Figure 1: Spectral trajectories and representative eigenmodes through an avoided crossing. (a) Eigenvalue trajectories $k$ versus deformation $\varepsilon$; markers A--F indicate sampled parameter values. The solid orange curve in (a) corresponds to mode 2, and the dashed cyan curve corresponds to mode 1. (b)--(m) Paired position $|\psi(r)|^2$ (left or upper) and momentum-space $(2\pi)^{-2}|\Psi(k)|^2$ (right or lower) intensity maps. Near the AC. center, the position intensity exhibits increased mixing and delocalization, while momentum-space signatures retain the characteristic ring/lobed structure of each eigenmode; the momentum-ring radius remains nearly constant at $\approx 22$.
• Figure 2: Wigner marginals for the two interacting modes near the avoided crossing. Panels (a)--(d) show Mode E in Figure \ref{['Figure-1']} and panels (e)--(h) show Mode B in Figure \ref{['Figure-1']} . For each mode the four marginal projections are presented in the same order: (a),(e) $f(x,k_y)$; (b),(f) $f(y,k_x)$; (c),(g) $f(x,k_x)=W(x,k_x)$ (reduced Wigner); (d),(h) $f(y,k_y)=W(y,k_y)$ (reduced Wigner). Accordingly, the mixed marginals $f(x,k_y)$ and $f(y,k_x)$ are always non-negative and reproduce Born-rule probability densities, whereas the reduced Wigner projections $f(x,k_x)$ and $f(y,k_y)$ may exhibit negative lobes (blue regions) that signal quantum interference and nonclassical phase-space structure. Black contour lines indicate the zero level of the reduced Wigner functions ($f=0$) and serve as a visual guide. A common colorbar applies to all panels.
• Figure 3: Local positive and negative phase space masses, $P(r)$ (green) and $N(r)$ (orange), shown as 3D surfaces over the configuration space $(x,y)$. Panels (a,b), (c,d), and (e,f) correspond to eigenmodes D, E and F from Fig. 1(a), while (g,h), (i,j) and (k,l) correspond to A, B and C. In each pair the left surface is $P(r)$ and the right is $N(r)$. Across all parameter points $P(r)$ is slightly larger in magnitude than $N(r)$, consistent with the global Wigner negativity. The spatial maps display magnitudes that tend to increase toward the configuration space origin $(r=0)$, indicating stronger local phase space mass near the center. These maps provide a spatial decomposition of the global nonclassicality measure $\mathcal{N}$.
• Figure 4: Momentum-space local positive and negative phase-space masses, $\widetilde{P}(k)$ (blue) and $\widetilde{N}(k)$ (orange), shown as intensity maps over momentum coordinates $(k_x,k_y)$. Panels (a,b), (c,d), and (e,f) correspond to eigenmodes D, E and F from Fig. \ref{['Figure-1']}(a), while (g,h), (i,j) and (k,l) correspond to A, B and C. In each pair the left image is $\widetilde{P}(k)$ and the right image is $\widetilde{N}(k)$. Although the integrated positive contribution slightly exceeds the negative one as in Fig. \ref{['Figure-3']}, the momentum-space maps exhibit a pronounced negative peak at the origin $(k=0)$, indicating that the dominant negative phase-space mass is concentrated at zero momentum. These plots complement the configuration-space decomposition of the global Wigner negativity $\mathcal{N}$.
• Figure 5: (a) Integrated positive mass $P=\int P(r)\,d^2r$ and (b) negative mass $\mathcal{N}=\int N(r)\,d^2r$ as functions of the deformation parameter $\epsilon$ for the avoided-crossing pair in Fig. 1(a). Both quantities peak near the AC., reflecting enhanced phase-space interference. The inset in (b) verifies the normalization $P-N=1$ across the full range of $\epsilon$.