Left-Right Husimi Representation of Chaotic Resonance States: Invariance and Factorization

TL;DR

This work analyzes the left-right Husimi representation for chaotic resonance states and establishes two key properties: quantum invariance, where the LR-Husimi density remains invariant under the closed classical dynamics in the semiclassical limit, and factorization, where the LR-Husimi density splits into a classical, smooth density times universal quantum fluctuations. The factorization is derived from a product of left and right resonance densities, yielding a LR-density ρ_γ^{LR} and LR-fluctuations η^{LR} with a universal distribution P^{LR}(η^{LR}) = (π^2/4) η^{LR} K_0((π/2) η^{LR}). Numerical verification across dielectric cavities, three-disk scattering, and quantum maps confirms the invariance and factorization, and introduces a coarse-grained LR-measure μ_γ^{LR} that converges to the LR-density in the semiclassical limit. The results provide a universal, semiclassical description of resonance-state structure in chaotic scattering and point to experimental observables via averaged LR-Husimi dynamics and LR-measures.

Abstract

For chaotic scattering systems we investigate the left-right Husimi representation, which combines left and right resonance states. We demonstrate that the left-right Husimi representation is invariant in the semiclassical limit under the corresponding closed classical dynamics, which we call quantum invariance. Furthermore, we show that it factorizes into a classical multifractal structure times universal quantum fluctuations. Numerical results for a dielectric cavity, the three-disk scattering system, and quantum maps confirm both the quantum invariance and the factorization.

Figures (10)

• Figure 1: LR-Husimi representation $\mathcal{H}^\mathrm{LR}$ of individual resonance states closest to the specified decay rate $\gamma$ for (a) a dielectric cavity, (b) the three-disk scattering system, (c) the standard map with partial escape, and (d) the standard map with full escape. System parameters are given in \ref{['sec:example_systems']}. In this and all following figures the average intensity is scaled to one (in (a) and (c) on the full phase space and in (b) and (d) on the chaotic saddle). Intensities greater than the maximal value of the colorbar are shown with darkest color.
• Figure 2: Visual comparison of the LR-Husimi representation $\mathcal{H}^\mathrm{LR}$ according to left-hand side (left) and right-hand side (right) of \ref{['Eq:LR_husimi_invariance']}. The integration is done over regions $A$ from (a) a fine $1600 \times 1600$ grid showing deviations from invariance due to stretching along the unstable direction, see zoom by factor 16, and (b) a coarse $50 \times 50$ grid showing less deviations from invariance (even though we choose a more sensitive range for the colorbar). The presented individual resonance state is closest to $\gamma=0.03$ for a dielectric cavity, see \ref{['fig:husimi_individual']}(a, second panel).
• Figure 3: Semiclassical convergence of left-hand side to right-hand side of \ref{['Eq:LR_husimi_invariance']} for a fixed size of regions $A$ of a $50 \times 50$ grid demonstrating quantum invariance. Shown is the Jensen-Shannon divergence $d_{\text{JS}}$ evaluated on this grid approaching zero in the semiclassical limit. (a-d) Four example systems with resonance states near five decay rates each, as in \ref{['fig:husimi_individual']}. Symbols as in \ref{['fig:JS_n_grid']}.
• Figure 4: Factorization of LR-Husimi representation $\mathcal{H}^\mathrm{LR}$ (left) according to \ref{['Eq:LR_Husimi_decomposition']} into the average $\langle \mathcal{H}^\mathrm{LR} \rangle_\gamma$ of resonance states close to decay rate $\gamma$ (middle) times LR-fluctuations $\eta^\mathrm{LR}$, \ref{['Eq:LR_fluctuations_numerical']}, (right) for (a) dielectric cavity at $\gamma = 0.053$, (b) three-disk scattering system at $\gamma = 1.0$, (c) standard map with partial escape at $\gamma = 0.055$, and (d) standard map with full escape at $\gamma = 0.5$.
• Figure 5: Convergence of coarse-grained LR-measure $\mu_\gamma^\mathrm{LR}$ for increasingly fine partitions with $n$ cells. Shown is the Jensen-Shannon divergence $d_{\text{JS}}$ (evaluated on a $50 \times 50$ grid) between measures from partitions $n$ and $n/4$ approaching zero. (a) Dielectric cavity for $\gamma \in \{ 0.011 \, (\gamma_{\mathrm{nat}}), 0.030, 0.053 \, (\gamma_{\mathrm{typ}}), 0.090, 0.122 \, (\gamma_{\mathrm{inv}}) \}$. (b) Three-disk scattering system for $\gamma \in \{0.436 \, (\gamma_{\mathrm{nat}}), 0.6, 1.0, 1.4, 1.8 \}$. (c) Standard map with partial escape for $\gamma \in \{0.22 \, (\gamma_{\mathrm{nat}}), 0.35, 0.55, 0.75, 0.88 \, (\gamma_{\mathrm{inv}}) \}$. (d) Standard map with full escape for $\gamma \in \{0.25 \, (\gamma_{\mathrm{nat}}), 0.35, 0.5, 0.75, 1.0 \}$. Symbols $\circ$, $+$, $\square$, $\times$, and $\diamond$ are used for increasing $\gamma$.