1 / f α noise in spectral fluctuations of quantum nonchaotic billiards

Let {εᵢ}ᵢ₌₁∞ represent the set of unfolded energy eigenvalues of a quantized billiard system. Then, the set {δₙ=εₙ₊₁-ε₁-n} may be considered as a finite time series of size M=nₘ-n₀, in which n plays the role of a discrete time and δₙ fluctuates around zero. These random fluctuations are known to exhibit the 1/fα noise characteristic, where α∈[1,2], the limit α=1(2) corresponding to a fully chaotic (regular) classical dynamics. Previously, the presence of 1/fα noise in the δₙ statistics has been demonstrated in a family of billiards for which the classical phase space is divided (Kolmogorov-Arnold-Moser systems). Here, the δₙ statistics is further explored, so as to include two new features, namely, (i) nonchaotic full ergodicity and (ii) symmetry. Four billiard families, classified as type P and type S, are considered. The type-P families are comprised of polygonal billiards, which are never chaotic, but may exhibit a mixing classical dynamics characterized quantitatively through the position autocorrelation function decay exponent σ. A type-S billiard has a smooth boundary and, correspondingly, a divided phase space with normalized chaotic volume ρc. Possible correlations between the spectral noise parameter α and the classical quantities, σ for type-P and ρc for type-S domains, are numerically investigated. Spearman's rank correlation coefficients support that the quantum spectral exponent α scales with the classical quantities. In addition, one family in each class is C₃ symmetric. In this case, singlets and doublets have different spectral statistics in the chaotic limit, namely, the singlet (doublet) subspectrum follows the Gaussian orthogonal ensemble (Gaussian unitary ensemble) of random matrices. Let α₁(α₂) be the noise exponent in a singlet (doublet) time series. Numerical calculations of Pearson's correlation coefficients show that α₁ and α₂ are compatible with each other for both type-S and type-P within symmetrical families in the regime 1<α<2.