Colored $1/f^{\alpha}$ noise and the Order to Chaos Transition in Quantum Mechanics

Abstract

The spectral statistic $\delta_n$ measures the fluctuations of the number of energy levels around its mean value. It has been shown that chaotic quantum systems display $1/f$ noise (pink noise) in the power spectrum $S(f)$ of the $\delta_n$ statistic, whereas integrable ones exibit $1/f^2$ noise (brown noise). These results have been explained on the basis of the random matrix theory and periodic orbit theory. Recently we have analyzed the order to chaos transition in terms of the power spectrum $S(f)$ by using the Robnik billiard (Phys. Rev. Lett. {\bf 94}, 084101 (2005)). We have numerically found a net power law $1/f^{\alpha}$, with $1\leq \alpha \leq 2$, at all the transition stages. Similar results have been obtained by Santhanam and Bandyopadhyay (Phys. Rev. Lett. {\bf 95}, 114101 (2005)) analyzing two coupled quartic oscillators and a quantum kicked top. All these numerical results suggest that the exponent $\alpha$ is related to the chaotic component of the classical phase space of the quantum billiard, but a satisfactory theoretical explanation is still lacking.