Kolmogorov stochasticity parameter as a measure of quantum chaos

Abstract

We propose the Kolmogorov stochasticity parameter, $\lambda$ for energy level spectra to classify quantum systems with corresponding classical dynamics ranging from integrable to chaotic. We also study the probability distribution function (PDF) of $\lambda$. Remarkably, the PDF of all the integrable systems studied here is the same and is found to be completely different from the PDF of chaotic systems. We also note that $\lambda_n$ for $n$ energy levels scales as $\lambda_n \sim n^{-\alpha}$. Furthermore, with $\alpha$, the stochastic probability (calculated from PDF) is seen to jump by about an order of magnitude as the systems turn chaotic.