Chaos-assisted tunneling and 1/ƒ^(α) spectral fluctuations in the order-chaos transition

Abstract

It has been shown that the spectral fluctuations of different quantum systems are characterized by 1/ƒ^(α) noise, with 1 </= "alpha" </= 2, in the transition from integrability to chaos. This result is not well understood. We show that chaos-assisted tunneling gives rise to this power-law behavior. We develop a random matrix model for intermediate quantum systems, based on chaos-assisted tunneling, and we discuss under which conditions it displays 1/ƒ^(α) noise in the transition from integrability to chaos. We conclude that the variance of the elements that connect regular with chaotic states must decay with the difference of energy between them. We compare the characteristics of the transition modeled in this way with what is obtained for the Robnik billiard.