Porter-Thomas fluctuations in complex quantum systems

TL;DR

This paper investigates Porter-Thomas fluctuations in complex quantum systems, revealing that coupling to decay channels can alter the effective degrees of freedom from one to two, with implications for understanding eigenstate decay rate distributions.

Contribution

It demonstrates that the effective number of degrees of freedom in Porter-Thomas fluctuations can change from one to two depending on system parameters, extending the understanding of GOE and GUE behaviors.

Findings

Coupling to decay channels can increase effective degrees of freedom from 1 to 2.

GUE-like fluctuations can occur under milder conditions than previously thought.

Analytic results are derived for certain parameter regimes.

Abstract

The Gaussian Orthogonal Ensemble (GOE) of random matrices has been widely employed to describe diverse phenomena in strongly coupled quantum systems. An important prediction is that the decay rates of the GOE eigenstates fluctuate according to the distribution for one degree of freedom, as derived by Brink and by Porter and Thomas. However, we find that the coupling to the decay channels can change the effective number of degrees of freedom from $\nu = 1$ to $\nu = 2$. Our conclusions are based on a configuration-interaction Hamiltonian originally constructed to test the validity of transition-state theory, also known as Rice-Ramsperger-Kassel-Marcus (RRKM) theory in chemistry. The internal Hamiltonian consists of two sets of GOE reservoirs connected by an internal channel. We find that the effective number of degrees of freedom $\nu$ can vary from one to two depending on the control parameter $\rho \Gamma$, where $\rho$ is the level density in the first reservoir and $\Gamma$ is the level decay width. The $\nu = 2$ distribution is a well-known property of the Gaussian Unitary ensemble (GUE); our model demonstrates that the GUE fluctuations can be present under much milder conditions. Our treatment of the model permits an analytic derivation for $\rho\Gamma \gtrsim 1$.