Parametric correlations and diffusion in quantum spectra

Abstract

The evolution of the quantum spectrum as the function of a control parameter can be mapped onto the molecular dynamics of a classical many-body system. This approach is used to study the parametric motion of energy levels in classically mostly regular and in mostly chaotic regimes. The parametric energy-level dynamics is described in terms of the velocity autocorrelation function and the friction kernel of a disordered many-body system. We show that the parametric dynamics is sensitively dependent on the underlying classical dynamics, and therefore provide new measures of quantum chaology. The transition from classically regular to chaotic motion manifests itself in the disappearance of parametric diffusion. Semiclassical arguments give analytic approximations for the spectrum of the velocity autocorrelation function, and these approximations are found to be consistent with numerical calculations for the vibronic motion of a triatomic molecule. We compare these autocorrelation functions with those found in gases, liquids, and glasses.

Global versus Local Billiard Level Dynamics: The Limits of Universality

“…originally introduced by Yang and Burgdörfer [6], and conjectured a universal behavior as long as the so-called zero-mode approximation holds, i.e., in the range where the energy fluctuations show random matrix behavior. For the velocity distribution Simons and Altshuler found a Gaussian behavior [5].…”

Experimental Determination of the Autocorrelation Function of Level Velocities for Microwave Networks Simulating Quantum Graphs

“…Instead of the original energy levels E i (X) one obtains the unfolded en- (1) where N av (E i (X)) is the average number of energy levels with the mean level spacing ∆ = 1 and for quantum graphs is given by the formula [47] …”

“…The parameter-dependent autocorrelation function c(x) of level velocities, where x is the rescaled external parameter X, was introduced about 20 years ago by [1,2]. It was shown that the autocorrelation function c(x) is a universal measure describing a quantum system which is classically chaotic.…”