Research Projects

Deviations from Ergodicity Via Parametric Variations

Research conducted by: Steven Tomsovic, Nicholas Cerruti, Arul Lakshminarayan, Srihari Keshavamurthy, Julie Lefebvre

Introduction

Quantum systems that are classical chaotic were conjectured to follow statistics from random matrix theory which assumes the independence of eigenvectors from eigenvalues.

Our work

By the use of parametric variations we observed systematic deviations. A detailed semiclassical analysis of the response of the eigenvalues to a perturbation related the level velocity variance to classical diffusion coefficients. We studied the standard map as a specific example, and thus the well-known oscillatory behavior of the diffusion coefficient with respect to the parameter is reflected exactly in the oscillations of the variance of the level velocities. A new correlation measure that sensitively probes phase space localization properties of eigenstates was also developed. The measure correlates level velocities with overlap intensities between the eigenstates and some localized state of interest. Random matrix theory predicts the absence of such correlations in chaotic systems whereas in the stadium billiard and quantum maps strong correlations were observed.

Figures

Illustration of ergodic behavior.
The upper square shows how the energy eigenvalues move as a function of some parameter, lambda. The lower square is a graphical representation of the strength function. Each small line segment is centered on an eigenvalue and its lambda value. The heights are proportional to the overlap intensity with a wavepacket. The level velocities and overlap intensities were produced using a Gaussian orthogonal ensemble.

Crossover decay. The upper panel demonstrates the agreement between the quantum standard map (solid curve) and the theoretical results (dashed curve). The middle and lower panel shows each piece of the fidelity (diagonal and off-diagonal) separately and the corresponding theoretical results (Gaussian and exponential). The parameters are N = 1000, lambda = 18 and epsilon =.0005.