Quantum response of weakly chaotic systems

TL;DR

This paper investigates the quantum response of weakly chaotic systems, revealing that their Hamiltonian matrices are sparse and differ from traditional random matrix predictions, affecting heating rates in optical billiards.

Contribution

It introduces a novel framework for understanding weak quantum chaos, highlighting the sparsity of Hamiltonian matrices and their impact on system response.

Findings

Hamiltonian matrices are sparse in weakly chaotic systems.

Sparsity characterized by parameters s and g.

Resistor network calculations relate to semi-linear response.

Abstract

Chaotic systems, that have a small Lyapunov exponent, do not obey the common random matrix theory predictions within a wide "weak quantum chaos" regime. This leads to a novel prediction for the rate of heating for cold atoms in optical billiards with vibrating walls. The Hamiltonian matrix of the driven system does not look like one from a Gaussian ensemble, but rather it is very sparse. This sparsity can be characterized by parameters $s$ and $g$ that reflect the percentage of large elements, and their connectivity respectively. For $g$ we use a resistor network calculation that has direct relation to the semi-linear response characteristics of the system.