An Exactly Solvable Model for the Integrability-Chaos Transition in Rough Quantum Billiards

TL;DR

This paper introduces an exactly solvable quantum model that describes how systems lose memory of their initial states during the transition from integrability to chaos, applicable to disordered quantum systems.

Contribution

The authors develop a simple, statistically solvable quantum model capturing the integrability-chaos transition with no selection rules, and validate it against numerical simulations.

Findings

Model accurately describes memory loss in quantum systems.

Good agreement with ab initio numerics for impurity lattices.

Provides a rigorous basis for recent theoretical relationships.

Abstract

A central question of dynamics, largely open in the quantum case, is to what extent it erases a system's memory of its initial properties. Here we present a simple statistically solvable quantum model describing this memory loss across an integrability-chaos transition under a perturbation obeying no selection rules. From the perspective of quantum localization-delocalization on the lattice of quantum numbers, we are dealing with a situation where every lattice site is coupled to every other site with the same strength, on average. The model also rigorously justifies a similar set of relationships recently proposed in the context of two short-range-interacting ultracold atoms in a harmonic waveguide. Application of our model to an ensemble of uncorrelated impurities on a rectangular lattice gives good agreement with ab initio numerics.