An exactly solvable model for the integrability–chaos transition in rough quantum billiards

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. The dynamics of isolated quantum systems can either be strongly correlated with their initial state, or chaotic, as they relax into thermal equilibrium. Olshanii et al . present a simple, exactly solvable model that captures the transition between these two limiting cases, and suggests it may have some universal features.