Quantum Chirikov criterion: Two particles in a box as a toy model for a quantum gas

TL;DR

This paper introduces a quantum version of Chirikov's criterion to analyze chaos in a simple two-particle quantum system, revealing how quantum effects modify classical resonance overlap conditions.

Contribution

It develops a quantum generalization of Chirikov's resonance overlap criterion for a two-particle quantum system, bridging classical chaos theory and quantum mechanics.

Findings

Quantum resonances are represented as low purity eigenstate patches.

Quantum effects exclude resonances smaller than a single eigenstate from chaos criteria.

Quantum analogs of classical KAM tori are identified as undestroyed eigenstate groups.

Abstract

We consider a toy model for emergence of chaos in a quantum many-body short-range-interacting system: two one-dimensional hard-core particles in a box, with a small mass defect as a perturbation over an integrable system, the latter represented by two equal mass particles. To that system, we apply a quantum generalization of Chirikov's criterion for the onset of chaos, i.e. the criterion of overlapping resonances. There, classical nonlinear resonances translate almost verbatim to the quantum language. Quantum mechanics intervenes at a later stage: the resonances occupying less than one Hamiltonian eigenstate are excluded from the chaos criterion. Resonances appear as contiguous patches of low purity unperturbed eigenstates, separated by the groups of undestroyed states -- the quantum analogues of the classical KAM tori.