Quantum chaos on the separatrix of the periodically perturbed Harper model

TL;DR

This paper investigates the quantum signatures of chaos in a periodically perturbed Harper model, linking classical chaotic dynamics with quantum eigenstates through phase space analysis and Floquet theory.

Contribution

It introduces a method to connect classical chaos near the separatrix with quantum eigenstates using Husimi distributions and energy dispersion analysis.

Findings

Floquet eigenstates resemble classical orbits with similar energies

Chaotic classical orbits correspond to ergodic Floquet eigenstates

Energy dispersion distinguishes between ergodic and integrable states

Abstract

We explore the relation between a classical periodic Hamiltonian system and an associated discrete quantum system on a torus in phase space. The model is a sinusoidally perturbed Harper model and is similar to the sinusoidally perturbed pendulum. Separatrices connecting hyperbolic fixed points in the unperturbed classical system become chaotic under sinusoidal perturbation. We numerically compute eigenstates of the Floquet propagator for the associated quantum system. For each propagator eigenstate we compute a Husimi distribution in phase space and an energy and energy dispersion from the expectation value of the unperturbed Hamiltonian operator. The Husimi distribution of each Floquet eigenstate resembles a classical orbit with a similar energy and similar energy dispersion. Chaotic orbits in the mixed classical system are related to Floquet eigenstates that appear ergodic. For a mixed regular and chaotic system, the energy dispersion can separate the Floquet eigenstates into ergodic and integrable subspaces. The width of a chaotic region in the classical system is estimated by integrating the perturbation along a separatrix orbit. We derive a related expression for the associated quantum system from the averaged perturbation in the interaction representation evaluated at states with energy close to the separatrix.