Evolution and exact eigenstates of a resonant quantum system

Abstract

We consider the generalized quantum Chirikov map under a resonance condition 2πħ=M/N. This is the quantum-resonance condition discovered by Casati et al. At the resonance, the quantum system reduces to a set of independent N×N unitary matrix eigenequations. We can reduce the evolution operator of the original system as a direct sum of these N×N unitary matrices. We then obtain the eigenstates and eigenenergies (the pseudoenergies) of the quantum map and illustrate their dependences on a number of parameters. We plot these eigenstates in the coherent-state representation and show that they follow closely the Kolmogorov-Arnol’d-Moser curves and other classical orbits.