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Physical Review Letters

Scars in Groups of Eigenstates in a Classically Chaotic System

Gonzalo G. de Polavieja, F. Borondo, R. M. Benito

Published Sep 19, 1994• 92 citations
Publisher

Abstract

A general method to construct wave functions highly localized on a given periodic orbit, using the information contained in the short term quantum dynamics of the system, is presented. The relationship with the Husimi's quasiprobability distribution in phase space is also discussed.
DOI
10.1103/physrevlett.73.1613

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The wave functions of the Dicke Hamiltonian, describing a spin coupled to a bosonic mode, are studied via Husimi distributions. A classical analogue of this system is also obtained. For several energy ranges studied, the Husimi distribution of the wave functions show the scar of simple periodic orbits when projected into the boson phase space. Surprisingly, these same distributions, when projected into the spin phase space, are spread through large regions. An explanation of this fact is given in terms of semiclassical theory and border effects associated with nonsemiclassical behaviour.
The Husimi distribution for the eigenstates of a classically nonintegrable two-dimensional system with mixed phase space was calculated in order to analyze its structure in terms of classical trajectories. Besides eigenfunctions concentrated on invariant tori or with support on chaotic regions of phase space, we also found others that combine different classical invariant sets (like a stable and an unstable periodic orbit) and families of eigenstates scarred by classically unstable periodic orbits.