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arXiv:nlin/0006017 · Physical Review Letters

Scars of the Wigner Function

Fabricio Toscano, Marcus A. M. de Aguiar, Alfredo M. Ozorio de Almeida

Published Jan 1, 2001• 12 citations
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Abstract

We propose a picture of Wigner function scars as a sequence of concentric rings along a two-dimensional surface inside a periodic orbit. This is verified for a two-dimensional plane that contains a classical hyperbolic orbit of a Hamiltonian system with 2 degrees of freedom. The stationary wave functions are the familiar mixture of scarred and random waves, but the spectral average of the Wigner functions in part of the plane is nearly that of a harmonic oscillator and individual states are also remarkably regular. These results are interpreted in terms of the semiclassical picture of chords and centers.

DOI
10.1103/physrevlett.86.59

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We propose a picture of Wigner function scars as a sequence of concentric rings along a two-dimensional surface inside a periodic orbit. This is verified for a two-dimensional plane that contains a classical hyperbolic orbit of a Hamiltonian system with 2 degrees of freedom. The stationary wave functions are the familiar mixture of scarred and random waves, but the spectral average of the Wigner functions in part of the plane is nearly that of a harmonic oscillator and individual states are also remarkably regular. These results are interpreted in terms of the semiclassical picture of chords and centers.
We study the relationship of the spectral form factor with quantum as well as classical probabilities to return. Defining a quantum return probability in phase space as a trace over the propagator of the Wigner function allows us to identify and resolve manifolds in phase space that contribute to the form factor. They can be associated with classical invariant manifolds such as periodic orbits, but also to nonclassical structures such as sets of midpoints between periodic points. In contrast to scars in wave functions, these features are not subject to the uncertainty relation and therefore need not show any smearing. They constitute important exceptions from a continuous convergence in the classical limit of the Wigner towards the Liouville propagator. We support our theory with numerical results for the quantum cat map and the harmonically driven quartic oscillator.
The two-center shell-model potential admits a two-component eigenfunction due to the presence of the spin-orbit coupling. We carry out a detailed numerical study of its variation with the shape of the potential and also investigate the applicability of the criteria commonly used to distinguish regular and irregular states in systems that possess a clear classical analog. Our results corroborate similar studies on spectral fluctuations. The occurrence of scarred states and a Gaussian amplitude distribution strongly indicate the presence of an underlying classical dynamics. Moreover, since the spin-up and spin-down spatial components of each eigenfunction are significantly dissimilar, there seem to be two sets of closed orbits that give rise to the same spectrum but influence the wave functions in different ways.
We study nonergodic behavior of an ensemble of quantum cat map with initially localized phase-space distributions. For this purpose, an amended version of the Wigner function on discretized phase torus is given, which shows meaningful classical-quantum correspondence in the semiclassical limit. We find the time averaged coarse-grained Wigner function is scarred and antiscared along some short periodic orbits, and the heights (depths) of these scars (antiscars) decrease linearly with when the initial phase-space distribution is fixed and the semiclassical limit is taken by letting with a prime and .
We extend the asymptotic boundary layer (ABL) method, originally developed for stable resonator modes, to the description of individual wave functions localized around unstable periodic orbits. The formalism applies to the description of scar states in fully or partially chaotic quantum systems, and also allows for the presence of smooth and sharp potentials, as well as magnetic fields. We argue that the separatrix wave function provides the largest contribution to the scars on a single wave function. This agrees with earlier results on the wave-function asymptotics and on the quantization condition of the scar states. Predictions of the ABL formalism are compared with the exact numerical solution for a strip resonator with a parabolic confinement potential and a magnetic field.
In this work we study the Wigner functions, which are the quantum analogues of the classical phase space density, and show how a full rigorous semiclassical scheme for all orders of ℏ can be constructed for them without referring to the actual coordinate space wavefunctions from which the Wigner functions are typically calculated. We find such a picture by a careful analysis around the stationary points of the main quantization equation, and apply this approach to the harmonic oscillator solving it for all orders of ℏ.