Beyond the First Recurrence in Scar Phenomena

Abstract

The scarring effect of short unstable periodic orbits up to times of the order of the first recurrence is well understood. Much less is known, however, about what happens past this short-time limit. By considering the evolution of a dynamically averaged wave packet, we show that the dynamics for longer times is controlled by only a few related short periodic orbits and their interplay.PACS numbers: 05.45.+b, 03.65.Ge, 03.65.Sq The study of the quantum manifestations of classical chaos is at present a topic of very active research interest [1]. Great advance came from random matrix theory (RMT) which provides an understanding of universal statistical properties of quantum spectra [2]. The most striking departure from RMT described so far in the literature is the phenomenon known as "scar" [3]. This term describes an anomalous localization of quantum probability density along unstable periodic orbits (PO) in classically chaotic systems. Heller showed [4] the importance of the first recurrences of POs in the scarring effect. In a subsequent work Tomsovic and Heller [5] demonstrated that the semiclassical propagation can be carried out with remarkable precission long after classical fine structure had developed on a scale much smaller thanh, by computing the corresponding correlation function, C scl (t), as a sum of contributions of the homoclinic excursions of the PO. This procedure is however cumbersome, since in general many orbits are needed to obtain converged results, and this number increases rapidly with time (see Ref.[5] for details). This picture is greatly simplified if alternatively considering the corresponding averaged dynamics for finite periods of time. In this case, as will be shown in this Letter, a structure of a few short POs emerges that govern the quantum dynamics for times past the first recurrence of the original PO.Understanding scarring can be tackled from two sides. One way is trying to disentangle the complexity involved in the distribution of individual levels in the spectra of classically chaotic systems [6][7][8]. For example, in Ref.[8] structures localized on short POs of the stadium were obtained, by considering state correlation diagrams and iteratively removing the parametric interactions (avoided crossings) between the involved eigenstates. The other way consists of approaching the problem in a much more straighforward fashion, by studying how the dynamics of POs induce scars in the eigenfunctions of the system.Heller's work provided a time-dependent view that shows how recurrences in the short time dynamics of a wave packet along the neighborhood of an isolated PO produces the accumulation of quantum probability density characteristic of scars. Very recently Kaplan and Heller [9] have shown how the use of coherent wave packet sums, decaying as the log-time, leads to enhanced scarring. Later, it was described by some of us [10] how (nonstationary) wave functions highly localized over POs can be constructed from finite time Fourier transform of wave packets. This method all...