Wave Function Intensity Statistics from Unstable Periodic Orbits

Abstract

We examine the effect of short unstable periodic orbits on wave function statistics in a classically chaotic system, and find that the tail of the wave function intensity distribution in phase space is dominated by scarring associated with the least unstable periodic orbits. In an ensemble average over systems with classical orbits of different instabilities, a power-law tail is found, in sharp contrast to the exponential prediction of random matrix theory. The calculations are compared with numerical data, and quantitative agreement is obtained. [S0031-9007 (98)05543-4] PACS numbers: 05.45. + b, 03.65.SqQuantum eigenstates of classically chaotic systems generically exhibit a phenomenon known as scarring, the enhancement of intensity along short unstable periodic orbits for some fraction of the wave functions. Scarring is a fascinating example of the influence of identifiable classical structures on stationary quantum properties and on long-time quantum transport in a classically ergodic system. The occurrence of scars is in some sense paradoxical, because classically, all such short-time information is destroyed at long times, and a classical probability distribution after being evolved for a sufficiently long time retains no memory of its initial state. Scarring is one of the most dramatic examples of a departure of quantum chaotic systems from the predictions of random matrix theory (RMT), according to which wave functions must be evenly distributed over phase space, up to quantum fluctuations. Scarring has now been observed experimentally in a variety of systems, including microwave cavities [1,2], tunnel junctions [3], and the hydrogen atom in a uniform magnetic field [4,5].Examples of scarring were observed numerically in [6], and a theory based on the semiclassical evolution of Husimi states near a periodic orbit was provided. Later work by Bogomolny [7] and Berry [8] involved calculations in coordinate space and Wigner phase space, respectively. All these works were based on the linearized dynamics around the unstable periodic orbit, and were thus, by construction, theories of the short-time behavior only. Yet to get a true understanding of the properties of individual eigenstates it is essential to understand the longtime quantum dynamics, including returns of amplitude to the original periodic orbit after undergoing excursions into other areas of phase space. In a recent paper [9], a formalism was developed for dealing with these nonlinear contributions to scarring, providing quantitative agreement of the theory with numerical results. This work used a measure of scarring based on Husimi intensities. (Recently Fishman, Agam, and others have provided interesting new perspectives on the problem of scarring, and have offered a measure of scarring related to, but somewhat different from ours [10]. A number of other authors have also made significant contributions in this area; we cannot list them all but a few recent references are provided in [11].) In this paper, we will apply a result previously obtained...

Scarring and the Statistics of Tunnelling

“…In [11], the distributions arising from such an analysis were shown to be determined in a simple way by the stability and action of a complex tunnelling orbit which crosses the potential barrier with minimum imaginary action. The resulting statistical distributions for the tunnelling rate agree well with numerically computed ensembles except when the real extension of the tunnelling orbit into the potential well is periodic; in that case, strong deviations from the RMT prediction are observed and it was proposed in [11] that these are due to the effect of scarring on wavefunction statistics as outlined in [13]. Additional evidence in support of this has subsequently been provided in [14].…”

“…We show here that in systems where the complexity of the internal dynamics derives from low-dimensional chaos, sufficiently strong deviation from the standard RMT statistics is possible that it dominates the statistics of tunnelling. These deviations were pointed out in [11] and are explained in detail here using the theory of scarring developed by Heller, Kaplan and coworkers [12,13].…”

“…The low-lying states of this tunnelling operator are approximated by the eigenstates of a harmonic oscillator. Following the linear scarring theory of Kaplan [13] we show that the components of chaotic states in this basis have variances which deviate from those of RMT and which must furthermore be correlated. We conjecture that the eigenfunction distribution is a nonisotropic Gaussian.…”

“…We conjecture that these distributions govern the wavefunction statistics in the presence of scarring when states are taken from a small window of eigenphases. A full justification of this would require a detailed analysis of long orbits contributing to (10), along the lines of the nonlinear theory of scarring in [13]. We will not offer such an analysis here and assume (12) on the grounds of simplicity alone.…”

“…1(a) for the map F ǫ with ǫ = 0.1. Following [13], we refer to this curve as the spectral envelope. We expect larger than average overlaps where C 00 (θ) > 1 and we will refer to this as the scarred region.…”