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arXiv:nlin/0607038 · Physical Review E

Wave packet autocorrelation functions for quantum hard-disk and hard-sphere billiards in the high-energy, diffraction regime

Arseni Goussev, J. R. Dorfman

Published Jul 17, 2006• 0 citations

Abstract

We consider the time evolution of a wave packet representing a quantum particle moving in a geometrically open billiard that consists of a number of fixed hard-disk or hard-sphere scatterers. Using the technique of multiple collision expansions we provide a first-principle analytical calculation of the time-dependent autocorrelation function for the wave packet in the high-energy diffraction regime, in which the particle's de Broglie wavelength, while being small compared to the size of the scatterers, is large enough to prevent the formation of geometric shadow over distances of the order of the particle's free flight path. The hard-disk or hard-sphere scattering system must be sufficiently dilute in order for this high-energy diffraction regime to be achievable. Apart from the overall exponential decay, the autocorrelation function exhibits a generally complicated sequence of relatively strong peaks corresponding to partial revivals of the wave packet. Both the exponential decay (or escape) rate and the revival peak structure are predominantly determined by the underlying classical dynamics. A relation between the escape rate, and the Lyapunov exponents and Kolmogorov-Sinai entropy of the counterpart classical system, previously known for hard-disk billiards, is strengthened by generalization to three spatial dimensions. The results of the quantum mechanical calculation of the time-dependent autocorrelation function agree with predictions of the semiclassical periodic orbit theory.

DOI: 10.1103/physreve.74.016204

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We examine the long-term time dependence of Gaussian wave packets in a circular infinite well (billiard) system and find that there are approximate revivals. For the special case of purely

m=0

states (central wave packets with no momentum) the revival time is ${T}_{\mathrm{rev}}^{(m=0)}=8\ensuremath{\mu}{R}^{2}/\ensuremath{\Elzxh}\ensuremath{\pi},$ where $\ensuremath{\mu}$ is the mass of the particle, and the revivals are almost exact. For all other wave packets, we find that ${T}_{\mathrm{rev}}^{(m\ensuremath{\ne}0)}=({\ensuremath{\pi}}^{2}{/2)T}_{\mathrm{rev}}^{(m=0)}\ensuremath{\approx}{5T}_{\mathrm{rev}}^{(m=0)}$ and the nature of the revivals becomes increasingly approximate as the average angular momentum or number of $m\ensuremath{\ne}0$ states is increased. The dependence of the revival structure on the initial position, energy, and angular momentum of the wave packet and the connection to the energy spectrum is discussed in detail. The results are also compared to two other highly symmetrical two-dimensional infinite well geometries with exact revivals, namely, the square and equilateral triangle billiards. We also show explicitly how the classical periodicity for closed orbits in a circular billiard arises from the energy eigenvalue spectrum, using a WKB analysis.

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We consider the semiclassical quantization of the Sinai billiard for disk radii R small compared with the wavelength . Via the application of the periodic orbit theory of diffraction we derive the semiclassical spectral determinant. The limitations of the derived determinant are studied by comparing it to the exact Korringa-Kohn-Rostoker (KKR) determinant, which we generalize here for the subspace. With the help of the Ewald resummation method developed for the full KKR determinant we transfer the complex diffractive determinant to a real form. The real zeros of the determinant are the quantum eigenvalues in semiclassical approximation. The essential parameter is the strength of the scatterer . Surprisingly, this can take any value between and within the range of validity of the diffractive approximation , causing strong perturbation despite the smallness of the disk. We study the statistics exhibited by spectra for fixed values of c . It is Poissonian for , provided the disk is placed inside a rectangle with irrational squared side ratio. For c = 0 we find a good agreement of the level spacing distribution with Gaussian orthogonal ensemble (GOE). The form factor and two-point correlation function are also similar to GOE, but with larger deviations. By varying the parameter c from 0 to the level statistics interpolates smoothly between these limiting cases. Any of these transitional level statistics can thus be found in the spectrum of a Sinai billiard with sufficiently small R while we go from the quantum to the semiclassical limit .

A semiclassical propagator is applied to describe the dynamics of circular-orbit wave packets which are initially well localized in three dimensions. A sum over classical Kepler trajectories for the wave-packet autocorrelation function is obtained which is extremely accurate well past the classical regime of the evolution regime of the wave packet. The nonclassical nature of the wave-packet evolution at long times is associated with the interference between amplitudes from classical paths at different energies. The time scale of the wave-packet revivals is related to the shearing rate of the corresponding classical ensemble.