Classical system underlying a diffracting quantum billiard

Natasha Leporé

Nicolas Pavloff, C. Schmit

Nicolas Pavloff, C. Schmit

We study diffractive effects in two-dimensional polygonal billiards. We derive an analytical trace formula accounting for the role of the nonclassical diffractive orbits in the quantum spectrum. As an illustration, the method is applied to a triangular billiard.

Arseni Goussev, J. R. Dorfman

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.

Zhang Ji-Quan, Yanhui Zhang, Zhou Hui +3 more

We analyze the quantum spectra of the two-dimensional annular billiard system based on the open-orbit quantum spectrum function, and respectively calculate the Fourier-transformed quantum spectra for various values of a parameter （in the case f=Rin/Rout）. The results show that the peak positions of quantum spectrum match with the lengths of classical orbits of particle motion very well ，and the semiclassical closed-orbit theory affords a good explanation. While the inner circle radius becomes comparaber with the de Brogile wavelength of the particle, the character of quantum spectrum changes essentially, turning out to be similar with the pattern of optical diffraction, which is a purely wavelike phenomenon and accords with Fresnel-Kirchhoff diffraction theorem. Our treatment provides a basis theory for researching the property of billiard system and microjunction transport, and opens a new way to investigate the diffraction of crystal and spectral analysis.

Peter Schmelcher, F. Lenz, Davron U. Matrasulov +2 more