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Physical Review A

Observation of quantum-classical correspondence from high-order transverse patterns

Yi‐Fan Chen, Yu-Pin Lan, K. F. Huang

Published Oct 6, 2003• 26 citations
Publisher

Abstract

We experimentally observe the formation of high-order transverse patterns in a laser resonator with a high degree of frequency degeneracy. It is found that the transverse patterns are well localized on the Lissajous orbits. The connection between the wave functions and the classical periodic orbits is analytically constructed by using the representation of SU(2) coherent states. With this connection, the observed transverse patterns are reconstructed very well. The nice reconstruction suggests that the laser resonators can be deliberately designed to attain a more thorough understanding of the quantum-classical connection.

DOI
10.1103/physreva.68.043803

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We experimentally observe the formation of high-order transverse patterns in a laser resonator with a high degree of frequency degeneracy. It is found that the transverse patterns are well localized on the Lissajous orbits. The connection between the wave functions and the classical periodic orbits is analytically constructed by using the representation of SU(2) coherent states. With this connection, the observed transverse patterns are reconstructed very well. The nice reconstruction suggests that the laser resonators can be deliberately designed to attain a more thorough understanding of the quantum-classical connection.
The geometry of classical dynamics in coupled oscillators with SU(2) transformations is explored and found to be relevant to a family of continuous-transformation orbits between Lissajous and trochoidal curves. The quantum wave-packet coherent states are derived analytically to correspond exactly to the transformation geometry of classical dynamics. By using the quantum wave-packet coherent states derived herein, stationary coherent states are constructed and are shown to possess spatial patterns identical to the transformation geometry between Lissajous and trochoidal orbits.
We extend the presentation of the SU(2) coherent states to analytically construct the wave function concentrated on high-order classical periodic orbits in a square billiard. With the constructed wave function, the localization of the wave pattern is found to be very efficient. We also analyze the vortices arising from the singular points of the quantum phase for the constructed coherent states. It is found that the wave interference gives rise to the appearance of vortex lattices in the probability current density associated with the high-order periodic orbits. Moreover, the topological charge of the vortex is in general nonintegral except for the periodic orbits with the same winding number to the sides of the square.
We study the connections between some specially entangled stationary states and the classical periodic trajectories of two non-interacting oscillators. The latter are well-known Lissajous figures, which are shown to run precisely over the apogees of their corresponding probability distributions in the above states. We propose in this work a very simple criterion enabling us to obtain the best agreement between the quantum and the classical images. Finally, our results are successfully applied to the interpretation of some experimentally observed wave patterns.
Abstract In the three‐dimensional (3D) transversely symmetric oscillator, there are plentiful degeneracies and gaps in the quantum energy spectrum as a function of the ratio of the transverse to longitudinal frequency. It is theoretically verified that while the SU(2) interaction destroys the original degeneracies, numerous new degeneracies and gaps emerge around the original degeneracies to form a similar fine energy spectrum. The classical trajectories at the emergent degeneracies are analyzed to be localized on the 3D parametric surfaces which are constituted by the topologically invariant curves in the transverse tomography. The quantum coherent states are exploited to develop the wave functions that correspond to the 3D geometric surfaces in classical dynamics. Furthermore, the wave structures of the stationary coherent states at small quantum numbers are explored and found to display peculiar patterns with symmetries related to classical trajectories.
A quantum scar-an enhancement of a quantum probability density in the vicinity of a classical periodic orbit-is a fundamental phenomenon connecting quantum and classical mechanics. Here we demonstrate that some of the eigenstates of the perturbed two-dimensional anisotropic (elliptic) harmonic oscillator are strongly scarred by the Lissajous orbits of the unperturbed classical counterpart. In particular, we show that the occurrence and geometry of these quantum Lissajous scars are connected to the anisotropy of the harmonic confinement, but unlike the classical Lissajous orbits the scars survive under a small perturbation of the potential. This Lissajous scarring is caused by the combined effect of the quantum (near) degeneracies in the unperturbed system and the localized character of the perturbation. Furthermore, we discuss experimental schemes to observe this perturbation-induced scarring.
We investigate the quantum signatures of classical nonlinear resonances by making the analytic connection between the quantum wave functions and the classical periodic orbits for the uncoupled systems. It is found that the highly efficient extension of the localized coherent states within the classical caustics is an intriguing phenomenon in mesoscopic systems with nonlinear resonances. With the theoretical analysis, we experimentally demonstrate that the laser resonator with an intracavity saturable absorber can be employed to visualize the wave patterns analogous to the quantum wave functions associated with Fermi resonance.