Correspondences between quantum and classical orbits Berry phases and Hannay angles for harmonic oscillator system

Abstract

On the basis of quantum-classical correspondence for two-dimensional anisotropic oscillator, we study quantum-classical correspondence for two-dimensional rotation and translation harmonic oscillator system from both quantum-classical orbits and geometric phases. Here, the two one-dimensional oscillators refer to a common harmonic oscillator and a rotation and translation harmonic oscillator. In terms of the generalized gauge transformation, we obtain the stationary Lissajous orbits and Hannay's angle. On the other hand, the eigenfunctions and Berry phases are derived analytically with the help of time-dependent gauge transformation. We may draw the conclusion that the nonadiabatic Berry phase in the original gauge is-n times the classical Hannay's angle, here n is the eigenfunction index. As a matter of fact, the quantum geometric phase and the classical Hannay's angle have the same nature according to Berry. Finally, by using the SU(2) coherent superposition of degenerate two-dimensional eigenfunctions for a fixed energy value, we construct the stationary wave functions and show that the spatial distribution of wave-function probability clouds is in excellent accordance with the classical orbits, indicating the exact quantum-classical correspondence. We also demonstrate the quantum-classical correspondences for the geometric phase-angle and the quantum-classical orbits in a unified form.