Semiclassical investigation of the revival phenomena in a one-dimensional system

Abstract

In a quantum revival, a localized wavepacket re-forms or "revives" into a compact reincarnation of itself long after it has spread in an unruly fashion over a region restricted only by the potential energy. This is a purely quantum phenomenon, which has no classical analog. Quantum revival, and Anderson localization, are members of a small class of subtle interference effects resulting in a quantum distribution radically different from the classical after long time evolution under classically nonlinear evolution. However it is not clear that semiclassical methods, which start with the classical density and add interference effects, are in fact capable of capturing the revival phenomenon. Here we investigate two different one dimensional systems, the infinite square well and Morse potential. In both cases, after a long time the underlying classical manifolds are spread rather uniformly over phase space and are correspondingly spread in coordinate space, yet the semiclassical amplitudes are able to destructively interfere over most of coordinate space and constructively interfere in a small region, correctly reproducing a quantum revival. Further implications of this ability are discussed.

Semiclassical evolution of quantum wave packets on the torus beyond the Ehrenfest time in terms of Husimi distributions

“…An interesting problem would be the calculation of semiclassical measures for more general potentials, for example, the Morse potential, as well as various multi-dimensional bounded domains and compact manifolds. Coherent states for the Morse potential were constructed in [5], the structure of revivals was studied in [36,55]. One can also consider quantum optimal control problems (with both coherent and incoherent controls [46,47]) in semiclassical long-time limit.…”

Quantum revivals of Morse oscillators and Farey-Ford geometry

“…An essential part of wave packet dynamics analysis of anharmonic systems is to predict if and when exact wave packet revival might occur. If T rev is a time for a Morse oscillator revival , then Wang and Heller [21] have shown where M is an integer. This revealed two facts about Morse oscillator dynamics.…”

“…The Morse oscillator potential Equation 1a is an anharmonic potential [16] used to describe covalent molecular bonding. Some dynamics of Morse states have been studied [17,18,19,20,21] as a model of vibrational anharmonicity.…”

Semiclassical description of wave packet revival

“…The key step was to use the Poisson transformation, which reshapes a semiclassical correlation function into a quantum-looking one, or vice versa. Similar analyses may be possible both for the Coulomb potential [6,7] and the Morse oscillator [15] (the latter is simpler, in principle, because its spectrum is quadratic in the quantum number [25]). …”

“…(This equation has already been used by Wang and Heller in their semiclassical study of the square well [15]. )…”