Localization of quantum wave packets

TL;DR

This paper investigates the semiclassical evolution of quantum wave packets, especially Gaussian states, under periodic Hamiltonian conditions, revealing conditions for wave packet revivals and exponential spreading in unstable classical systems.

Contribution

It extends the propagation theorem to analyze wave packet behavior with periodic Hessians, identifying conditions for revivals and recurrences, including unbounded and unstable classical motions.

Findings

Wave packet width remains small up to Ehrenfest time under periodic Hessian conditions.

Conditions for classical revivals of wave packets are established.

Exponential spreading occurs in classically unstable systems.

Abstract

We study the semiclassical propagation of squeezed Gau{\ss}ian states. We do so by considering the propagation theorem introduced by Combescure and Robert \cite{CR97} approximating the evolution generated by the Weyl-quantization of symbols $H$. We examine the particular case when the Hessian $H^{\prime\prime}(X_{t})$ evaluated at the corresponding solution $X_{t}$ of Hamilton's equations of motion is periodic in time. Under this assumption, we show that the width of the wave packet can remain small up to the Ehrenfest time. We also determine conditions for ``classical revivals'' in that case. More generally, we may define recurrences of the initial width. Some of these results include the case of unbounded classical motion. In the classically unstable case we recover an exponential spreading of the wave packet as in \cite{CR97}.