How do wave packets spread? Time evolution on Ehrenfest time scales

TL;DR

This paper extends time-dependent WKB theory to accurately describe the long-time evolution of localized quantum states, including their transition to delocalized states at Ehrenfest times, using metaplectic operators and phase space analysis.

Contribution

It introduces a novel method combining metaplectic operators with WKB theory for long-time propagation of localized states, providing uniform descriptions at Ehrenfest times.

Findings

Uniform approximation for wave packet scattering near critical energy

Description of wave packet transition from reflection to transmission

Illustration of wave packet delocalization along unstable manifolds

Abstract

We derive an extension of the standard time dependent WKB theory which can be applied to propagate coherent states and other strongly localised states for long times. It allows in particular to give a uniform description of the transformation from a localised coherent state to a delocalised Lagrangian state which takes place at the Ehrenfest time. The main new ingredient is a metaplectic operator which is used to modify the initial state in a way that standard time dependent WKB can then be applied for the propagation. We give a detailed analysis of the phase space geometry underlying this construction and use this to determine the range of validity of the new method. Several examples are used to illustrate and test the scheme and two applications are discussed: (i) For scattering of a wave packet on a barrier near the critical energy we can derive uniform approximations for the transition from reflection to transmission. (ii) A wave packet propagated along a hyperbolic trajectory becomes a Lagrangian state associated with the unstable manifold at the Ehrenfest time, this is illustrated with the kicked harmonic oscillator.