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

Abstract

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.

Quantum revival patterns from classical phase-space trajectories

“…in which the initial distribution's center, moving with angular velocity ω c , has performed a full revolution around the origin [4]. For t 2 , for instance, we have already exceeded T E and the superposition of multiple interferences masks the relationship to the underlying classical structure.…”

“…It was not until the seminal work of Tomsovic and Heller [2], however, that semiclassical approximations were shown to describe intricate phenomena in non-trivial time regimes. Indeed, such approximations remained valid for longer than the previously established threshold for accuracy, the Ehrenfest time, at which classical structure finer than a quantum cell starts to develop [1,3,4]. Motivated by this success, semiclassical methods have since been applied for systems with ever-increasing complexity, testing the limit of what one would consider to be exclusively quantum [5][6][7].…”

Strong scarring of logarithmic quasimodes

“…Let us mention that the singularity of the evolution at the time T λ /3 can be avoided without using a QNF. In [37] the authors manage to propagage a coherent state up to the delocalization time t ≈ T λ , even when the unstable manifold is curved. The trick consists in changing the representation of these states: instead of excited Gaussian states, the authors represent them as WKB states, whose phases may be nonquadratic.…”

Symplectic semiclassical wave packet dynamics II: non-Gaussian states

“…On the other hand, using (25) and (26) as well as L j χ n , L k χ n = δ jk n j N (B, δ), we can evaluate the third term as follows:…”