Long-time propagation of coherent states in a normally hyperbolic setting

TL;DR

This paper develops a method to analyze the long-time evolution of coherent states in semiclassical quantum mechanics within a normally hyperbolic dynamical setting, extending previous approximations to Ehrenfest times.

Contribution

It introduces a new approach to describe propagated states as a combination of WKB and squeezed states near hyperbolic invariant manifolds, allowing analysis up to longer times.

Findings

Extended the validity of coherent state propagation to Ehrenfest time scales.

Described states as WKB in transverse directions and squeezed along invariant manifolds.

Provided conditions on classical flow for accurate long-time quantum evolution.

Abstract

We present a method to find asymptotics for the evolution of coherent states (or Gaussian wavepackets with standard deviation $\sqrt{h}$) under semiclassical Schr\"odinger's equation for a given Hamiltonian. These results extend the work of Combescure and Robert, in which the evolution of coherent states can be approximated in the limit $h\to 0$ with deformed Gaussian wavepackets called squeezed coherent states. The description with squeezed states holds for times $t$ that can go to infinity as $h\to 0$, under the constraint $|t|\leq |\log h|/(6\lambda_0)$ where $\lambda_0$ is the maximal Lyapunov exponent of the classical dynamics. The breakdown of this approximation at time $|\log h|/(6\lambda_0)$ is related to the bending of evolved wavepackets: once propagated states spread at a scale $h^{1/3}$, squeezed states no longer provide an appropriate description. To obtain a representation of propagated states valid up to times $|t|\leq C|\log h|$ with a larger $C$ (for instance, up to Ehrenfest's time $|\log h|/(2\lambda_0)$ where spreading on macroscopic scales is allowed), we make additional assumptions on the flow $\Phi_t$ associated to the classical dynamics, imposing constraints on directions of elongation. Namely, we work in a neighborhood of a normally hyperbolic $\Phi_t$-invariant submanifold $K$, on which the dynamics is considered as slow in comparison with its transverse directions, along which $\Phi_t$ is assumed to be hyperbolic. In this context, we describe the propagated state as a WKB state in transverse directions and a squeezed state along $K$. This description emphasizes the fact that propagated states should no longer be thought of as microlocalized on a point, but rather on an isotropic submanifold (corresponding to transverse unstable directions). Guillemin, Uribe, and Wang presented a similar class of wavefunctions microlocalized on an isotropic submanifold.