Semiclassical propagation of coherent states and wave packets: hidden saddles

TL;DR

This paper advances semiclassical methods for wave packet dynamics by establishing a theoretical foundation for locating both visible and hidden saddle trajectories, crucial for understanding tunneling phenomena in multi-degree systems.

Contribution

It extends existing techniques to identify hidden saddle points, providing a more comprehensive framework for semiclassical propagation in complex systems.

Findings

Theoretical foundation for hidden saddle trajectories.

Extension of methods to locate hidden saddles.

Clarification of tunneling interpretation in wave packet dynamics.

Abstract

Semiclassical methods are extremely important in the subjects of wave packet and coherent state dynamics. Unfortunately, these essentially saddle point approximations are considered nearly impossible to carry out in detail for systems with multiple degrees of freedom due to the difficulties of solving the resulting two-point boundary value problems. However, recent developments have extended the applicability to a broader range of systems and circumstances. The most important advances are first to generate a set of real reference trajectories using appropriately reduced dimensional spaces of initial conditions, and second to feed that set into a Newton-Raphson search scheme to locate the $exposed$ complex saddle trajectories. The arguments for this approach were based mostly on intuition and numerical verification. In this paper, the methods are put on a firmer theoretical foundation and then extended to incorporate saddles $hidden$ from Newton-Raphson searches initiated with real trajectories. This hidden class of saddles is relevant to tunneling-type processes, but a hidden saddle can sometimes contribute just as much as or more than an exposed one. The distinctions between hidden and exposed saddles clarifies the interpretation of what constitutes tunneling for wave packets and coherent states in the time domain.