Complexified phase spaces, initial value representations, and the accuracy of semiclassical propagation

TL;DR

This paper develops a complexified phase-space initial value representation for semiclassical propagators, demonstrating its equivalence to the Herman-Kluk method, analyzing its stability, and revealing new phenomena in Kerr system dynamics.

Contribution

It introduces a caustic-free IVR based on phase-space complexification that is equivalent to the Herman-Kluk propagator and compares its stability with other IVRs.

Findings

Herman-Kluk propagator matches the complexified IVR.

Root-search bypass leads to numerical instability except in H-K.

Observed phenomena include half-Ehrenfest time effects and caustic stickiness.

Abstract

Using phase-space complexification, an Initial Value Representation (IVR) for the semiclassical propagator in position space is obtained as a composition of inverse Segal-Bargmann (S-B) transforms of the semiclassical coherent state propagator. The result is shown to be free of caustic singularities and identical to the Herman-Kluk (H-K) propagator, found ubiquitously in physical and chemical applications. We contrast the theoretical aspects of this particular IVR with the van Vleck-Gutzwiller (vV-G) propagator and one of its IVRs, often employed in order to evade the non-linear "root-search" for trajectories required by vV-G. We demonstrate that bypassing the root-search comes at the price of serious numerical instability for all IVRs except the H-K propagator. We back up our theoretical arguments with comprehensive numerical calculations performed using the homogeneous Kerr system, about which we also unveil some unexpected new phenomena, namely: (1) the observation of a clear mark of half the Ehrenfest's time in semiclassical dynamics; and (2) the accumulation of trajectories around caustics as a function of increasing time (dubbed "caustic stickiness"). We expect these phenomena to be more general than for the Kerr system alone.