Complex classical paths in quantum reflections and tunneling

TL;DR

This paper utilizes the Picard-Lefschetz formalism to analyze real-time quantum propagators in a symmetric barrier, revealing how complex classical paths and singularity crossings influence tunneling and interference patterns.

Contribution

It introduces a detailed analysis of classical paths, caustics, and singularity crossings in the real-time propagator for the symmetric Rosen-Morse barrier, advancing understanding of quantum tunneling.

Findings

Interference patterns are organized by caustics and Stokes phenomena.

Singularity crossings are crucial for describing quantum tunneling.

Complex classical paths are explicitly characterized as a function of initial and final positions.

Abstract

The real-time propagator of the symmetric Rosen-Morse, also known as the symmetric modified P\"oschl-Teller, barrier is expressed in the Picard-Lefschetz path integral formalism using real and complex classical paths. We explain how the interference pattern in the real-time propagator and energy propagator is organized by caustics and Stoke's phenomena, and list the relevant real and complex classical paths as a function of the initial and final position. We discover the occurrence of singularity crossings, where the analytic continuation of the complex classical path no longer satisfies the boundary value problem and needs to be analytically continued. Moreover, we demonstrate how these singularity crossings play a central role in the real-time description of quantum tunneling.