Unraveling the bounce: a real time perspective on tunneling

TL;DR

This paper explores quantum tunneling in one dimension using real-time path integrals, revealing how the bounce solution emerges and connecting it with the WKB approximation, thus providing a new perspective on tunneling phenomena.

Contribution

It introduces a real-time path integral approach to tunneling, explicitly demonstrating the emergence of the bounce solution and its relation to the WKB approximation.

Findings

Analytic structure of the action in complex plane

Bounce solution as a parameterization of energy expansion

Reproduction of WKB approximation in real-time framework

Abstract

We study tunneling in one-dimensional quantum mechanics using the path integral in real time, where solutions of the classical equation of motion live in the complex plane. Analyzing solutions with small (complex) energy, relevant for constructing the wave function after a long time, we unravel the analytic structure of the action, and show explicitly how the imaginary time bounce arises as a parameterization of the lowest order term in the energy expansion. The real time calculation naturally extends to describe the wave function in the free region of the potential, reproducing the usual WKB approximation. The extension of our analysis to the semiclassical correction due to fluctuations on the saddle is left for future work.