Quantum tunnelling, real-time dynamics and Picard-Lefschetz thimbles

TL;DR

This paper applies the Picard-Lefschetz thimble approach to quantum tunnelling, demonstrating its effectiveness in accurately reproducing quantum mechanical results and highlighting limitations of classical approximations.

Contribution

It introduces a two-stage path integral method using Picard-Lefschetz thimbles for real-time quantum dynamics, specifically applied to quantum tunnelling.

Findings

Picard-Lefschetz method matches quantum mechanical results

Classical-statistical approximation fails for tunnelling

Unique thimbles exist for each initial condition in the approach

Abstract

We follow up the work, where in light of the Picard-Lefschetz thimble approach, we split up the real-time path integral into two parts: the initial density matrix part which can be represented via an ensemble of initial conditions, and the dynamic part of the path integral which corresponds to the integration over field variables at all later times. This turns the path integral into a two-stage problem where, for each initial condition, there exits one and only one critical point and hence a single thimble in the complex space, whose existence and uniqueness are guaranteed by the characteristics of the initial value problem. In this paper, we test the method for a fully quantum mechanical phenomenon, quantum tunnelling in quantum mechanics. We compare the method to solving the Schr\"odinger equation numerically, and to the classical-statistical approximation, which emerges naturally in a well-defined limit. We find that the Picard-Lefschetz result matches the expectation from quantum mechanics and that, for this application, the classical-statistical approximation does not.