A new picture of quantum tunneling in the real-time path integral from Lefschetz thimble calculations

TL;DR

This paper uses Lefschetz thimble calculations to reveal how quantum tunneling in real-time path integrals can be understood through complex saddle points, confirmed by Monte Carlo simulations and linked to measurable weak values.

Contribution

It introduces a novel approach to describe quantum tunneling in real-time path integrals via complex saddle points identified by Picard-Lefschetz theory, supported by numerical simulations.

Findings

Complex saddle points contribute to quantum tunneling.

Monte Carlo simulations confirm the role of complex saddle points.

Weak values of position operator are linked to tunneling phenomena.

Abstract

It is well known that quantum tunneling can be described by instantons in the imaginary-time path integral formalism. However, its description in the real-time path integral formalism has been elusive. Here we establish a statement that quantum tunneling can be characterized in general by the contribution of complex saddle points, which can be identified by using the Picard-Lefschetz theory. We demonstrate this explicitly by performing Monte Carlo simulations of simple quantum mechanical systems, overcoming the sign problem by the generalized Lefschetz thimble method. We confirm numerically that the contribution of complex saddle points manifests itself in a complex ``weak value'' of the Hermitian coordinate operator $\hat{x}$ evaluated at time $t$, which is a physical quantity that can be measured by experiments in principle. We also discuss the transition to classical dynamics based on our picture.