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

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 x̂\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \hat{x} $$\end{document} 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.