Toward quantum tunneling from excited states: Recovering imaginary-time instantons from a real-time analysis

TL;DR

This paper develops a real-time path integral approach to quantum tunneling from excited states, avoiding Wick rotation and recovering instanton solutions through a complex Hamiltonian regularization.

Contribution

It introduces a novel real-time method for analyzing tunneling from excited states, generalizing instanton concepts without Wick rotation, applicable to broader quantum systems.

Findings

Regularized path integral with complex Hamiltonian yields complex stationary solutions.

Analytic solutions match boundary conditions in relevant limits.

Approach reproduces instanton-like solutions without imaginary time.

Abstract

We revisit the path integral description of quantum tunneling and lay the groundwork for its generalization to excites states through real-time path integral techniques. For clarity, we focus on the simple toy model of a point particle in a double-well potential, for which we perform all steps explicitly. Instead of performing the familiar Wick rotation from physical to imaginary time -- which is inconsistent with the requisite boundary conditions when treating tunneling from states other than the false vacuum -- we regularize the path integral by adding an infinitesimal complex contribution to the Hamiltonian, while keeping time strictly real. We find that this gives rise to a complex stationary-phase solution, in agreement with recent insights from Picard-Lefshitz theory. We then show that there exists a class of analytic solutions for the corresponding equations of motion, which can be made to match the appropriate boundary conditions in the physically relevant limits of a vanishing regulator and an infinite physical time. We provide a detailed discussion of this non-trivial limit. We find that, for systems without an explicit time-dependence, our approach reproduces the picture of an instanton-like solution defined on a finite Euclidean-time interval. Lastly, we discuss the generalization of our approach to broader classes of systems, for which it serves as a reliable framework for high-precision calculations.