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

Abstract: We revisit the path integral description of quantum tunneling and show how it can be generalized to excited states. 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 excited states - 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.