Quantum tunneling from excited states: Recovering imaginary-time instantons from a real-time analysis
arxiv(2024)
摘要
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.
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