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Bulletin of the Lebedev Physics Institute

Gutzwiller Trace Formula for One-Dimensional Systems with Tunneling

Alexey A. Radkevich

Published Nov 1, 2025• 0 citations
Publisher

Abstract

Abstract Using the path integral formalism, quantum-mechanical tunneling is described by instantons—solutions to the equations of system motion in imaginary time. However, in imaginary time, this formalism is suitable only for studying the characteristics of a system in thermodynamic equilibrium and is not applicable to significantly nonequilibrium phenomena. Therefore, it is necessary to develop an approach that takes into account instanton solutions within the framework of the real-time path integral. This paper takes a step in this direction by generalizing the Gutzwiller trace formula, derived within the real-time formalism, for the density of states of a quantum-mechanical system in a tunnel-type potential. It is shown that to obtain real values of the level splitting in a symmetric double-well potential, it is necessary to take into account approximate solutions to the equations of motion in imaginary time, corresponding to multi-instanton configurations.

DOI
10.3103/s1068335625602468

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Abstract Using the path integral formalism, quantum-mechanical tunneling is described by instantons—solutions to the equations of system motion in imaginary time. However, in imaginary time, this formalism is suitable only for studying the characteristics of a system in thermodynamic equilibrium and is not applicable to significantly nonequilibrium phenomena. Therefore, it is necessary to develop an approach that takes into account instanton solutions within the framework of the real-time path integral. This paper takes a step in this direction by generalizing the Gutzwiller trace formula, derived within the real-time formalism, for the density of states of a quantum-mechanical system in a tunnel-type potential. It is shown that to obtain real values of the level splitting in a symmetric double-well potential, it is necessary to take into account approximate solutions to the equations of motion in imaginary time, corresponding to multi-instanton configurations.
We revisit the path integral description of quantum tunneling and its generalization 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-Lefshetz 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 nontrivial limit. We find that, for systems without an explicit time-dependence, our approach reproduces the picture of an instantonlike 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.
The complex-time formalism is developed in the framework of the path-integral formalism, to be used for analysis of the quantum tunneling phenomena. We show that subleading complex-time saddle-points do not account for the right WKB result. Instead, we develop a reduction formula, which enables us to construct Green functions from simple components of the potential, for which saddle-point method is applicable. This method leads us to the valid WKB result, which incorporates imaginary-time instantons and bounces, as well as the real-time boundary conditions.
Starting from trace formulae for the tunnelling splittings (or decay rates) analytically continued in the complex time domain, we obtain explicit semiclassical expansions in terms of complex trajectories that are selected with appropriate complex-time paths. We show how this instanton-like approach, which takes advantage of an incomplete Wick rotation, accurately reproduces tunnelling effects not only in the usual double-well potential but also in situations where a pure Wick rotation is insufficient, for instance dynamical tunnelling or resonant tunnelling. Even though only one-dimensional autonomous Hamiltonian systems are quantitatively studied, we discuss the relevance of our method for multidimensional and/or chaotic tunnelling.
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 x^ \hat{x} x^ 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.
The Gutzwiller trace formula establishes a profound connection between the quantum spectrum and classical periodic orbits. However, its application is limited by its reliance on the semiclassical saddle point approximation. In this work, we explore the full quantum version of the trace formula using the Lefschetz thimble method by incorporating complexified periodic orbits. Upon complexification, classical real periodic orbits are transformed into cycles on compact Riemann surfaces. Our key innovation lies in the simultaneous complexification of the periods of cycles, resulting in a fully quantum trace formula that accounts for all contributions classified by the homology classes of the associated Riemann surfaces. This formulation connects the quantum spectrum to contributions across all complex time directions, encompassing all relevant homology classes. Our approach naturally unifies and extends two established methodologies: periodic orbits in real time, as in Gutzwiller's original work, and quantum tunneling in imaginary time, as in the instanton method.