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Instantons revisited: dynamical tunnelling and resonant tunnelling

Jérémy Le Deunff, Amaury Mouchet

Published Apr 8, 2010• 0 citations
OpenAlex

Abstract

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.

Related papers

Instantons re-examined: Dynamical tunneling and resonant tunneling

Jérémy Le Deunff, Amaury Mouchet

Starting from trace formulas for the tunneling 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 instantonlike approach, which takes advantage of an incomplete Wick rotation, accurately reproduces tunneling effects not only in the usual double-well potential but also in situations where a pure Wick rotation is insufficient, for instance dynamical tunneling or resonant tunneling. Even though only one-dimensional autonomous Hamiltonian systems are quantitatively studied, we discuss the relevance of our method for multidimensional and/or chaotic tunneling.

arXiv:0911.4093•Physical Review E•Apr 8, 2010•26 citations

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

Thomas Steingasser, David I. Kaiser

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.

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Multidimensional tunneling in terms of complex classical mechanics: Wave functions, energy splittings, and decay rates in nonintegrable systems

Shoji Takada

A semiclassical theory of multidimensional tunneling is formulated to calculate the tunneling wave function, energy splitting in a double well and decay rate constant from a metastable state. First, the tunneling wave function is calculated by analytic continuation of a quantized torus prepared in analytic forms using either the Chapman–Garrett–Miller method or the Birkhoff–Gustavson normal form method. For a weakly nonintegrable system, tunnelings are confirmed to be classified into two qualitatively different domains; pure tunneling in the I region and mixed tunneling in the C region. Semiclassical wave functions agree with quantum mechanical ones within a few percent both in classically allowed and tunneling regions. Breakdown of this simple picture is exemplified for cases of relatively strong couplings. Second, expressions of the tunneling energy splitting in a double well potential and the decay rate from a metastable state are derived. The wave function near a well formulated above is connected with the semiclassical Green’s function in the deep tunneling region. The latter is expressed by complex trajectories which start from the complex quantized torus. A preliminary numerical comparison with the quantum mechanical value is also done for the energy splitting.

The Journal of Chemical Physics•Mar 8, 1996•28 citations

Semiclassical tunneling splittings for arbitrary vibrational states in multidimensional double wells

Kenneth G. Kay

, 042116 (2017)] is applied to calculate tunneling splittings for arbitrary vibrational states of model two-dimensional double-well systems. Cases in which the classical dynamics for the wells is chaotic, mixed, and regular are considered. A perturbative treatment, based on the condition of small tunneling amplitudes, is found to be sufficiently accurate for the cases studied and is applied for most of the calculations. Treatments that approximate certain imaginary-time trajectories in the classically forbidden region by linearization about a variety of judiciously selected reference trajectories yield good results for all systems treated. These calculations can be greatly simplified by approximating all imaginary-time trajectories as linearizations about a single reference trajectory. A simple way to determine optimal reference trajectories for this purpose is presented. It is found that their use yields splittings of satisfactory accuracy for the cases studied.

The Journal of Chemical Physics•Oct 9, 2018•2 citations

Gutzwiller Trace Formula for One-Dimensional Systems with Tunneling

Alexey A. Radkevich

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.

Bulletin of the Lebedev Physics Institute•Nov 1, 2025•0 citations

Semiclassical approach to states near the potential barrier top

Viktor A. Benderskii, E.V. Vetoshkin, Efim I Kats

Within the framework of the instanton approach, we present analytical results for the following model problems: (i) particle penetration through a parabolic potential barrier, where the instanton solution practically coincides with the exact (quantum) one; (ii) descriptions of highly excited states in anharmonic potentials of two types: double-well X 4 and decay X 3 . For the former potential, the instanton method accurately reproduces not only single-well and double-well quantization, but also a crossover region (in contrast to the standard WKB approach that fails to describe the crossover behavior); for the latter potential, the instanton method allows studying the resonance broadening and collapse phenomena. We also investigate resonance tunneling that plays a relevant role in many semiconducting devices. We show that the instanton approach gives exact (quantum) results in a broad range of energies. Applications of the method and of the results are applicable to various systems in physics, chemistry, and biology exhibiting double-level behavior and resonance tunneling.

arXiv:cond-mat/0203510•Journal of Experimental and Theoretical Physics•Oct 1, 2002•7 citations
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.
Starting from trace formulas for the tunneling 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 instantonlike approach, which takes advantage of an incomplete Wick rotation, accurately reproduces tunneling effects not only in the usual double-well potential but also in situations where a pure Wick rotation is insufficient, for instance dynamical tunneling or resonant tunneling. Even though only one-dimensional autonomous Hamiltonian systems are quantitatively studied, we discuss the relevance of our method for multidimensional and/or chaotic tunneling.
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.
A semiclassical theory of multidimensional tunneling is formulated to calculate the tunneling wave function, energy splitting in a double well and decay rate constant from a metastable state. First, the tunneling wave function is calculated by analytic continuation of a quantized torus prepared in analytic forms using either the Chapman–Garrett–Miller method or the Birkhoff–Gustavson normal form method. For a weakly nonintegrable system, tunnelings are confirmed to be classified into two qualitatively different domains; pure tunneling in the I region and mixed tunneling in the C region. Semiclassical wave functions agree with quantum mechanical ones within a few percent both in classically allowed and tunneling regions. Breakdown of this simple picture is exemplified for cases of relatively strong couplings. Second, expressions of the tunneling energy splitting in a double well potential and the decay rate from a metastable state are derived. The wave function near a well formulated above is connected with the semiclassical Green’s function in the deep tunneling region. The latter is expressed by complex trajectories which start from the complex quantized torus. A preliminary numerical comparison with the quantum mechanical value is also done for the energy splitting.
, 042116 (2017)] is applied to calculate tunneling splittings for arbitrary vibrational states of model two-dimensional double-well systems. Cases in which the classical dynamics for the wells is chaotic, mixed, and regular are considered. A perturbative treatment, based on the condition of small tunneling amplitudes, is found to be sufficiently accurate for the cases studied and is applied for most of the calculations. Treatments that approximate certain imaginary-time trajectories in the classically forbidden region by linearization about a variety of judiciously selected reference trajectories yield good results for all systems treated. These calculations can be greatly simplified by approximating all imaginary-time trajectories as linearizations about a single reference trajectory. A simple way to determine optimal reference trajectories for this purpose is presented. It is found that their use yields splittings of satisfactory accuracy for the cases studied.
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.
Within the framework of the instanton approach, we present analytical results for the following model problems: (i) particle penetration through a parabolic potential barrier, where the instanton solution practically coincides with the exact (quantum) one; (ii) descriptions of highly excited states in anharmonic potentials of two types: double-well X 4 and decay X 3 . For the former potential, the instanton method accurately reproduces not only single-well and double-well quantization, but also a crossover region (in contrast to the standard WKB approach that fails to describe the crossover behavior); for the latter potential, the instanton method allows studying the resonance broadening and collapse phenomena. We also investigate resonance tunneling that plays a relevant role in many semiconducting devices. We show that the instanton approach gives exact (quantum) results in a broad range of energies. Applications of the method and of the results are applicable to various systems in physics, chemistry, and biology exhibiting double-level behavior and resonance tunneling.