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Physical Review E

Instanton and noninstanton tunneling in periodically perturbed barriers: Semiclassical and quantum interpretations

Kin’ya Takahashi, Kensuke S. Ikeda

Published Nov 13, 2012• 11 citations
Publisher

Abstract

In multidimensional barrier tunneling, there exist two different types of tunneling mechanisms, instanton-type tunneling and noninstanton tunneling. In this paper we investigate transitions between the two tunneling mechanisms from the semiclassical and quantum viewpoints taking two simple models: a periodically perturbed Eckart barrier for the semiclassical analysis and a periodically perturbed rectangular barrier for the quantum analysis. As a result, similar transitions are observed with change of the perturbation frequency ω for both systems, and we obtain a comprehensive scenario from both semiclassical and quantum viewpoints for them. In the middle range of ω, in which the plateau spectrum is observed, noninstanton tunneling dominates the tunneling process, and the tunneling amplitude takes the maximum value. Noninstanton tunneling explained by stable-unstable manifold guided tunneling (SUMGT) from the semiclassical viewpoint is interpreted as multiphoton-assisted tunneling from the quantum viewpoint. However, in the limit ω→0, instanton-type tunneling takes the place of noninstanton tunneling, and the tunneling amplitude converges on a constant value depending on the perturbation strength. The spectrum localized around the input energy is observed, and there is a scaling law with respect to the width of the spectrum envelope, i.e., the width ∝ℏω. In the limit ω→∞, the tunneling amplitude converges on that of the unperturbed system, i.e., the instanton of the unperturbed system.

DOI
10.1103/physreve.86.056206

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In multidimensional barrier tunneling, there exist two different types of tunneling mechanisms, instanton-type tunneling and noninstanton tunneling. In this paper we investigate transitions between the two tunneling mechanisms from the semiclassical and quantum viewpoints taking two simple models: a periodically perturbed Eckart barrier for the semiclassical analysis and a periodically perturbed rectangular barrier for the quantum analysis. As a result, similar transitions are observed with change of the perturbation frequency ω for both systems, and we obtain a comprehensive scenario from both semiclassical and quantum viewpoints for them. In the middle range of ω, in which the plateau spectrum is observed, noninstanton tunneling dominates the tunneling process, and the tunneling amplitude takes the maximum value. Noninstanton tunneling explained by stable-unstable manifold guided tunneling (SUMGT) from the semiclassical viewpoint is interpreted as multiphoton-assisted tunneling from the quantum viewpoint. However, in the limit ω→0, instanton-type tunneling takes the place of noninstanton tunneling, and the tunneling amplitude converges on a constant value depending on the perturbation strength. The spectrum localized around the input energy is observed, and there is a scaling law with respect to the width of the spectrum envelope, i.e., the width ∝ℏω. In the limit ω→∞, the tunneling amplitude converges on that of the unperturbed system, i.e., the instanton of the unperturbed system.
Tunneling essentially different from instanton-type tunneling, say noninstanton tunneling, is studied from both semiclassical and quantum viewpoints. Taking a periodically perturbed rounded-off step potential for which the instanton-type tunneling is substantially prohibited, we analyze change of the tunneling probability with change of the perturbation frequency based on the stable-unstable manifold-guided tunneling (SUMGT) theory, which we have recently introduced. In the large and small limits of the frequency, the tunneling rate rapidly decays, but it is markedly enhanced in an intermediate range. We will also make a quantum interpretation of the noninstanton tunneling by using an exactly solvable model--a periodically perturbed right-angled step potential. Analysis with this model shows that SUMGT is considered as a sort of photoassisted tunneling through a large energy gap induced with absorbing a huge number quanta, which is completely different from the instanton-type tunneling. Both approaches from the semiclassical and quantum viewpoints complement each other to cause a better understanding of noninstanton tunneling.
In recent papers we demonstrated a new class of barrier tunneling phenomena which can be observed in periodically driven one dimensional barrier systems (i.e., 1.5D barrier tunneling systems). In such systems the tunneling component is in general accompanied by remarkable interference fringe patterns as the perturbation strength exceeds a certain characteristic value. Such a phenomenon is quite general and is not restricted to periodically driven 1D systems, and it may be also observed in intrinsic two dimensional autonomous systems. The complex-domain semiclassical theory developed by ourselves was applied to the problem, and we succeeded in reconstructing every detail of the fringed tunneling wavefunction only by using the complexified classical tunneling trajectories. This fact greatly motivates us to investigate the mechanism of the tunneling in terms of “classical mechanics”. The later part of the present paper is devoted to asserting the generality of the “classical mechanism” of the fringed tunneling. We develop detailed semiclassical analyses in the complex domain and demonstrate that the key trajectories contributing to the fringed tunneling are quite different from the instanton-like trajectories, which works only in unperturbed or weakly-perturbed regimes. Indeed, such trajectories are traveling in the complex phase space being guided by the complexified stable and unstable manifolds which exhibit heteroclinic-like entanglement with the incoming wave manifolds.
The reflection and transmission of a wave on a two-dimensional barrier is studied quantum mechanically and semiclassically. The simple model consists of two degrees of freedom. One of them contains a barrier (of the Eckart type); the second allows only a bound motion in discrete quantum states. These two degrees of freedom are coupled. The transition probabilities for transitions between the different quantum states can, in the model considered here, be evaluated analytically, both quantum mechanically and semiclassically. Several of the so called uniform semiclassical approximations and also the initial value integral representation for the semiclassical transition amplitudes are considered and compared to the exact quantum mechanical results. Some new insight on the semiclassical treatment of the tunneling process is gained, specially concerning the proper choice of the integration path for the initial value integral representation of the transition amplitudes.
We investigate the tunneling mechanism in a rounded-step potential under a time-periodic perturbation, where instanton tunneling is substantially prohibited. The tunneling probability exhibits a characteristic sawtoothlike structure as a function of the inverse Planck's constant 1/ℏ. This structure arises from the multiquanta absorption tunneling mechanism, specifically owing to the replacement of the dominant harmonic channel at every edge. Indeed, the replacement causes a sudden change in tunneling probability. Furthermore, the interference between the dominant and subdominant harmonic channels plays a crucial role in forming detailed properties, specifically probability oscillations superposed on the sawtooth structure. By employing a semiclassical approach, we show that the sawtooth structure is reproduced by the superposition of many complex branches, i.e., complex classical trajectories over many periods of the perturbation. The semiclassical decomposition of the wave operator to many complex branches is analogous to frequency decomposition and reproduces the sawtooth structure. We analyze the convergence of the semiclassical results to the quantum calculations as the number of time periods increases and elucidate the underlying mechanism of constructing the sawtooth structure from the viewpoint of semiclassics. Furthermore, the semiclassical method reproduces the probability oscillations induced by the interference between dominant and subdominant harmonic channels.