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arXiv:nucl-th/0112060 · Physical Review Letters

Anomalously Long Passage through a Rounded-Off-Step Potential due to a New Mechanism of Multidimensional Tunneling

Kin’ya Takahashi, Kensuke S. Ikeda

Published Dec 12, 2006• 18 citations
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Abstract

The fully complex domain semiclassical theory based upon the complexified stable-unstable manifold theory, which we have developed in our recent studies, is successfully applied to explain anomalous tunneling phenomena numerically observed in a periodically modulated round-off-step potential. Numerical experiments show that tunneling through the oscillating step potential is characterized by a spatially nondecaying tunneling tail and an anomalously slow relaxation. The key is the existence of a critical trajectory exhibiting singular behavior, and the analysis of neighboring trajectories around it reproduces the essence of such anomalous phenomena.

DOI
10.1103/physrevlett.97.240403

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The fully complex domain semiclassical theory based upon the complexified stable-unstable manifold theory, which we have developed in our recent studies, is successfully applied to explain anomalous tunneling phenomena numerically observed in a periodically modulated round-off-step potential. Numerical experiments show that tunneling through the oscillating step potential is characterized by a spatially nondecaying tunneling tail and an anomalously slow relaxation. The key is the existence of a critical trajectory exhibiting singular behavior, and the analysis of neighboring trajectories around it reproduces the essence of such anomalous phenomena.
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.
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.
It was found recently that processes of multidimensional tunneling are generally described at high energies by unstable semiclassical trajectories. We study two observational signatures related to the instability of trajectories. First, we find an additional power-law dependence of the tunneling probability on the semiclassical parameter as compared to the standard case of potential tunneling. The second signature is a substantial widening of the probability distribution over final-state quantum numbers. These effects are studied using a modified semiclassical technique which incorporates stabilization of the tunneling trajectories. The technique is derived from first principles. We obtain expressions for the inclusive and exclusive tunneling probabilities in the case of unstable semiclassical trajectories. We also investigate the 'phase transition' between the cases of stable and unstable trajectories across certain 'critical' values of energy. Finally, we derive the relation between the semiclassical probabilities of tunneling from the low-lying and highly excited initial states. This puts on firm ground a conjecture made previously in the semiclassical description of collision-induced tunneling in field theory.