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arXiv:quant-ph/0202145 · Physical Review A

Euclidean resonance and quantum tunneling

Boris I. Ivlev

Published Jul 17, 2002• 16 citations
arXivHTMLPDF

Abstract

The extremely small probability of tunneling through an almost classical potential barrier may become not small under the action of the specially adapted nonstationary signal which selects the certain particle energy ${E}_{R}.$ For particle energies close to this value, the tunneling rate is not small during a finite interval of time and has a very sharp peak at the energy ${E}_{R}.$ After entering inside the barrier, the particle emits electromagnetic quanta and exits the barrier with a lower energy. The signal amplitude can be much less compared to the field of the static barrier. This phenomenon can be called the Euclidean resonance since the under-barrier motion occurs in imaginary time. The resonance may stimulate chemical and biochemical reactions in a selective way by adapting the signal to a certain particular chemical bond. The resonance may be used in search of the soft-\ensuremath{\alpha}-particle decay for which a conventional observation is impossible due to an extremely small decay rate.

DOI
10.1103/physreva.66.012102

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The extremely small probability of tunneling through an almost classical potential barrier may become not small under the action of the specially adapted nonstationary signal which selects the certain particle energy ER.{E}_{R}.ER​. For particle energies close to this value, the tunneling rate is not small during a finite interval of time and has a very sharp peak at the energy ER.{E}_{R}.ER​. After entering inside the barrier, the particle emits electromagnetic quanta and exits the barrier with a lower energy. The signal amplitude can be much less compared to the field of the static barrier. This phenomenon can be called the Euclidean resonance since the under-barrier motion occurs in imaginary time. The resonance may stimulate chemical and biochemical reactions in a selective way by adapting the signal to a certain particular chemical bond. The resonance may be used in search of the soft-\ensuremath{\alpha}-particle decay for which a conventional observation is impossible due to an extremely small decay rate.
Quantum tunneling through an almost classical potential barrier can be strongly enhanced by a nonstationary field so that the penetration through the barrier becomes not exponentially small. This constitutes an extremely unusual phenomenon of quantum physics called Euclidean resonance. A certain nonstationary barrier is proposed with a very low WKB tunneling rate. The quantum dynamics of this barrier is mapped on resonant tunneling across a static double barrier with a resonant level inside. The real penetration through the dynamical barrier is not exponentially small providing an example of Euclidean resonance. Therefore, the Schoedinger equation allows solutions of the type of Euclidean resonance. The counterintuitive phenomenon of Euclidean resonance is a dynamical analogue of static resonant tunneling.
Quantum tunneling from a thin wire or a thin film through a static potential barrier in a zero magnetic field is studied. The wire or the film should satisfy a condition of transverse quantization of levels and be inhomogeneous. Depending on a form of the inhomogeneity the tunneling scenario can dramatically differ from the conventional scheme of an exponential decay of a probability density inside the barrier. This happens due to interference of various underbarrier paths, which are collected at some points giving rise to a trans-barrier state of a large amplitude. As a result, the tunneling probability through an almost classical barrier can be not exponentially small. This is a phenomenon of Euclidean resonance studied earlier for tunneling across nonstationary barriers.
The process of quantum tunneling of particles in various physical systems can be effectively controlled even by a weak and slow varying in time electromagnetic signal, especially if to adapt its shape to a particular system. During an under-barrier motion of a particle, such a signal provides a ``coherent'' assistance of tunneling by the multiquanta absorption resulting in a strong enhancement of the tunneling probability. The semiclassical approach based on trajectories in the complex time is developed for tunneling in a nonstationary field. Enhancement of tunneling occurs when a singularity of the signal coincides in position at the complex time plane with a singularity of the classical Newtonian trajectory of the particle. The developed theory is also applicable to the over-barrier reflection of particles and to reflection of classical waves (electromagnetic, hydrodynamic, etc.) from a spatially-smooth medium.
Quantum tunneling through a nonstationary barrier is studied analytically and by a direct numerical solution of Schr\"odinger equation. Both methods are in agreement and say that the main features of the phenomenon can be described in terms of classical trajectories which are solutions of Newton's equation in complex time. The probability of tunneling is governed by analytical properties of a time-dependent perturbation and the classical trajectory in the plane of complex time. Some preliminary numerical calculations of Euclidean resonance (an easy penetration through a classical nonstationary barrier due to an underbarrier interference) are presented.
Numerical simulations of quantum tunneling with time-dependent barriers show that there is a resonance, with a marked increase in the transmission coefficient. For a raised cosine potential, and for low energies with square barriers, the resonance occurs when a modulation quantum can take a tunneling particle to the top of the barrier. For energies near the top of a square barrier the resonance may be understood by hypothesizing that a tunneling particle may travel from end to end of the barrier until it is ultimately either transmitted or reflected.
Tunneling is a supreme quantum effect. Every introductory text [1] on quantum mechanics gives the paradigm example of a particle tunneling through a one-dimensional potential barrier despite having a total energy less than the barrier height. Indeed, the reader typically works through a number of exercises, all involving one-dimensional potential barriers of one form or another modeling several key physical phenomena ranging from atom transfer reactions to the decay of α-particles [2]. However, one seldom encounters coupled multidimenisonal tunneling in such texts since an analytical solution of the Schro¨dinger equation in such cases is not possible. Interestingly, the richness and complexity of the tunneling phenomenon manifest themselves in full glory in the case of multidimensional systems [3]. Thus, for instance, the usual one-dimensional expectation of increasing tunneling splittings as one approaches the barrier top from below is not necessarily true as soon as one couples another bound degree of freedom to the tunneling coordinate. In the context of molecular reaction dynamics, multidimensional tunneling can result in strong mode-specificity and fluctuations in the reaction rates [4]. In fact, a proper description of tunneling of electrons and hydrogen atoms is absolutely essential [5,6] even in molecular systems as large as enzymes and proteins. Although one usually assumes tunneling effects to be significant in molecules involving light atom transfers, it is worth pointing out that neglecting the tunneling of even a heavy atom like carbon is the difference between a reaction occurring or not occurring. In particular, one can underestimate rates by nearly hundred orders of magnitude [7]. Interestingly, and perhaps paradoxically, several penetrating insights into the nature and mechanism of multidimensional barrier tunneling have been obtained from a phase-space perspective [8,9]. The contributions by Creagh, Shudo and Ikeda, and Takahashi in the present volume provide a detailed account of the latest advances in the phasespace-based understanding of multidimensional barrier tunneling.