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arXiv:quant-ph/0407163

Mapping of Euclidean resonance on static resonant tunneling

Boris I. Ivlev

Published Jul 20, 2004• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.quant-ph/0407163

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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.
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 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.
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.
Quantum tunneling across a static potential barrier in a static magnetic field is very sensitive to an analytical form of the potential barrier. Depending on that, the oscillatory structure of the modulus of the wave function can be formed in the direction of tunneling. Due to an underbarrier interference, the probability of tunneling through a higher barrier can be larger than through a lower one. For some barriers the quantum interference of underbarrier cyclotron paths results in a strong enhancement of tunneling. This occurs in the vicinity of the certain magnetic field and is referred to as Euclidean resonance. This strongly contrasts to the Wentzel, Kramers, and Brillouin type tunneling which occurs with no magnetic field.
Since the early days of quantum mechanics, tunneling has been recognized as one of the hallmarks of the wave character of microscopic physics. The possibility of a quantum particle to penetrate an energetic barrier represents certainly one of the most spectacular implications of quantum theory and has led to various applications in nuclear, atomic, and molecular physics as well as in mesoscopic science. Typical scenarios in which tunneling manifests are the escape route of a quantum particle from a quasibounded region, the transition between two or more symmetry-related, but classically disconnected wells (which we shall focus on in the following), as well as scattering or transport through potential barriers. The spectrum of scenarios becomes even richer when the concept of tunneling is generalized to any kind of classically forbidden transitions in phase space, that is, to transitions that are not necessarily inhibited by static potential barriers but by some other constraints of the underlying classical dynamics (such as integrals of motion). Such “dynamical tunneling” processes arise frequently in molecular systems [1] and were realized with cold atoms propagating in periodically modulated optical lattices [2-4]. Moreover, the electromagnetic analog of dynamical tunneling was also obtained with microwaves in billiards [5].
The problem of inter-band tunneling in a semiconductor (Zener breakdown) in a nonstationary and homogeneous electric field is solved exactly. Using the exact analytical solution, the approximation based on classical trajectories is studied. A new mechanism of enhanced tunneling through static non-one-dimensional barriers is proposed in addition to well known normal tunneling solely described by a trajectory in imaginary time. Under certain conditions on the barrier shape and the particle energy, the probability of enhanced tunneling is not exponentially small even for non-transparent barriers, in contrast to the case of normal tunneling.