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arXiv:0806.1554

Underbarrier interference and Euclidean resonance

Boris I. Ivlev

Published Jun 10, 2008• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.0806.1554

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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.
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 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.
The usual WKB analysis for quantum tunneling applies when the tunneling action is large, as it is for tall, wide potential barriers. In contrast we analyze tunneling when the action is small, as it is for tunneling across a tall, thin barrier. We develop a perturbative analysis where the control parameter is the inverse of the area under the potential barrier and apply our technique to several examples in 1+11+11+1 dimensions. In resonant situations for bound particles we find that the tunneling probability grows with time as ∝t2\propto t^2∝t2, while in non-resonant situations it grows linearly with time. We evaluate not only the tunneling probability but also the time-dependent tunneling wavefunction for a particle that escapes to infinity, {\it i.e.} from a quasi-bound state to the continuum.