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Journal of Applied Physics

Mechanism for resonance in the interaction of tunneling particles with modulation quanta

Mark J. Hagmann

Published Jul 1, 1995• 49 citations
Publisher

Abstract

Numerical simulations of quantum tunneling with time-dependent barriers show that there is a resonance, with a marked increase in the tunneling current. For square barriers the resonance occurs when the tunneling particles absorb modulation quanta and the length of the barrier is a multiple of one-half de Broglie wavelengths. The resonance has a similar mechanism with triangular barriers. However, the relationship is more complex because the absorption and emission of modulation quanta takes place throughout the full length of the barrier, whereas this exchange only occurs at the ends of a square barrier.

DOI
10.1063/1.360667

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Numerical simulations of quantum tunneling with time-dependent barriers show that there is a resonance, with a marked increase in the tunneling current. For square barriers the resonance occurs when the tunneling particles absorb modulation quanta and the length of the barrier is a multiple of one-half de Broglie wavelengths. The resonance has a similar mechanism with triangular barriers. However, the relationship is more complex because the absorption and emission of modulation quanta takes place throughout the full length of the barrier, whereas this exchange only occurs at the ends of a square barrier.
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.
The frequency-dependent transmission coefficient of a particle tunneling through a time-dependent barrier is calculated taking into account the quantum interference effect. We consider a rectangular barrier of height V0 with a time-dependent modulation of single frequency ω. For small modulation amplitude V1, our result is exact up to order V21 for any barrier width. In the low transmission region, our result is consistent with those obtained using a semiclassical method. We find that quantum interference (two modulation quanta process) makes a significant contribution to the tunneling current. In the static approximation, ωτ≪1, where τ is the barrier transit time, the contribution to the current from interference process is to cancel one-half of the tunneling current from first-order process (one modulation quantum process). To the lowest order in power series of V1/ℏω, the tunneling current is reduced due to interference.
This paper revisited quantum tunneling dynamics through a square double-barrier potential. We emphasized the similarity of tunneling dynamics through double-barrier and that of optical Fabry--Peˊ\acute{e}eˊrot (FP) interferometer. Based on this similarity, we showed that the well-known resonant tunneling can also be interpreted as a result of matter multi-wave interference, analogous to that of FP interferometer. From this analogy, we also got an analytical finesse formula of double-barrier. Compared with that obtained numerically for a specific barrier configuration, we found that this formula works well for resonances at "deep tunneling region". Besides that, we also calculated standing wave spectrum inside the well of double barriers and phase time of double-barrier tunneling. The wave number spectrums of standing wave and phase time show another points of view on resonance. From semi-numerical calculations, we interpreted the peak of phase time at resonance as resonance life time, which coincides at least in order of magnitude with that obtained from uncertainty principle. Not to our surprise, phase time of double-barrier tunneling also saturates at long barrier length limit l→∞l\rightarrow\inftyl→∞ as that of tunneling through a single barrier, and the limits are the same.
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.
We have used several different methods to solve the one-dimensional, time-dependent Schrödinger equation for a sinusiodally modulated barrier. Analytical solutions are given for the case in which the time-dependent part of the potential has several different forms, but is independent of position. For more general fields, Floquet's theorem is used to write the wavefunction as a summation of components which have different energies as a result of the absorption and emission of modulation quanta. A system of coupled ordinary differential equations is obtained, which is then solved numerically using shooting methods. The examples show a resonance in which the tunneling current is markedly increased. For square barriers, this resonance occurs when the particles absorbing modulation quanta are above the barrier, and the length of the barrier is an integer multiple of one-half the de Broglie wavelength. The existence of the resonance is confirmed by asymptotic solutions for large and small frequencies. Examples suggest that it may be possible to make a microwave power amplifier by illuminating a field emitter array with an amplitude-modulated laser operating near the new resonance. © 1995 John Wiley & Sons, Inc.
One of several contradictory existing results for the time a tunneling particle interacts with its barrier is confirmed, by considering tunneling through a time-modulated barrier. At low modulation frequencies the traversing particle sees a static barrier. At high frequencies the particle tunnels through the time-averaged potential, but can do it inelastically, losing or gaining modulation quanta. The transition between the two regimes yields $\ensuremath{\int}\mathrm{dx}{[\frac{m}{2(V\ensuremath{-}E)}]}^{\frac{1}{2}}$ for the traversal time.