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Applied Physics Letters

Resonance due to the interaction of tunneling particles with modulation quanta

Mark J. Hagmann

Published Feb 13, 1995• 25 citations
Publisher

Abstract

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.

DOI
10.1063/1.114189

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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 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.
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.
Some aspects of quantum tunnelling of a composite particle are clarified by a simple model. We calculate the probability that a couple of point particles bound to each other by a square well potential with a hard core tunnel through a rectangular potential barrier. It is shown that resonance structures appear in the transmission spectrum in some cases, which reflects the existence of quasi-bound-states of the centre of mass motion around the potential barrier. It is also shown that an inelastic tunnelling occurs in which the transitions in the relative motion are induced by the tunnelling of the centre of mass.
We calculate the tunneling process of a Dirac particle across two square barriers separated by a distance d, as well as the scattering by a double-cusp barrier, where the centers of the cusps are separated by a distance larger than their screening lengths. Using the scattering matrix formalism, we obtain the transmission and reflection amplitudes for the scattering processes of both configurations. We show that the presence of transmission resonances modifies the Lorentzian shape of the energy resonances and induces the appearance of additional maxima in the transmission coefficient in the range of energies where transmission resonances occur. We calculate the Wigner time delay and show how their maxima depend on the position of the transmission resonance.
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.
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.
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.