Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Applied Physics Letters

Quantum interference in time-dependent tunneling

Chao Zhang, Narkis Tzoar

Published Nov 14, 1988• 10 citations
Publisher

Abstract

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.

DOI
10.1063/1.100339

Related papers

Virtual Process in Time-dependent Tunneling

Chao Zhang, Narkis Tzoar

The frequency-dependent transmission coefficient for a particle tunneling through a time-dependent barrier is calculated taking into account inelastic and virtual processes. We consider a barrier of arbitrary shape with a time-dependent modulation of single frequency ω. For small modulation amplitudes V 1 , our result is exact up to order V 1 2 . The result is obtained in the low transmission limit and is consistent with the exact calculation for a rectangular barrier. We find that the virtual processes can make a significant contribution to the tunneling current, especially at low ω. In the static approximation ωτ << 1, where τ is the barrier transit time, the contribution of the second order virtual process is equal to one half of the tunneling current from the first order inelastic processes.

Physica Scripta•Jan 1, 1989•2 citations

Traversal Time for Tunneling

Μ. Büttiker, Rolf W. Landauer

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.

Physical Review Letters•Dec 6, 1982•1,117 citations

Quantum mechanical tunneling through a time-dependent barrier

Jiu-Yuan Ge, John Z. H. Zhang

We present a numerical investigation of quantum mechanical tunneling process in a time-dependent (fluctuating) barrier using a one dimensional model of Eckart barrier. The tunneling probability is calculated for two cases in which (1) the height of the barrier is undergoing harmonic oscillation with frequency ω and (2) the location of the barrier is undergoing harmonic oscillation with frequency ω. It is observed in both cases that the quantum mechanical tunneling probability exhibits a maximum as a function of the oscillating frequency ω between the low and high frequency limits. The physical origin and process underlying this resonantlike phenomenon are proposed in this paper based on the current model study.

The Journal of Chemical Physics•Nov 15, 1996•23 citations

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

Mark J. Hagmann

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.

Journal of Applied Physics•Jul 1, 1995•49 citations

Resonance due to the interaction of tunneling particles with modulation quanta

Mark J. Hagmann

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.

Applied Physics Letters•Feb 13, 1995•25 citations

Modulated barrier approach to the interaction time in tunneling for arbitrary potentials

Th. Martin, Rolf W. Landauer

The time that a transmitted particle interacts with a tunneling barrier is evaluated in one of the principal existing approaches by considering a small oscillatory perturbation added to the static barrier. The modulation frequency, at which a serious deviation from quasistatic behavior appears, is indicative of the traversal time. This approach is extended here, to a long region including the barrier, and utilizing a modulation potential which is spatially uniform throughout this region, except in the incident region ahead of the actual barrier, where it is made sufficiently smooth to avoid reflections. The resulting traversal time is \ensuremath{\Elzxh}\ensuremath{\partial} ln[\ensuremath{\partial}t/\ensuremath{\partial}E]/\ensuremath{\partial}E, where the complex transmission coefficient t(E) is defined as the ratio of the wave function at the far end to that at the point of incidence. This answer applies to arbitrary potentials and to all energies, including both propagation and tunneling in a single framework. This result is closely related to, but not identical to, that obtained by another earlier approach in which the potential is fixed, but the incident amplitude is modulated.

Physical Review A•Mar 1, 1993•33 citations
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.
The frequency-dependent transmission coefficient for a particle tunneling through a time-dependent barrier is calculated taking into account inelastic and virtual processes. We consider a barrier of arbitrary shape with a time-dependent modulation of single frequency ω. For small modulation amplitudes V 1 , our result is exact up to order V 1 2 . The result is obtained in the low transmission limit and is consistent with the exact calculation for a rectangular barrier. We find that the virtual processes can make a significant contribution to the tunneling current, especially at low ω. In the static approximation ωτ << 1, where τ is the barrier transit time, the contribution of the second order virtual process is equal to one half of the tunneling current from the first order inelastic processes.
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.
We present a numerical investigation of quantum mechanical tunneling process in a time-dependent (fluctuating) barrier using a one dimensional model of Eckart barrier. The tunneling probability is calculated for two cases in which (1) the height of the barrier is undergoing harmonic oscillation with frequency ω and (2) the location of the barrier is undergoing harmonic oscillation with frequency ω. It is observed in both cases that the quantum mechanical tunneling probability exhibits a maximum as a function of the oscillating frequency ω between the low and high frequency limits. The physical origin and process underlying this resonantlike phenomenon are proposed in this paper based on the current model study.
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 time that a transmitted particle interacts with a tunneling barrier is evaluated in one of the principal existing approaches by considering a small oscillatory perturbation added to the static barrier. The modulation frequency, at which a serious deviation from quasistatic behavior appears, is indicative of the traversal time. This approach is extended here, to a long region including the barrier, and utilizing a modulation potential which is spatially uniform throughout this region, except in the incident region ahead of the actual barrier, where it is made sufficiently smooth to avoid reflections. The resulting traversal time is \ensuremath{\Elzxh}\ensuremath{\partial} ln[\ensuremath{\partial}t/\ensuremath{\partial}E]/\ensuremath{\partial}E, where the complex transmission coefficient t(E) is defined as the ratio of the wave function at the far end to that at the point of incidence. This answer applies to arbitrary potentials and to all energies, including both propagation and tunneling in a single framework. This result is closely related to, but not identical to, that obtained by another earlier approach in which the potential is fixed, but the incident amplitude is modulated.