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

Physical Review Letters

Closed Formulas for Tunneling Time in Superlattices

Pedro Pereyra

Published Feb 21, 2000• 94 citations
Publisher

Abstract

New formulas for an exact and simple evaluation of the phase time associated with the passage of electrons or photons through a finite superlattice are derived. This time, named here the superlattice-tunneling time tau(n), exhibits a resonant-band structure or a superluminal phase time behavior depending on whether the particle's energy lies in a band or a gap. In the band, tau(n) remains larger than the free motion time tau(f), while in the gap it can be less than tau(f) (with strong substrate effects), but it is larger than the single-cell phase time tau(1). Extremely good agreements with optical-pulse and superluminal delay times measured by Spielmann et al. and Steinberg et al. are found, including the superlattice-tunneling-time limit and the substrate effects. Conditions for measurements of earlier electron arrival are also analyzed.

DOI
10.1103/physrevlett.84.1772

Related papers

Measurement of the single-photon tunneling time

Aephraim M. Steinberg, Paul G. Kwiat, R. Y. Chiao

Using a two-photon interferometer, we have measured the time delay for a photon to tunnel across a barrier consisting of a 1.1-\ensuremath{\mu}m-thick 1D photonic band-gap material. The peak of the photon wave packet appears on the far side of the barrier 1.47\ifmmode\pm\else\textpm\fi{}0.21 fs earlier than it would if it were to travel at the vacuum speed of light c. Although the apparent tunneling velocity (1.7\ifmmode\pm\else\textpm\fi{}0.2)c is superluminal, this is not a genuine signal velocity, and Einstein causality is not violated. The measured tunneling time is consistent with the group delay (``phase time''), but not with the semiclassical time.

Physical Review Letters•Aug 2, 1993•837 citations

Relativistic tunnelling time for electronic wave packets

Óscar del Barco, V. Gasparian

We have analyzed the influence of the wave packet size in the relativistic tunnelling time τ and its uncertainty �τ when it traverses a given potential barrier. The analytical expressions obtained for both magnitudes confirm that the size of the incident pulse has a significant effect on the tunnelling process. This effect is greater for short pulses, compared with the length of the barrier. For the evanescent zone, we have derived an analytical expression for τ with a good limit of validity. This expression constitutes a value tool to calculate the relativistic tunnelling time as a function of the incident wave packet with a good limit of validity. Superluminal propagation is found in this region but with a large value of the uncertainty �τ compared with the tunnelling time itself. We can conclude that the probability of superluminal propagation is practically negligible in the evanescent region. In respect to the Klein zone, we have derived an analytical expression for τ that depends on the size of the incident

Journal of Physics A Mathematical and Theoretical•Nov 30, 2010•6 citations

Quantum-mechanical tunneling time and its relation to the Tsu-Esaki formula

Marc M. Cahay, T. Dichiaro, P. Thanikasalam +1 more

Various approaches have been used to calculate the quantum-mechanical tunneling time through potential barriers including the phase-delay method first introduced by Bohm and Wigner, Buttiker's analysis of the Larmor clock and its generalization, the dwell time of Smith, and numerical studies of wavepacket propagation through potential barriers among others. Most of those previous estimates have only dealt with the tunneling time through simple obstacles, including delta-potential scatterers and simple rectangular barriers under zero bias condition. Even for these simple cases, the agreement among the various estimates is far from being satisfactory. In this manuscript, the transmission line technique is employed to solve the time-dependent Schrodinger equation and an expression of the quantum-mechanical tunneling time is derived for an arbitrary potential profile under non zero bias condition. An exact analytical expression of the tunneling time through a rectangular barrier is derived and shown to be identical to the one obtained recently by Spiller et. al. using the Bohm's quantum potential approach. The tunneling time through resonant tunneling structures under zero bias is also calculated and is shown to be minimum at the quasi-boundstate energy. Finally, the quantum-mechanical tunneling time through the emitter-base junction of a typical heterojunction bipolar transistor is shown to be larger than its semiclassical counterpart.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Sep 3, 1992•3 citations

An analysis of the resonant tunneling‐time based on the presence‐time formalism

Óscar del Barco

Abstract We analyze the tunneling‐time of a spatial wave packet which traverses a resonant structure by means of the presence‐time formalism. With this method we can evaluate the finite size effects of the corresponding energy probability distribution in the tunneling‐time. We show that these effects are not negligible and reduce the tunneling‐time by 25%. An expression for the resonant tunneling‐time is derived, 3ħ/2Γr, where Γr is the width of the resonance. We confirm numerically this result for two resonant structures: a superlattice with n cells and square‐barrier periodic potential, and an asymmetrical rectangular double‐barrier potential. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

physica status solidi (b)•Jan 17, 2007•2 citations

On Times and Speeds of Time-Dependent Quantum and Electromagnetic Tunneling

М. В. Давидович

Abstract This is a review of studies of quantum tunneling, which is described by the one-dimensional Schrödinger equation, and electromagnetic tunneling, where “superluminal” velocities and times of tunneling are considered. Integral and integrodifferential equations have been presented to describe tunneling. According to these equations, superluminal motion is impossible. The paradoxical Hartman effect has been discussed and explained. It has been shown that the velocity of passage of a particle in a beam through a barrier in the case of steady-state and time-dependent quantum tunneling is equal to the velocity of its incidence on the barrier and quasiphotons inside any layer of matter carry the energy always at a subluminal velocity. However, the tunneling time of a single particle or a photon is meaningless.

Journal of Experimental and Theoretical Physics•Jan 1, 2020•12 citations

Tunneling time in the case of frustrated total reflection

Daniela Mugnai, Anedio Ranfagni, Laura Ronchi

The tunneling time is here investigated by means of an electromagnetic model, for a system where a gap, between two parallel planes, acts as a classically-forbidden region for an impinging pulse with incidence angle larger than the critical angle. In all cases of frustrated total reflection we obtain a superluminal behavior both for phase and group delays.

arXiv:physics/0111192•Nov 27, 2001•2 citations
New formulas for an exact and simple evaluation of the phase time associated with the passage of electrons or photons through a finite superlattice are derived. This time, named here the superlattice-tunneling time tau(n), exhibits a resonant-band structure or a superluminal phase time behavior depending on whether the particle's energy lies in a band or a gap. In the band, tau(n) remains larger than the free motion time tau(f), while in the gap it can be less than tau(f) (with strong substrate effects), but it is larger than the single-cell phase time tau(1). Extremely good agreements with optical-pulse and superluminal delay times measured by Spielmann et al. and Steinberg et al. are found, including the superlattice-tunneling-time limit and the substrate effects. Conditions for measurements of earlier electron arrival are also analyzed.
Using a two-photon interferometer, we have measured the time delay for a photon to tunnel across a barrier consisting of a 1.1-\ensuremath{\mu}m-thick 1D photonic band-gap material. The peak of the photon wave packet appears on the far side of the barrier 1.47\ifmmode\pm\else\textpm\fi{}0.21 fs earlier than it would if it were to travel at the vacuum speed of light c. Although the apparent tunneling velocity (1.7\ifmmode\pm\else\textpm\fi{}0.2)c is superluminal, this is not a genuine signal velocity, and Einstein causality is not violated. The measured tunneling time is consistent with the group delay (``phase time''), but not with the semiclassical time.
We have analyzed the influence of the wave packet size in the relativistic tunnelling time τ and its uncertainty �τ when it traverses a given potential barrier. The analytical expressions obtained for both magnitudes confirm that the size of the incident pulse has a significant effect on the tunnelling process. This effect is greater for short pulses, compared with the length of the barrier. For the evanescent zone, we have derived an analytical expression for τ with a good limit of validity. This expression constitutes a value tool to calculate the relativistic tunnelling time as a function of the incident wave packet with a good limit of validity. Superluminal propagation is found in this region but with a large value of the uncertainty �τ compared with the tunnelling time itself. We can conclude that the probability of superluminal propagation is practically negligible in the evanescent region. In respect to the Klein zone, we have derived an analytical expression for τ that depends on the size of the incident
Various approaches have been used to calculate the quantum-mechanical tunneling time through potential barriers including the phase-delay method first introduced by Bohm and Wigner, Buttiker's analysis of the Larmor clock and its generalization, the dwell time of Smith, and numerical studies of wavepacket propagation through potential barriers among others. Most of those previous estimates have only dealt with the tunneling time through simple obstacles, including delta-potential scatterers and simple rectangular barriers under zero bias condition. Even for these simple cases, the agreement among the various estimates is far from being satisfactory. In this manuscript, the transmission line technique is employed to solve the time-dependent Schrodinger equation and an expression of the quantum-mechanical tunneling time is derived for an arbitrary potential profile under non zero bias condition. An exact analytical expression of the tunneling time through a rectangular barrier is derived and shown to be identical to the one obtained recently by Spiller et. al. using the Bohm's quantum potential approach. The tunneling time through resonant tunneling structures under zero bias is also calculated and is shown to be minimum at the quasi-boundstate energy. Finally, the quantum-mechanical tunneling time through the emitter-base junction of a typical heterojunction bipolar transistor is shown to be larger than its semiclassical counterpart.
Abstract We analyze the tunneling‐time of a spatial wave packet which traverses a resonant structure by means of the presence‐time formalism. With this method we can evaluate the finite size effects of the corresponding energy probability distribution in the tunneling‐time. We show that these effects are not negligible and reduce the tunneling‐time by 25%. An expression for the resonant tunneling‐time is derived, 3ħ/2Γr, where Γr is the width of the resonance. We confirm numerically this result for two resonant structures: a superlattice with n cells and square‐barrier periodic potential, and an asymmetrical rectangular double‐barrier potential. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
Abstract This is a review of studies of quantum tunneling, which is described by the one-dimensional Schrödinger equation, and electromagnetic tunneling, where “superluminal” velocities and times of tunneling are considered. Integral and integrodifferential equations have been presented to describe tunneling. According to these equations, superluminal motion is impossible. The paradoxical Hartman effect has been discussed and explained. It has been shown that the velocity of passage of a particle in a beam through a barrier in the case of steady-state and time-dependent quantum tunneling is equal to the velocity of its incidence on the barrier and quasiphotons inside any layer of matter carry the energy always at a subluminal velocity. However, the tunneling time of a single particle or a photon is meaningless.