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

Unified Time Analysis of Photon and (Nonrelativistic) Particle Tunnelling, and the Superluminal Group-Velocity Problem

Jacek Jakiel, V. S. Olkhovsky, Erasmo Recami

Published Apr 30, 2001• 0 citations
Publisher

Abstract

A unified approach to the time analysis of tunnelling of nonrelativistic particles is presented, in which Time is regarded as a quantum-mechanical observable, canonically conjugated to Energy. The validity of the Hartman effect (independence of the Tunnelling Time of the opaque barrier width, with Superluminal group velocities as a consequence) is verified for all the known expressions of the mean tunnelling time. Moreover, the analogy between particle and photon tunnelling is suitably exploited. On the basis of such an analogy, an explanation of some recent microwave and optics experimental results on tunnelling times is proposed. Attention is devoted to some aspects of the causality problem for particle and photon tunnelling.

DOI
10.15161/oar.it/sr930-2tm32

Related papers

Developments in time analysis of particle and photon tunnelling

V. S. Olkhovsky, A. Agresti

A compact analysis of development and prospects in the study of the tunnelling evolution is given. A new systematization of various approaches to defining tunnelling times in the light of time as a quantum mechanical observable is proposed. The problem of superluminal group velocities, without violations of special relativity, is also taken in account. Then a particular attention is devoted to the presentation of new results on the analogy between particle and photon tunnelling and analysis of the causality validity during tunnelling. [PACS nos. 03.40.Kf, 03.50.De, 41.20.Jb, 41.20.Bt, 42.25.Bs, 03.30.+p, 03.65.-w].

arXiv:physics/9612004•Dec 6, 1996•0 citations

The validity of the Hartman effect for all kinds of mean tunnelling times

Jacek Jakiel, V. S. Olkhovsky, Erasmo Recami

It is shown that the Hartman and Fletcher effect is valid for all known expressions for the mean tunnelling time in various nonrelativistic approaches in the cases of barriers of finite width without absorption. Then it is shown that the Hartmann effect is not valid for the mean-square fluctuation tunnelling time. On the base of this effect, taking the analogy between nonrelativistic-particle and photon tunnelling into account, one can explain the observed superluminal effective (group) velocities in photon tunnelling experiments without violations of Einstein causality.

Mysteries, Puzzles, and Paradoxes in Quantum Mechanics•Jan 1, 1999•0 citations

Developments in time analysis of particle and photon tunneling

V. S. Olkhovsky, A. Agresti

A compact analysis of development and prospects in the study of the tunnelling evolution is given. A new systematization of various approaches to defining tunnelling times in the light of time as a quantum mechanical observable is proposed. The problem of superluminal group velocities, without violations of special relativity, is also taken in account. Then a particular attention is devoted to the presentation of new results on the analogy between particle and photon tunnelling and analysis of the causality validity during tunnelling. [PACS nos. 03.40.Kf, this http URL, 41.20.Jb, this http URL, this http URL, 03.30.+p, 03.65.-w].

•Dec 6, 1996•3 citations

Do single photons tunnel faster than light?

Herbert G. Winful

Experiments done with single photons in the early 1990's produced a surprising result: that single photons pass through a photonic tunnel barrier with a group velocity faster than the vacuum speed of light. This result has stimulated intense discussions related to causality, the speed of information transfer, the nature of barrier tunneling and the meaning of group velocity. The superluminality of tunneling photons is now textbook material, although the authors note that controversy still remains. Another paradoxical result, known as the Hartman effect, is that the tunneling time of the photons becomes independent of barrier length in the limit of opaque barriers. In this paper we examine the meaning of group velocity in the context of barrier tunneling. We ask whether a single tunneling photon can be described by a group velocity and whether the short group delays imply superluminal group velocity. We resolve the paradox of the Hartman effect and show that the predicted and measured group delays are not transit times but photon lifetimes.

arXiv:0708.3889•Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Aug 26, 2007•4 citations

Relativistic tunneling through two successive barriers

Jose Tadeu Lunardi, Luiz A. Manzoni

We study the relativistic quantum mechanical problem of a Dirac particle tunneling through two successive electrostatic barriers. Our aim is to study the emergence of the so-called generalized Hartman effect, an effect observed in the context of nonrelativistic tunneling as well as in its electromagnetic counterparts and which is often associated with the possibility of superluminal velocities in the tunneling process. We discuss the behavior of both the phase (or group) tunneling time and the dwell time, and show that in the limit of opaque barriers the relativistic theory also allows the emergence of the generalized Hartman effect. We compare our results with the nonrelativistic ones and discuss their interpretation.

arXiv:0708.3507•Physical Review A•Oct 16, 2007•22 citations

On Superluminal Photonic Tunnelling

Günter Nimtz, Astrid Haibel, A. A. Stahlhofen

Evanescent modes or the tunnelling process are characterized by an imaginary wave number. These special solutions of the Helmholtz and of the Schrödinger equations first noticed in connection with the total reflection were said to have no physical meaning about 200 years ago. Last century the tunnelling problem in quantum mechanics has been described by the phase time approach. The phase time approach yields the group velocity of a wave packet (see e.g. Hartman, 1962). Nowadays the phase time approach is used in network analyzers to determine the group velocity of an electromagnetic wave in devices. We have reported about superluminal signal velocities in photonic tunnelling. It was observed that the superluminal photonic tunnelling time data measured in the time domain are in agreement with the calculated phase time data. This result is also in agreement with the definition of the barrier transition time given in the Federal Standard 1037 C (NTIA, USA 2000). Thus we have added further credibility to the assumption that the photonic tunnelling results are representative for quantum mechanical tunnelling of particles.

Kluwer Academic Publishers eBooks•Dec 23, 2005•2 citations
A unified approach to the time analysis of tunnelling of nonrelativistic particles is presented, in which Time is regarded as a quantum-mechanical observable, canonically conjugated to Energy. The validity of the Hartman effect (independence of the Tunnelling Time of the opaque barrier width, with Superluminal group velocities as a consequence) is verified for all the known expressions of the mean tunnelling time. Moreover, the analogy between particle and photon tunnelling is suitably exploited. On the basis of such an analogy, an explanation of some recent microwave and optics experimental results on tunnelling times is proposed. Attention is devoted to some aspects of the causality problem for particle and photon tunnelling.
A compact analysis of development and prospects in the study of the tunnelling evolution is given. A new systematization of various approaches to defining tunnelling times in the light of time as a quantum mechanical observable is proposed. The problem of superluminal group velocities, without violations of special relativity, is also taken in account. Then a particular attention is devoted to the presentation of new results on the analogy between particle and photon tunnelling and analysis of the causality validity during tunnelling. [PACS nos. 03.40.Kf, 03.50.De, 41.20.Jb, 41.20.Bt, 42.25.Bs, 03.30.+p, 03.65.-w].
It is shown that the Hartman and Fletcher effect is valid for all known expressions for the mean tunnelling time in various nonrelativistic approaches in the cases of barriers of finite width without absorption. Then it is shown that the Hartmann effect is not valid for the mean-square fluctuation tunnelling time. On the base of this effect, taking the analogy between nonrelativistic-particle and photon tunnelling into account, one can explain the observed superluminal effective (group) velocities in photon tunnelling experiments without violations of Einstein causality.
A compact analysis of development and prospects in the study of the tunnelling evolution is given. A new systematization of various approaches to defining tunnelling times in the light of time as a quantum mechanical observable is proposed. The problem of superluminal group velocities, without violations of special relativity, is also taken in account. Then a particular attention is devoted to the presentation of new results on the analogy between particle and photon tunnelling and analysis of the causality validity during tunnelling. [PACS nos. 03.40.Kf, this http URL, 41.20.Jb, this http URL, this http URL, 03.30.+p, 03.65.-w].
Experiments done with single photons in the early 1990's produced a surprising result: that single photons pass through a photonic tunnel barrier with a group velocity faster than the vacuum speed of light. This result has stimulated intense discussions related to causality, the speed of information transfer, the nature of barrier tunneling and the meaning of group velocity. The superluminality of tunneling photons is now textbook material, although the authors note that controversy still remains. Another paradoxical result, known as the Hartman effect, is that the tunneling time of the photons becomes independent of barrier length in the limit of opaque barriers. In this paper we examine the meaning of group velocity in the context of barrier tunneling. We ask whether a single tunneling photon can be described by a group velocity and whether the short group delays imply superluminal group velocity. We resolve the paradox of the Hartman effect and show that the predicted and measured group delays are not transit times but photon lifetimes.
We study the relativistic quantum mechanical problem of a Dirac particle tunneling through two successive electrostatic barriers. Our aim is to study the emergence of the so-called generalized Hartman effect, an effect observed in the context of nonrelativistic tunneling as well as in its electromagnetic counterparts and which is often associated with the possibility of superluminal velocities in the tunneling process. We discuss the behavior of both the phase (or group) tunneling time and the dwell time, and show that in the limit of opaque barriers the relativistic theory also allows the emergence of the generalized Hartman effect. We compare our results with the nonrelativistic ones and discuss their interpretation.
Evanescent modes or the tunnelling process are characterized by an imaginary wave number. These special solutions of the Helmholtz and of the Schrödinger equations first noticed in connection with the total reflection were said to have no physical meaning about 200 years ago. Last century the tunnelling problem in quantum mechanics has been described by the phase time approach. The phase time approach yields the group velocity of a wave packet (see e.g. Hartman, 1962). Nowadays the phase time approach is used in network analyzers to determine the group velocity of an electromagnetic wave in devices. We have reported about superluminal signal velocities in photonic tunnelling. It was observed that the superluminal photonic tunnelling time data measured in the time domain are in agreement with the calculated phase time data. This result is also in agreement with the definition of the barrier transition time given in the Federal Standard 1037 C (NTIA, USA 2000). Thus we have added further credibility to the assumption that the photonic tunnelling results are representative for quantum mechanical tunnelling of particles.