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

Journal of Modern Optics

Superluminal tunnelling through successive barriers: Does QM predict infinite group-velocities?

Erasmo Recami

Published Apr 1, 2004• 34 citations
Publisher

Abstract

The phenomenon of one-dimensional non-resonant tunnelling is analyzed through two or more successive (opaque) potential barriers, separated by intermediate free regions R, just by exploiting the relevant solutions to the Schroedinger equation. The total traversal time has been shown by us to be independent not only of the barrier widths (the so-called ‘Hartman effect’), but also of the R-widths: so that the effective group velocity in the regions R, between two successive barriers, can be regarded as practically infinite. Such a prediction has been theoretically confirmed and generalized (as well as interpreted in terms of ‘super-oscillations’) by Aharonov et al. A recent experiment by Longhi et al. supported the predictions by considering two successive gratings in an optical fibre, that is, by having recourse to two ‘classical barriers’ (which allow simulating the tunnelling, due to the known formal identity between the Schrödinger and the Helmholtz equation).

DOI
10.1080/09500340408233605

Related papers

Tunneling Through Two Successive Barriers and the Hartman (Superluminal) Effects

V. S. Olkhovsky, Erasmo Recami, Giovanni Salesi

We study the phenomenon of one-dimensional non-resonant tunneling through two successive potential barriers, separated by an intermediate free region R, by analyzing the relevant solutions to the Schroedinger equation. We find that the total traversal time does not depend not only on the barrier widths (the so-called 'Hartman effect'), but also on the R width: so that the effective velocity in the region R, between the two barriers, can be regarded as infinite. This agrees with the results known from the corresponding waveguide experiments, which simulated the tunnelling experiment herein considered due to the formal identity between the Schroedinger and the Helmholtz equation.

•Mar 10, 2000•0 citations

Superluminal tunneling through two successive barriers

V. S. Olkhovsky, Erasmo Recami, Giovanni Salesi

We study the phenomenon of one-dimensional non-resonant tunneling through two successive (opaque) potential barriers, separated by an intermediate free region , by analyzing the relevant solutions to the Schroedinger equation. We find that the total traversal time does not depend not only on the barrier widths (the so-called "Hartman effect"), but also on the width: so that the effective velocity in the region , between the two barriers, can be regarded as practically infinite. This agrees with the results known from the corresponding waveguide experiments, which simulated the tunneling experiment herein considered due to the known formal identity between the Schroedinger and the Helmholtz equation.

arXiv:quant-ph/0002022•Europhysics Letters (EPL)•Mar 1, 2002•118 citations

Apparent superluminality and the generalized Hartman effect in double-barrier tunneling

Herbert G. Winful

Recent papers suggest that tunneling wave packets traverse the region of allowed propagation between two potential barriers with superluminal group velocity and in a time independent of barrier separation. This phenomenon has been termed the "generalized Hartman effect" and extended to multiple barriers. Here we show that this delay time is not a transit time but a cavity lifetime. It does not imply superluminal velocity. Reported experimental verifications of this effect are reinterpreted.

Physical Review E•Oct 18, 2005•33 citations

Tunneling Times and "Superluminal" Tunneling: A brief Review

V. S. Olkhovsky, Erasmo Recami

In the First Part of this paper [that was submitted for pub. in 1991 and appeared in print in Phys. Reports 214 (1992) 339] we critically review the main theoretical definitions and calculations of the sub-barrier tunnelling and reflection times. Moreover, within conventional QM, we propose a new definition of such durations, on the basis of a recent general formalism of ours. At last, we discuss a surprising result: that QM predicts tunnelling through opaque barriers to take place with Superluminal group-velocities. In the Second Part [that appeared in print later, in J. de Physique-I 5 (1995) 1351] we, e.g., present the results of some numerical calculations (based on our equations) on the penetration and return times during tunnelling INSIDE a rectangular potential barrier, and set forth suitable definitions also of the variances, or dispersions, for the transmission and reflection time-durations. The numerical evaluations confirm that our approach implied, and implies, the existence of Superluminal tunnelling (that we called "Hartman effect"): an effect that in these days is receiving --at Cologne, Berkeley, Florence and Vienna-- indirect, but interesting, experimental verifications. More detailed reviews of the same topics are in the LANL Archive cond-mat/9802126 (and in an Appendix to physics/9712051, where the data are summarized of the experiments that --in four different sectors of physics-- seem to indicate the existence of Superluminal motions).

arXiv:cond-mat/9802162•Feb 15, 1998•1 citation

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

Relativistic tunneling through two "transparent" successive barriers

MASSIMO GERMANO

In the case of tunneling of relativistic particles, differently from the nonrelativistic case, a limit of "transparent" barrier can also lead to an apparent "superluminal" behavior when considering the phase time. In this limit, the restricting condition of "opaque" barrier of the nonrelativistic case is avoided, nevertheless, the very thin width of a single barrier to obtain this ?transparent? limit can result in a problem itself, for probing the effect. A combination of two successive transparent barriers can show an apparent "superluminal" behavior along a macroscopic arbitrary distance "L". Two solutions for energy E above and below the potential square barrier V are found, for both solutions there the apparent superluminal behavior is possible above a threshold of free travelling group velocity (energy) and dependent on the ratio barriers length free path as function of the ratio group velocity - speed of light.

arXiv:1909.07708•TURKISH JOURNAL OF PHYSICS•Jan 1, 2022•0 citations
The phenomenon of one-dimensional non-resonant tunnelling is analyzed through two or more successive (opaque) potential barriers, separated by intermediate free regions R, just by exploiting the relevant solutions to the Schroedinger equation. The total traversal time has been shown by us to be independent not only of the barrier widths (the so-called ‘Hartman effect’), but also of the R-widths: so that the effective group velocity in the regions R, between two successive barriers, can be regarded as practically infinite. Such a prediction has been theoretically confirmed and generalized (as well as interpreted in terms of ‘super-oscillations’) by Aharonov et al. A recent experiment by Longhi et al. supported the predictions by considering two successive gratings in an optical fibre, that is, by having recourse to two ‘classical barriers’ (which allow simulating the tunnelling, due to the known formal identity between the Schrödinger and the Helmholtz equation).
We study the phenomenon of one-dimensional non-resonant tunneling through two successive potential barriers, separated by an intermediate free region R, by analyzing the relevant solutions to the Schroedinger equation. We find that the total traversal time does not depend not only on the barrier widths (the so-called 'Hartman effect'), but also on the R width: so that the effective velocity in the region R, between the two barriers, can be regarded as infinite. This agrees with the results known from the corresponding waveguide experiments, which simulated the tunnelling experiment herein considered due to the formal identity between the Schroedinger and the Helmholtz equation.
We study the phenomenon of one-dimensional non-resonant tunneling through two successive (opaque) potential barriers, separated by an intermediate free region , by analyzing the relevant solutions to the Schroedinger equation. We find that the total traversal time does not depend not only on the barrier widths (the so-called "Hartman effect"), but also on the width: so that the effective velocity in the region , between the two barriers, can be regarded as practically infinite. This agrees with the results known from the corresponding waveguide experiments, which simulated the tunneling experiment herein considered due to the known formal identity between the Schroedinger and the Helmholtz equation.
Recent papers suggest that tunneling wave packets traverse the region of allowed propagation between two potential barriers with superluminal group velocity and in a time independent of barrier separation. This phenomenon has been termed the "generalized Hartman effect" and extended to multiple barriers. Here we show that this delay time is not a transit time but a cavity lifetime. It does not imply superluminal velocity. Reported experimental verifications of this effect are reinterpreted.
In the First Part of this paper [that was submitted for pub. in 1991 and appeared in print in Phys. Reports 214 (1992) 339] we critically review the main theoretical definitions and calculations of the sub-barrier tunnelling and reflection times. Moreover, within conventional QM, we propose a new definition of such durations, on the basis of a recent general formalism of ours. At last, we discuss a surprising result: that QM predicts tunnelling through opaque barriers to take place with Superluminal group-velocities. In the Second Part [that appeared in print later, in J. de Physique-I 5 (1995) 1351] we, e.g., present the results of some numerical calculations (based on our equations) on the penetration and return times during tunnelling INSIDE a rectangular potential barrier, and set forth suitable definitions also of the variances, or dispersions, for the transmission and reflection time-durations. The numerical evaluations confirm that our approach implied, and implies, the existence of Superluminal tunnelling (that we called "Hartman effect"): an effect that in these days is receiving --at Cologne, Berkeley, Florence and Vienna-- indirect, but interesting, experimental verifications. More detailed reviews of the same topics are in the LANL Archive cond-mat/9802126 (and in an Appendix to physics/9712051, where the data are summarized of the experiments that --in four different sectors of physics-- seem to indicate the existence of Superluminal motions).
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.
In the case of tunneling of relativistic particles, differently from the nonrelativistic case, a limit of "transparent" barrier can also lead to an apparent "superluminal" behavior when considering the phase time. In this limit, the restricting condition of "opaque" barrier of the nonrelativistic case is avoided, nevertheless, the very thin width of a single barrier to obtain this ?transparent? limit can result in a problem itself, for probing the effect. A combination of two successive transparent barriers can show an apparent "superluminal" behavior along a macroscopic arbitrary distance "L". Two solutions for energy E above and below the potential square barrier V are found, for both solutions there the apparent superluminal behavior is possible above a threshold of free travelling group velocity (energy) and dependent on the ratio barriers length free path as function of the ratio group velocity - speed of light.