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United Nations Educational, Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS MISLEADING PRESUMPTION OF A GENERALIZED HARTMAN EFFECT

Herbert P. Simanjuntak, Pedro Pereyra, Abdus Salam

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

We analyze different examples to show that the so-called generalized Hartman effect is an erroneous presumption. The results obtained for electron tunneling and transmission of electromagnetic waves through superlattices and Bragg gratings show clearly the resonant character of the phase time behavior where a generalized Hartman effect is expected. A reinterpretation of the experimental results in double Bragg gratings is proposed.

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We analyze different examples to show that the so-called generalized Hartman effect is an erroneous presumption. The results obtained for electron tunneling and transmission of electromagnetic waves through superlattices and Bragg gratings show clearly the resonant character of the phase time behavior where a generalized Hartman effect is expected. A reinterpretation of the experimental results in double Bragg gratings is proposed.
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.
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.
Abstract In this letter we prove explicitly that if the Hartman effect exists for an arbitrary “unit cell” potential, then it also exists for a periodic system constructed using the same “unit cell” potential repeatedly. We further show that if the Hartman effect exists, the tunneling time in the limiting case of a sufficiently thick “unit cell” potential is the same as that of its periodic system. This is true for any arbitrary value of the intervening gap between the consecutive “unit cell” of the periodic system. Thus, the generalized Hartman effect always occurs for a general potential constructed using multiple copies of single potential which shows Hartman effect.
An explicit expression is obtained for the phase-time corresponding to tunneling of a (non-relativistic) particle through two rectangular barriers, both in the case of resonant and in the case of non-resonant tunneling. It is shown that the behavior of the transmission coefficient and of the tunneling phase-time near a resonance is given by expressions with "Breit-Wigner type" denominators. By contrast, it is shown that, when the tunneling probability is low (but not negligible), the non-resonant tunneling time depends on the barrier width and on the distance between the barriers only in a very weak (exponentially decreasing) way: This can imply in various cases, as well known, the highly superluminal tunneling associated with the so-called "generalized Hartman Effect": but we are now able to improve and modify the mathematical description of such an effect, and to compare more in detail our results with the experimental data for non-resonant tunneling of photons . Finally, as a second example, the tunneling phase-time is calculated, and compared with the available experimental results, in the case of the quantum-mechanical tunneling of neutrons through two barrier-filters at the resonance energy of the set-up. Our analysis appears to explain satisfactorily both sets of experiments.
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.