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Laser Physics

Confronting the Hartman effect with data from frustrated total internal reflection (FTIR)

Jose C. Martinez

Published Jun 28, 2006• 0 citations
Publisher

Abstract

Over 40 years ago, Hartman noted that the tunneling time τ of a particle through a barrier becomes independent of width for thick barriers. Lately, the Hartman effect has been seen as a support for superluminal tunneling time. By interpreting the reflection and transmission amplitudes in terms of multiple reflection series, we show that τ is linear in barrier width for thin barriers and may be associated with actual traversal time; for thick barriers, τ saturates to the Hartman value because of the suppression of all but the first term of the series due to the smallness of the tunneling factor. For large widths, τ cannot be identified with the propagation time but may be associated with a time to penetrate to a characteristic depth into the barrier, which is independent of width. We discuss data from frustrated internal reflection experiments, which support this view.

DOI
10.1134/s1054660x06070176

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Over 40 years ago, Hartman noted that the tunneling time τ of a particle through a barrier becomes independent of width for thick barriers. Lately, the Hartman effect has been seen as a support for superluminal tunneling time. By interpreting the reflection and transmission amplitudes in terms of multiple reflection series, we show that τ is linear in barrier width for thin barriers and may be associated with actual traversal time; for thick barriers, τ saturates to the Hartman value because of the suppression of all but the first term of the series due to the smallness of the tunneling factor. For large widths, τ cannot be identified with the propagation time but may be associated with a time to penetrate to a characteristic depth into the barrier, which is independent of width. We discuss data from frustrated internal reflection experiments, which support this view.
The mechanism of superluminal traversal time is investigated from the viewpoint of interference between multiple finite wave packets, due to the multiple reflections inside the well or barrier. In the case of potential well traveling that is classically allowed, each of the successively transmitted constituents is delayed by a subluminal time. When the thickness of the well is much smaller in comparison with a characteristic length of the incident wave packet, the reshaped wave packet with superluminal traversal time in transmission maintains the pro. le of the incident wave packet. In the case of potential barrier tunneling that is classically forbidden, though each of the successively transmitted constituents is delayed by a time that is independent of the barrier thickness, the interference between multiple transmitted constituents explains the barrier thickness dependence of the traversal time for thin barriers and its barrier thickness independence for thick barriers. This manifests the nature of the Hartman effect. Copyright (c) EPLA, 2008.
The group delay in frustrated total internal reflection (FTIR) is shown to equal the dwell time plus a self-interference delay. The self-interference delay originates from the energy stored in the overlap between incident and reflected waves above the interface. The group delay in FTIR for finite beams is thus shown to be the lifetime of stored energy in the gap between the prisms and in the partial standing wave above the gap. As in other manifestations of tunneling the saturation of group delay with barrier length (Hartman effect) is explained by the saturation of stored energy in the region defined. No superluminal propagation is implied.
A new approach to the problem of scattering the plane electromagnetic TE wave on a homogeneous dielectric layer is presented, which does not predict, unlike the standard model, the Hartman effect for the tunneling time in the case of scattering the TE wave in the regime of a frustrated total internal refection (FTIR). The basic idea of this approach is that a correct definition of the tunneling velocity and time is possible if only the dynamics of both its subprocesses — transmission and reflection — is known at all stages of the scattering process investigated. It is shown that the Wigner (group) tunneling time was introduced without taking into account of this requirement, and, as a consequence, both this characteristic itself and the associated with it Hartman effect have no relation to transferring the light energy through the layer. The dwell transmission time, which is directly related to it, does not lead to the Hartman effect.
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.
The phenomenon of quantum tunneling challenges classical intuitions about particle behavior and offers profound insights into the nature of quantum entities. This article examines tunneling through the lens of an effect predicted by Thomas E. Hartman in 1962: the tunneling time-delay becomes independent of the barrier width as the latter increases, implying superluminal effective velocities that appear to conflict with relativistic constraints. Exploiting the inherent symmetries of transmitted and reflected time-delays, our analysis shows that a tunneling particle, like its reflected counterpart, completely avoids the barrier region. However, since it can be detected on the transmitted side of the barrier, the only logical conclusion is that the classical notion of spatiality is violated during the tunneling process. In other words, quantum tunneling and the associated Hartman effect strongly support the idea that quantum non-locality should be interpreted as quantum non-spatiality. The latter, according to the conceptuality interpretation of quantum mechanics, expresses the conceptual-like nature of quantum entities, which during measurement transition from an abstract state to a maximally concrete one, in full compatibility with the Hartman effect.