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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Stochastic model for tunneling processes: The question of superluminal behavior

Anedio Ranfagni, Daniela Mugnai

Published Jul 1, 1995• 31 citations
Publisher

Abstract

A stochastic model for tunneling processes that is based on a close correspondence between quantum relativistic and telegrapher's equations is proposed along the lines of a procedure similar to the one already adopted by Kac [Rocky Mountain J. Math 4, 497 (1974)]. It is hypothesized that reversals of motion imply the inversion of the (imaginary) time rather than of the space, as in the ordinary allowed processes. In this way, superluminal behavior can be predicted in limiting cases. The plausibility of the superluminal motions observed is then discussed on the basis of a signal analysis along the lines of the Sommerfeld-Brillouin criterion which has been suitably modified.

DOI
10.1103/physreve.52.1128

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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).

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A stochastic model for tunneling processes that is based on a close correspondence between quantum relativistic and telegrapher's equations is proposed along the lines of a procedure similar to the one already adopted by Kac [Rocky Mountain J. Math 4, 497 (1974)]. It is hypothesized that reversals of motion imply the inversion of the (imaginary) time rather than of the space, as in the ordinary allowed processes. In this way, superluminal behavior can be predicted in limiting cases. The plausibility of the superluminal motions observed is then discussed on the basis of a signal analysis along the lines of the Sommerfeld-Brillouin criterion which has been suitably modified.
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).
This is an analysis of some aspects of an old but still controversial topic, superluminal quantum tunneling. Some features of quantum tunneling described in literature, such as definition of the tunneling time and a frequency range of a signal, are discussed. The argument is presented that claim of superluminal signaling allegedly observed in frustrated internal reflection experiment was based on the wrong interpretation of the tunneling process. A thought experiment similar to that in the Tolman paradox is discussed. It shows that a new factor, attenuation, comes in the interplay between tunneled signals and macroscopic causality.
Abstract Wavepacket tunneling, in the relativistic limit, is studied via solutions to the Dirac equation for a square barrier potential. Specifically, the arrival time distribution (the time-dependent flux) is computed for wavepackets initiated far away from the barrier, and whose momentum is well below the threshold for above-barrier transmission. The resulting distributions exhibit peaks at shorter times than those of photons with the same initial wavepacket transmitting through a vacuum. However, this apparent superluminality in time is accompanied by very low transmission probabilities. We discuss these observations, and related observations by other authors, in the context of published objections to the notion that tunneling can be superluminal in time. We find that many of these objections are not consistent with our observations, and conclude that post-selected (for transmission) distributions of arrival times can be superluminal. However, the low probability of tunneling means a photon will most likely be seen first and therefore the superluminality does not imply superluminal signaling.
Within the framework of the Kershaw stochastic model (1964) and of the hypothesis on the space stochasticity, equations of motion for two relativistic stochastic particles are obtained which coincide in form with the equations of Cufaro Petroni and Vigier (1979). In the nonrelativistic limit, these equations of free motion reduce to the usual two-particle Schrodinger equation, the imaginary part of which corresponds to a Hamilton-Jacobi equation for two particles interacting through a nonlocal quantum potential.
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.