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.
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.
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.
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.
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).
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.
This book explores the theoretical and philosophical implications of transcending the speed of light limit, building upon Florentin Smarandache's "no speed barrier" hypothesis. This collection of works develops the hypothesis that no physical entity is fundamentally constrained by a speed barrier, suggesting a framework for extending classical and relativistic physics into the realms of Superluminal Physics and Instantaneous Physics. The book’s core technical argument centers on the phenomenon of quantum tunneling and the Hartman effect, where the tunneling time appears independent of barrier width, leading to seemingly superluminal results. The authors propose a novel interpretation of the Hartman effect, suggesting that it is a manifestation of the multivaluedness of solutions to the Schrödinger equation, implying a quantum entity can exist in multiple locations simultaneously and thus appear to tunnel instantaneously. To provide a plausible mechanism for this superluminal effect, the book considers soliton tunnelling, specifically the Falaco soliton, which is presented as a known exact solution to the three-dimensional Navier-Stokes equations and is proposed as a realistic mechanism for quantum tunneling. Ultimately, the research seeks to find a general theory, such as the S-Multispace Theory, that unites physics at subluminal, relativistic, superluminal, and instantaneous speeds. This theoretical path is explored as an alternative for deep space and interstellar travel, moving beyond conventional concepts like the Alcubierre warp drive or wormholes in General Relativity.
Experiments at Berkeley and elsewhere which show that the process of tunneling is apparently superluminal will be reviewed. Conflicting theories for the tunneling time will be compared with experiment. The tunneling particle in the Berkeley experiments was the photon. The measurement of the tunneling time utilized a two-photon light source (spontaneous parametric down-conversion), a Hong-Ou-Mandel interferometer, and a coincidence counter of photon pairs. The tunnel barrier consisted of a photonic-bandgap medium excited at midgap. We find that the peak of a tunneling single-photon wave packet appeared on the far side of the barrier 1.47 ± 0.21 fs earlier than the peak of a wave packet which traveled an equal distance in air. Group velocities which exceed c can also occur in transparent media with inverted atomic populations. Tachyon-like excitations, i.e., collective normal modes of atoms coupled to electromagnetic radiation which possess tachyon-like dispersion relations, occur in these media. However, relativistic causality is not violated by any of these superluminal phenomena.