Has Superluminal Tunneling Been Observed?

Earlier this month, in Science (Vol. 322, p. 1525), P. Eckle et al. reported an experiment from which they conclude that an electron can tunnel through the potential barrier of an He atom in practically no time. This would indeed be a fundamental discovery, but I think the analysis of their technically brilliant experiment is faulty. It may instead represent further evidence that the interpretation of an electron as a particle is simply wrong. There had already been earlier claims of superluminal tunneling, but some were based on doubtful definitions of clocks, while most of them used questionable concepts or classical pictures (particles) for their interpretation.

Before discussing this new experiment, let me remind you of a mathematical paper by G. Hegerfeldt (Phys. Rev. Lett. 72, 596, 1994), who claimed that an atom in a state that is formed at some definite time by means of a quantum jump can _immediately_ cause an effect in another atom at an arbitrary distance with very small but finite probability.

Understandably, this claim caused a considerable stir in the media when it was published, although it is partly trivial and partly wrong. Since the eigenstate of a bound electron has an infinitely long, exponentially decreasing tail in the forbidden external region, there is some probability that it (or its electromagnetic force) does not have to travel the distance to the other atom in order to interact with it. What is physically wrong in the considered scenario is the assumption that the bound state can form instantaneously and exactly in a quantum jump. Physicists should know that a time-dependent process has to be described by the time-dependent Schrödinger equation (or its relativistic equivalent). Therefore, the tails of the wave function require sufficient time to _form_, and the suggested experiment would fail.

In the new experiment, a similar mistake seems to be responsible for its proposed interpretation. In this case the authors assume (directly or indirectly) that the electron is initially inside the potential barrier of the He atom. They then lower the barrier at a precisely defined time, and they observe the emitted electron immediately thereafter (at a time earlier than it would be required to travel through the barrier region at light speed). Here they neglect again the exponential tail of the wave function that existed even before the potential barrier was lowered (regarded as the "onset of efficient tunneling").

The details of the experiment are complex, so I cannot exclude that other aspects may be important, too, for explaining the result in a realistic manner. However, every physics student with a computer can numerically calculate the time-dependence of the exact wave function (with an initially present tail) in a one-dimensional model after the barrier has been lowered. Even more than that: because of the linearity of the Schrödinger equation, one can do this calculation separately for all _parts_ of the wave function (internal part, tail in the forbidden region, or any other chosen position interval), and thereafter superpose their final states.

This might be the most informative part of the story! For a relativistic wave equation it must confirm the light speed limit for all corresponding wave fronts ("signal speed") in accordance with the Sommerfeld theorem. (Relativistically, sharp wave fronts may require negative frequencies, which would have to be interpreted as contributions from particle-antiparticle pairs in the particle picture--but this exact case is here not necessarily relevant.) If the wave function describes reality, its propagation represents the only physical motion in the experiment. The peak velocity of the wave packet does not represent any causal motion under a potential barrier, while the rest of the rich debate about tunneling times in the literature is concerned with particle metaphysics.

So why all the fuss about tunneling times if the results can be readily predicted? The reason is presumably that most physicists once learned, and still believe, that "the wave function only describes probabilities for positions (or other properties) of particles." If the electron, as a particle, does not possess a trajectory ("because of the uncertainty relations") it must acausally hop from here to there--possibly at superluminal speed. However, this hopping of postulated particles depends on artificial definitions. Even the usual pragmatic probability interpretation in terms of particles would not allow superluminal effects in this way. So the true scandal is that such misleading interpretations are readily accepted (or uncritically commented) by reputed journals, such as PRL, Science or Nature. As a related example, consider the hype about so-called "quantum teleportation", which is but another quantum misnomer.

There is in fact no evidence for the existence of particles in any form. The occurrence of spots on a screen or clicks of a detector can be dynamically described in quantum mechanical terms by means of wave functions, taking into account environmental decoherence (see here for a recent account). Although physicists should know that the wave function is defined in configuration space, it is often exclusively understood as a spatial wave (thereby giving rise to the pseudo-concept of a "wave-particle dualism"). However, these two kinds of spaces are isomorphic only in the case of single mass points, such as the centers of mass of more or less complex objects. This very special case applies to most practicable experiments (scattering experiments or single particle problems)--so one often finds the conclusion that "the wave function loses its meaning at the final detector".

This pragmatism is really astonishing in view of the fact that there are solid state and other quantum physicists who have studied the physics of complex systems, which must include measurement devices, in much detail in terms of many-particle wave functions. As I understand the new experiment, the measurement selects ("projects out", branches into) that partial wave in which the electron was already deep in the barrier region when the barrier was lowered.