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Journal of Modern Optics

Superluminal tunnelling times as weak values

Yakir Aharonov, Noam Erez A, Benni Reznik

Published Jan 1, 2003• 31 citations
Publisher

Abstract

We consider the tunnelling particle as a pre- and post-selected system and prove that the tunnelling time is the expectation value of the position of a ‘clock’ degree of freedom weakly coupled to it. Such a value, called a ‘weak value’, typically falls outside the eigenvalue spectrum of the operator. The appearance of unusual weak values has been associated with a unique interference structure called ‘superoscillations’ (band-limited functions which on a finite interval, approximate functions with spectra well outside their band). It is proposed that superoscillations play an important role in the interferences which give rise to superluminal effects. To demonstrate that, we consider a certain simple tunnelling barrier which allows a wave packet to travel in zero time and negligible distortion, a distance arbitrarily longer than the width of the wave packet. The peak is shown to result from a superoscillatory superposition at the tail. Similar reasoning applies to the dwell time. For this system, both the Wigner time (related to the group velocity) and a clock time correspond to superluminal velocities.

DOI
10.1080/09500340308234558

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We consider the tunnelling particle as a pre- and post-selected system and prove that the tunnelling time is the expectation value of the position of a ‘clock’ degree of freedom weakly coupled to it. Such a value, called a ‘weak value’, typically falls outside the eigenvalue spectrum of the operator. The appearance of unusual weak values has been associated with a unique interference structure called ‘superoscillations’ (band-limited functions which on a finite interval, approximate functions with spectra well outside their band). It is proposed that superoscillations play an important role in the interferences which give rise to superluminal effects. To demonstrate that, we consider a certain simple tunnelling barrier which allows a wave packet to travel in zero time and negligible distortion, a distance arbitrarily longer than the width of the wave packet. The peak is shown to result from a superoscillatory superposition at the tail. Similar reasoning applies to the dwell time. For this system, both the Wigner time (related to the group velocity) and a clock time correspond to superluminal velocities.
A simple model of a quantum clock is applied to the old and controversial problem of how long a particle takes to tunnel through a quantum barrier. The model has the advantage of yielding sensible results for energy eigenstates and does not require the use of time-dependent wave packets. Although the treatment does not forbid superluminal tunneling velocities, there is no implication of faster-than-light signaling because only the transit duration is measurable, not the absolute time of transit. A comparison is given with the weak-measurement post-selection calculations of Steinberg.
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.
Superoscillations, namely regions where a band-limited function f (x) varies faster than the fastest of its Fourier components k, generate the illusion that the Fourier content is ?supershifted? so as to lie outside the spectrum of the function. The relation between supershifts and superoscillations, central to the quantum weak measurements scheme, is explored in terms of two different representations of the local Fourier transform in the ?phase space? (x, k). The Wigner function W(x, k), regarded as a function of k for fixed x, inherits the band-limited property of f (x). Neverthless, its local k average can lie outside the spectrum because W, although real, posesses negative values. The local Wigner average of k equals the local wavenumber at x (local weak value of momentum), defined as the phase variation k loc(x)?=?? x arg f (x). By contrast, the Husimi function H(x, k), i.e. the windowed Fourier transform with window width L, corresponding to squeezing of the coherent state associated with (x, k) (and representing the pointer wavefunction after a weak measurement), is positive-definite. But it is not band-limited, and the local Husimi average of k equals k loc if L is small enough. These properties are illustrated numerically with two superoscillatory functions.
In the context of superluminal propagation of wave packets through potential barriers, the tunneling speed is usually characterized by the Wigner velocity. We propose an alternative speed that takes into account the interference between the incoming and the reflected waves and leads to a better estimation of arrival time for a wave packet entering the tunneling region. This arrival time is derived by an extrapolation from inside the barrier. The analytical theory is based on the stationary phase approximation whose validity is justified by a comparison with the numerical solution of the time-dependent Dirac equation.