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Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

Why does relativity allow quantum tunnelling to ‘take no time’?

D. Sokolovski

Published Feb 8, 2004• 13 citations
Publisher

Abstract

In quantum tunnelling, what appears to be an infinitely fast barrier traversal can be explained in terms of an Aharonov–like weak measurement of the tunnelling time, in which the role of the pointer is played by the particle's own coordinate. A relativistic wave packet is shown to be reshaped through a series of subluminal shifts which together produce an anomalous ‘superluminal’ result.

DOI
10.1098/rspa.2003.1222

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In quantum tunnelling, what appears to be an infinitely fast barrier traversal can be explained in terms of an Aharonov–like weak measurement of the tunnelling time, in which the role of the pointer is played by the particle's own coordinate. A relativistic wave packet is shown to be reshaped through a series of subluminal shifts which together produce an anomalous ‘superluminal’ result.
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.
We exploit the analogy between tunnelling across a potential barrier and Aharonov's weak measurements to resolve the long standing paradox between the impossibility to exceed the speed of light and the seemingly 'superluminal' behaviur of the tunnelling particle in the barrier. We demonstrate that 'superluminality' occurs when the value of the duration tau\\tautau spent in the barrier is uncertain, whereas when tau\\tautau is known accurately, no 'superluminal' behaviur is observed. In all cases only subluminal durations contribute to the transmission across the barrier.
We exploit the analogy between the transfer of a pulse across a scattering medium and Aharonov's weak measurements to resolve the long standing paradox between the impossibility to exceed the speed of light and the seemingly ``superluminal'' behavior of a tunneling particle in the barrier or a photon in a ``fast-light'' medium. We demonstrate that superluminality occurs when the value of the duration $\ensuremath{\tau}$ spent in the barrier is uncertain, whereas when $\ensuremath{\tau}$ is known accurately, no superluminal behavior is observed. In all cases only subluminal durations contribute to the transmission which precludes faster-than-light information transfer, as observed in a recent experiment.
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.
We exploit the analogy between tunnelling across a potential barrier and Aharonov's weak measurements to resolve the long standing paradox between the impossibility to exceed the speed of light and the seemingly 'superluminal' behaviur of the tunnelling particle in the barrier. We demonstrate that 'superluminality' occurs when the value of the duration τττ spent in the barrier is uncertain, whereas when τττ is known accurately, no 'superluminal' behaviur is observed. In all cases only subluminal durations contribute to the transmission, which precludes faster-than-light information transfer, as shown by a recent experiment.