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Journal de Physique I

Traversal Time as a Function of the Size of the Wavepacket

J. Ruiz, M. Ortuño, Emilio Cuevas, V. Gasparian

Published May 1, 1997• 3 citations
Publisher

Abstract

We calculate numerically the transmission coefficient and the traversal time for finite Gaussian wavepackets as a function of their widths. We consider electromagnetic waves crossing a slab and a periodic structure. We find that the periodic structure can be crossed at superluminal velocities for wavenumbers in the optical gap and sizes larger than the thickness of the system. The corresponding transmission coefficients are very small. The long wavepacket limit of the traversal time coincides with previous analytical results for the real component of the interaction time. The imaginary component of this time only affects the change in size of the wavepacket.

DOI
10.1051/jp1:1997182

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We calculate numerically the transmission coefficient and the traversal time for finite Gaussian wavepackets as a function of their widths. We consider electromagnetic waves crossing a slab and a periodic structure. We find that the periodic structure can be crossed at superluminal velocities for wavenumbers in the optical gap and sizes larger than the thickness of the system. The corresponding transmission coefficients are very small. The long wavepacket limit of the traversal time coincides with previous analytical results for the real component of the interaction time. The imaginary component of this time only affects the change in size of the wavepacket.
An analytical treatment of a propagating wavepacket incident on a transient barrier reveals an effect in which, for a particular time interval, the time-varying transmission probability exceeds that for the free propagation of the wavepacket. We show that this effect can be interpreted semiclassically. This effect is quantified and it is shown that its magnitude is in one-to-one correspondence with the strength of the barrier, a feature that has the potential to be used in a scheme for key generation. It is found that the speed with which the information about the barrier perturbation propagates across the wavepacket can exceed the group velocity of the wavepacket. An application to the speed-up of entanglement generation is also considered.
The mechanism of superluminal traversal time is investigated from the viewpoint of interference between multiple finite wave packets, due to the multiple reflections inside the well or barrier. In the case of potential well traveling that is classically allowed, each of the successively transmitted constituents is delayed by a subluminal time. When the thickness of the well is much smaller in comparison with a characteristic length of the incident wave packet, the reshaped wave packet with superluminal traversal time in transmission maintains the pro. le of the incident wave packet. In the case of potential barrier tunneling that is classically forbidden, though each of the successively transmitted constituents is delayed by a time that is independent of the barrier thickness, the interference between multiple transmitted constituents explains the barrier thickness dependence of the traversal time for thin barriers and its barrier thickness independence for thick barriers. This manifests the nature of the Hartman effect. Copyright (c) EPLA, 2008.
We report on microwave measurements of barrier traversal time of electromagnetic wave packets in an undersized waveguide. This time was significantly shorter than the ratio L/cL/cL/c, with LLL the barrier length and ccc the velocity of light in vacuum. Consequently the transport velocity for the wave packet inside the barrier has to be superluminal, i.e. larger than ccc.
Traversal time in tunneling effect for ultrasonic waves in tapered waveguides is derived considering its analogy with quantum and electromagnetic waves tunneling. If, as traversal time, the so called phase time is considered, the ultrasonic wavepacket shows the equivalent in acoustics of superluminality i.e. the derived velocity , crosses the limit of bulk transverse ultrasonic waves in the medium of the waveguide that is the equivalent of c in the quantum and electromagnetic cases. Graphs clearly illustrating this so called Hartman effect are obtained confirming the experimental results in the three different fields.
We introduce a magnetic clock approach to measure the traversal and reflection times of an electromagnetic wave through a slab, when multiple reflections are taken into account. The traversal time is a complex quantity whose real component is proportional to the Faraday rotation and whose imaginary component is proportional to the degree of ellipticity. We conclude that the complex traversal time found for an electron by several different methods is not a consequence of the quantum nature of the particle, but due to the wavelike character of the entity involved.
Much experimental evidence of superluminal phenomena has been available by electromagnetic wave propagation experiments, with the results showing that the phase time describes the barrier traversal time. Based on the extrapolated phase time approach and numerical methods, we show that, in contrast to the ordinary Bessel waves of real argument, the group velocities of modified Bessel waves are superluminal. We obtain the following results. The group velocities increase with the increase of propagation distance, which is similar to the evanescent plane-wave cases. For large wave numbers, the group velocities fall off as the wave numbers increase, which is similar to the evanescent plane-wave cases. For small wave numbers, the group velocities increase with the increase of wave numbers, this is different from the evanescent plane-wave cases.