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arXiv:quant-ph/9603018

On the concept of the tunneling time

M. S. Marinov, Bilha Segev

Published Mar 13, 1996• 2 citations
arXivHTMLPDF

Abstract

Asymptotic time evolution of a wave packet describing a non-relativistic particle incident on a potential barrier is considered, using the Wigner phase-space distribution. The distortion of the trasmitted wave packet is determined by two time-like parameters, given by the energy derivative of the complex transmission amplitude. The result is consistent with various definitions of the tunneling time (e.g. the Büttiker-Landauer time, the complex time and Wigner's phase time). The speed-up effect and the negative dispersion are discussed, and new experimental implications are considered.

DOI
10.48550/arxiv.quant-ph/9603018

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Physica Scripta•May 20, 2025•2 citations
Asymptotic time evolution of a wave packet describing a non-relativistic particle incident on a potential barrier is considered, using the Wigner phase-space distribution. The distortion of the trasmitted wave packet is determined by two time-like parameters, given by the energy derivative of the complex transmission amplitude. The result is consistent with various definitions of the tunneling time (e.g. the Büttiker-Landauer time, the complex time and Wigner's phase time). The speed-up effect and the negative dispersion are discussed, and new experimental implications are considered.
An interesting analytical expression is obtained for the coordinate probability distribution of a wave packet transmitted through a potential barrier. In the large-time asymptotics the distribution is given in terms of the transmission amplitude and the phase-space quasidistribution of the initial (not necessarily pure) state. The distortion of a (broad) wave packet due to the barrier penetration is determined by two timelike parameters, given by the energy derivative of the complex transmission amplitude. The resulting change of the packet shape reveals a dependence on the detector position and incorporates a possible speed-up effect and a negative dispersion.
We consider the evolution of a wave packet that is made up of a group of the wave functions describing the stationary scattering process and tunnels through a one-dimensional potential of arbitrary form. As the main characteristics of the time difference of the tunnelling process, use is made of the propagation speed of the wave-packet maximum. We show that the known Hartman formula for the tunnelling time corresponds to the wave packet with a wavenumber-uniform spectral composition in the case, when the phase and transmission coefficient modulus dispersions are taken into account only in the linear approximation. The amplitude of the main peak of the transmitted wave intensity is proven to be independent of the tunnelling time and is determined by the transmission coefficient of the spectral component at the carrier frequency and the spectral width of the wave packet. In the limit of an infinitely wide potential barrier the amplitude of the wave-packet maximum is shown to tend to zero slower than the tunnelling time tends to its asymptotic value, i.e., indeed we deal with the paradox of an infinitely large propagation speed of a wave disturbance through the barrier.
We reexamine the Salecker-Wigner-Peres clock formalism and show that it can be directly applied to the phenomenon of tunneling. Then we apply this formalism to the determination of the tunneling time of a nonrelativistic wave packet, sharply concentrated around a tunneling energy, incident on a symmetric double-barrier potential. In order to deepen the discussion about the generalized Hartman effect, we consider the case in which the clock runs only when the particle can be found inside the region between the barriers and show that, whenever the probability to find the particle in this region is non-negligible, the corresponding time (which in this case turns out to be a dwell time) increases with barrier spacing.
Analytic expressions are given for the time spent by a particle tunneling through a potential barrier. The expressions are derived for an incident wave packet which is initially Gaussian, centered about a point an arbitrary distance away from a rectangular potential barrier and moving toward the barrier with constant average velocity. Upon collision with the barrier, the packet splits into a transmitted and a reflected packet. The resultant transmission time is positive, nonzero and in principle measurable. Although the transmission time becomes quite large as the incident kinetic energy becomes very small, in general, for nonzero incident momentum and finite potential barriers which are neither very thick nor very thin, the transmission times are less than the time that would be required for the incident particle to travel a distance equal to the barrier thickness. The transmission times for metal-insulator-metal thin film sandwiches, given approximately by δt3=ℏ/(Efφ)12, where Ef is the Fermi energy of the metal, and φ the vacuum work function, are of the order of 10−16 sec, compared to RC time constants of about 10−13 sec.
Abstract Tunneling time as an important quantity to characterize tunneling process has not been well defined yet in quantum mechanics. In this paper, we adopt the definition of tunneling time given in [S.-C. Li, New J. Phys. 24, 083033 (2022)] to investigate the time spent by a macroscopic quantum wave packet passing through a Gaussian barrier. We show a general scenario of tunneling times in different energy regimes and find that it has a good agreement with the scenario for a rectangular barrier, which verifies the validity of the above definition of tunneling time for the Gaussian barrier. The influence of both incident energy of wave packet and barrier parameters on tunneling time is discussed thoroughly. The connection between tunneling time and tunneling probability for the Gaussian barrier is established as well. In particular, we use the product of the variances of position and momentum to quantify the deformation of wave packet caused by tunneling process and demonstrate that different behaviors of the variances strongly support our oft-emphasized division of energy regimes.