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Physical Review A

Barrier penetration by wave packets and the tunneling times

M. S. Marinov, Bilha Segev

Published May 1, 1997• 22 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.55.3580

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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.
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.
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.
Localized wave-packet tunneling through a one-dimensional potential barrier is investigated with respect to the distribution of tunneling times. Specifically, we construct a quantity--- termed the ``tunneled flux''---which has properties of a tunneling-time probability distribution. The tunneled flux is determined completely in terms of the time evolution of an initially localized wave packet. The tunneled flux and the corresponding tunneling-time distributions are investigated analytically via a semiclassical approximation. Additionally, numerical studies are performed with both the semiclassical and quantal versions of the tunneled flux. The semiclassical calculation qualitatively reproduces numerical tunneled fluxes, converging quantitatively as \ensuremath{\Elzxh}\ensuremath{\rightarrow}0. It also provides a wealth of insight into the nature of tunneling processes. Our principle conclusions are as follows: (i) There is an essential disparity between the time- and energy-domain pictures of wave-packet tunneling; (ii) the tunneling-time distribution can be modeled as an exponential of a skewed Lorentzian function, with a width governed primarily by the phase-space asymmetry of the initial wave packet; and (iii) tunneling is faster than simple classical motion from one side of the barrier to another, even assuming an instantaneous transit between the turning points. Effectively, the barrier acts as a filter for the high-momentum components of the initial wave packet.
Dispersion associated with the complex band structure of a periodic potential can be used to explain single-electron wave packet tunneling through a finite-sized tunnel barrier. The imaginary k-space dispersion in a tunnel barrier may be viewed as a filter that controls the arrival time, shape, and group velocity of a transmitted wave packet. This k-space filter can be designed to increase or decrease the group velocity of the transmitted pulse and a transmitted peak-pulse probability arrival time can be engineered to occur before or after that of an electron wave packet propagating in the absence of the tunnel barrier.
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.
The transmission of wave packets through barriers by tunneling is studied in detail by the method of quantum molecular dynamics. The distribution of the arrival times of a tunneling packet in front of and behind a barrier and the momentum distribution function of the packet are calculated. The average position and average momentum of the packet and their spread are investigated. It is found that below the barrier a part of the packet is reflected, and a Gaussian barrier increases the average momentum of the transmitted packet and its spread in momentum space.