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International Conference on Mathematical Methods in Electromagnetic Theory

Tunneling of rectangular wave packages through resonant quantum systems

N. A. Ivanov, V. V. Skalozub

Published Aug 1, 2014• 0 citations
Publisher

Abstract

Tunneling of wave packages through resonant quantum systems presented in an S-matrix form is investigated. An arbitrary incident wave package can be written as the superposition of the Gaussian-form pulses and for latter ones the exact analytic solution was obtained already. As a practically interesting case, the transition of the rectangular wave package through a one-resonant level system is considered. Using developed recently modified saddle-points method the transition of the wave package having a Gaussian and a rectangular form is investigated. The time delay, argument and amplitude of the outgoing wave are calculated. The dependence of tunneling time on the width and energy of resonance levels, and the momentum of incident wave-package are illustrated. The results are compared with the ones obtained for tunneling of the one Gaussian package. Possible applications are discussed.

DOI
10.1109/mmet.2014.6928744

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Tunneling of wave packages through resonant quantum systems presented in an S-matrix form is investigated. An arbitrary incident wave package can be written as the superposition of the Gaussian-form pulses and for latter ones the exact analytic solution was obtained already. As a practically interesting case, the transition of the rectangular wave package through a one-resonant level system is considered. Using developed recently modified saddle-points method the transition of the wave package having a Gaussian and a rectangular form is investigated. The time delay, argument and amplitude of the outgoing wave are calculated. The dependence of tunneling time on the width and energy of resonance levels, and the momentum of incident wave-package are illustrated. The results are compared with the ones obtained for tunneling of the one Gaussian package. Possible applications are discussed.
Tunneling of wave packages through resonant quantum systems presented in an S-matrix form is investigated. An arbitrary incident wave package can be written as the superposition of the Gaussian-form pulses and for latter ones the exact analytic solution was obtained already. As a practically interesting case, the transition of the rectangular wave package through the one-resonant level system is considered. Using developed recently modified saddle-points method the time-delay (Δt) of the wave pulses having Gaussian and rectangular form transmitted through a quantum system with resonance level is investigated. The time delay, argument and amplitude of the outgoing wave are calculated. The dependency Δt from width and energy of resonance levels, and the momentum of incident wave-package are illustrated. The results are compared with the ones obtained for tunneling of the one Gaussian package. Possible applications are discussed.
A modified saddle-point method is used to investigate the process of propagation of a wave packet through a quantum diode. A scattering matrix is constructed for the structure in question. The case of tunnelling of a packet with a Gaussian envelope through the diode is considered in detail. The time delay and the shape of the wave packet transmitted are calculated. The dependence of the delay time on the characteristics of the input packet and the internal characteristics of the quantum diode is studied. Possible applications of the results obtained are discussed.
In this paper, we discuss the tunneling time of a quantum particle through a rectangular barrier. The reflection and transmission times associated with the wave packets representing the particle are discussed. By using an initial Gaussian momentum distribution, we carry out a comparative analysis of the stationary phases of the incident, reflected, and transmitted wave packets leading to the reflection and transmission times $\ensuremath{\Delta}{t}_{R}$ and $\ensuremath{\Delta}{t}_{T},$ respectively. In the present treatment of this old and very known problem we take into account the deformations of the reflected and transmitted momentum distributions. These deformations produce a dependence of the reflection and transmission times on the location of the initial wave packet. In a parallel calculation, by numerically monitoring the time evolution of the system, we characterize a reflection and a transmission time. Such times agree with the ones obtained via the stationary phase method.
The reflection of wave packets from a quantum well bounded by rectangular potential barriers, and which has a tunneling transmission resonance, is discussed. Numeric solutions to Schrödinger’s time-dependent equation show that the shape of the probability function ψψ* for a reflected wave packet can be quite different from that of the incident packet. Either a symmetric double hump or an asymmetric shape are possible for the reflected ψψ*, depending on the sharpness of the resonance. These shapes are explained analytically in terms of the reflection coefficient.
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.