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Journal of Physics A Mathematical and General

Tunnelling of narrow Gaussian packets through delta potentials

Mauro Antonio ANDREATA, V. V. Dodonov

Published Jan 28, 2004• 24 citations
Publisher

Abstract

We consider transmission and reflection of narrow Gaussian wave packets by delta potentials in the cases of constant and specific (inverse linear) time-dependent strength. Both transmitted and reflected packets exhibit some ‘squeezing’ in the momentum probability distributions. Several different definitions of the transmission time are introduced and compared.

DOI
10.1088/0305-4470/37/6/031

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We consider transmission and reflection of narrow Gaussian wave packets by delta potentials in the cases of constant and specific (inverse linear) time-dependent strength. Both transmitted and reflected packets exhibit some ‘squeezing’ in the momentum probability distributions. Several different definitions of the transmission time are introduced and compared.
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 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.
Quantum tunneling in one spatial dimension in the presence of time-dependent potentials is investigated theoretically. First, a general multichannel formulation of the problem is given for harmonic time-dependence. The case of an oscillatory delta-function potential of constant strength is discussed at length and specific numerical calculations are presented for the reflection and transmission coefficients. Then other exactly solvable time-dependent potentials are obtained by carrying out successive unitary transformations. As an example of this class, a delta-function potential with time-dependent strength and boundary conditions is studied. Finally tunneling time-delay for time-dependent potentials is formulated in the multichannel situation, and also the concept of first passage time for the decay of a wave packet confined to one well of a double well potential is generalized for the case of time-dependent barrier and boundary conditions.
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 We examine the tunnelling process of incident Hermite-Gaussian wavepackets which feature several nodes and peaks. An analytic formula for the transmitted wave function is developed which applies to arbitrary incident wave functions and potential barriers with known transmission amplitudes. This formula permits the accurate numerical calculation of the transmitted wave packet, even when the transmission probability is extremely small. For square potential barriers, the shape of the transmitted packet, its relationship to the shape of the incident packet, the momentum of the transmitted packet, and the duration of the tunnelling process are investigated. The shape of the reconstructed transmitted wave packet is also studied for the smooth repulsive Pöschl-Teller potential.