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arXiv:cond-mat/0212527 · physica status solidi (b)

Transmission through a Quantum Dynamical Delta Barrier

Tobias Brandes, J. E. M. Robinson

Published Nov 1, 2002• 12 citations
arXivHTMLPDF

Abstract

We discuss electron scattering in a one-dimensional delta barrier potential with either a time-dependent coupling constant (classical model) or a coupling constant that is linear in a boson coordinate (quantum model). We find an exact continued fraction solution and Fano like resonances in the transmission coefficient. In the quantum model, energies for perfect transmission exist below the first sideband threshold.

DOI
10.1002/1521-3951(200211)234:1<378::aid-pssb378>3.0.co;2-#

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Journal of Physics A Mathematical and General•Jun 7, 1988•5 citations
We discuss electron scattering in a one-dimensional delta barrier potential with either a time-dependent coupling constant (classical model) or a coupling constant that is linear in a boson coordinate (quantum model). We find an exact continued fraction solution and Fano like resonances in the transmission coefficient. In the quantum model, energies for perfect transmission exist below the first sideband threshold.
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.
We study the dynamics of the classical and quantum mechanical scattering of a wave packet from an oscillating barrier. Our main focus is on the dependence of the transmission coefficient on the initial energy of the wave packet for a wide range of oscillation frequencies. The behavior of the quantum transmission coefficient is affected by tunneling phenomena, resonances, and kinematic effects emanating from the time dependence of the potential. We show that when kinematic effects dominate (mainly in intermediate frequencies), classical mechanics provides very good approximation of quantum results. In that frequency region, the classical and quantum transmission coefficients are in optimal agreement. Moreover, the transmission threshold (i.e., the energy above which the transmission coefficient becomes larger than a specific small threshold value) is found to exhibit a minimum. We also consider the form of the transmitted wave packet and we find that for low values of the frequency the incoming classical and quantum wave packet can be split into a train of well-separated coherent pulses, a phenomenon that admits purely classical kinematic interpretation.
Abstract In this paper, we propose a pedagogical presentation of the Lippmann–Schwinger equation as a powerful tool, so as to obtain important scattering information. In particular, we consider a one-dimensional system with a Schrödinger-type free Hamiltonian decorated with a sequence of N attractive Dirac delta interactions. We first write the Lippmann–Schwinger equation for the system and then solve it explicitly in terms of an N × N matrix. Then, we discuss the reflection and the transmission coefficients for an arbitrary number of centres and study the threshold anomaly for the N = 2 and N = 4 cases. We also study further features like the quantum metastable states and resonances, including their corresponding Gamow functions and virtual or antibound states. The use of the Lippmann–Schwinger equation simplifies our analysis enormously and gives exact results for an arbitrary number of Dirac delta potentials.
The analytic nature of the transmission coefficient for a δ-function barrier makes it a useful tool to examine a variety of technologically important applications, such as photoemission from semiconductors with an alkali coating, the examination of tunneling times for wave packets incident on a barrier, and for parameterizing tunneling through the narrow barrier of a normal-superconducting point contact. The analytic model of a δ-function barrier inside a confining well is extended to the finite height and width rectangular barrier (a delta-function sequence). Methods to exactly evaluate the eigenstates are given and their dependencies are examined. The time evolution of a superposition of the lowest eigenstates is considered for barriers having comparable Gamow tunneling factors so as to quantify the impact of barrier height and shape on time evolution in a simple and exact system and, therefore, serve as a proxy for tunneling time. Last, density profiles and associated quantum potentials are examined for coupled wells to show changes induced by weaker and wider barriers.
Scattering by a one-dimensional potential V=V(x) which is constant outside (-a, a) (a>0) is investigated. Exact expressions for the transmission and reflection coefficients are obtained in terms of f 0 (a), f 1 (a), f' O (a) and f' 1 (a), where f 0 and f 1 are the solutions in (-a, a) satisfying the boundary conditions f 0 (-a)=1, f 0 '(-a)=0, f 1 (-a)=0 and f 1 '(-a)=1. These expressions are used to show that the first-order WKB approximation conserves particles. Numerical results are obtained from these expressions for the transmission and reflection probabilities by (i) using the WKB formulae and (ii) solving the time-independent Schrodinger equation numerically in (-a, a). The time T for the centre of a wavepacket to traverse the interval (-a, a) is also obtained in terms of f 0 (a), f 1 (a), f 0 '(a) and f 1 '(a), and some of its properties studied. In particular, it is shown that T takes the classical form in the classical limit.