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Physical Review Letters

Delay Time and the Hartman Effect in Quantum Tunneling

Herbert G. Winful

Published Dec 23, 2003• 234 citations
Publisher

Abstract

A general relation between the group delay and the dwell time is derived for quantum tunneling. It is shown that the group delay is equal to the dwell time plus a self-interference delay. The Hartman effect in quantum tunneling is explained on the basis of saturation of the integrated probability density (or number of particles) under the barrier.

DOI
10.1103/physrevlett.91.260401

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Tunneling time,as a fundamental concept in tunneling effect,plays an important basic concept to understand quantum theory.In a current concept,dwell time introduces some effects,which are hard to be explained,such as the Hartman effect.After carefully studying on Larmor time and dwell time,we explore the reason could be flux density.Considering the tunneling as an stable solution,we replace the incident flux with stable flux.By which,one reasonable time is obtained and the tunneling time is also equivalent to Larmor time divided by transmission probability.

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A general relation between the group delay and the dwell time is derived for quantum tunneling. It is shown that the group delay is equal to the dwell time plus a self-interference delay. The Hartman effect in quantum tunneling is explained on the basis of saturation of the integrated probability density (or number of particles) under the barrier.
I show that the MacColl-Hartman effect, namely, the saturation of the group delay time of sub-barrier quantum tunneling as a function of the barrier width, comes from the saturating behavior of a more fundamental concept - the phase of the stationary wave function. The explanation of saturation is given based on the decomposition of the stationary wave function into the spectrum of wave numbers and formulation of the initial condition for the direction of propagation of the incident matter wave. It is also shown that the saturation plateau of MacColl and Hartman actually doesn't continue indefinitely, but has a finite length. After the plateau, the sub-barrier tunneling time monotonically increases with increasing width of the potential, and this applies both to the maximum's of the wave packet and to the average tunneling time.
A general relation between group delay and stored electric and magnetic energies is presented for two-port networks. It generalizes the results of Dicke to situations where electric and magnetic stored energies differ. The general result is applied to tunneling evanescent waves in cutoff waveguides. It is shown explicitly that the group delay is equal to the dwell time plus a self-interference delay which is proportional to the net reactive stored energy. The Hartman effect, the saturation of group delay with length in cutoff waveguides, is explained on the basis of saturation of stored energy with guide length. It is pointed out that the anomalously short delays observed in tunneling experiments are not propagation delays and should not be associated with superluminal velocities. A strictly luminal energy velocity is derived and a method is suggested for the measurement of dwell time and energy velocity.
Tunneling time,as a fundamental concept in tunneling effect,plays an important basic concept to understand quantum theory.In a current concept,dwell time introduces some effects,which are hard to be explained,such as the Hartman effect.After carefully studying on Larmor time and dwell time,we explore the reason could be flux density.Considering the tunneling as an stable solution,we replace the incident flux with stable flux.By which,one reasonable time is obtained and the tunneling time is also equivalent to Larmor time divided by transmission probability.