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Journal of Physics A Mathematical and General

Relationship between dwell, transmission and reflection tunnelling times

Mário Goto, Hiromi Iwamoto, Verissimo Manoel de Aquino, Valdir C. Aguilera-Navarro, Donald H. Kobe

Published Mar 2, 2004• 23 citations
Publisher

Abstract

The transmission tunnelling time through a barrier is often defined as the probability that the particle is in the barrier divided by the transmission probability current density. The dwell time, the time the particle stays in the barrier, is commonly defined as the probability that the particle is in the barrier divided by the incident probability current density. We define the reflection tunnelling time in a similar way by dividing the probability that the particle is in the barrier by the reflection current density. By conservation of probability, the reciprocal of the dwell time is equal to the sum of the reciprocals of the transmission and reflection times. This relation is illustrated by calculating the different tunnelling times for a rectangular barrier.

DOI
10.1088/0305-4470/37/11/005

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The transmission tunnelling time through a barrier is often defined as the probability that the particle is in the barrier divided by the transmission probability current density. The dwell time, the time the particle stays in the barrier, is commonly defined as the probability that the particle is in the barrier divided by the incident probability current density. We define the reflection tunnelling time in a similar way by dividing the probability that the particle is in the barrier by the reflection current density. By conservation of probability, the reciprocal of the dwell time is equal to the sum of the reciprocals of the transmission and reflection times. This relation is illustrated by calculating the different tunnelling times for a rectangular barrier.
As was shown in quant-ph/0405028, the state of a tunneling particle can be uniquely presented as a coherent superposition of two states to describe alternative sub-processes, transmission and reflection. In this paper, on the basis of the stationary wave functions for these sub-processes, we give new definitions of the dwell times for transmission and reflection. In the case of rectangular potential barriers the dwell times are obtained explicitly. In contrast with the well-known Büttiker's dwell-time, our dwell time for transmission increases exponentially, for the under-barrier tunneling, with increasing the barrier's width. By our approach the well-known Hartman effect is rather an artifact resulted from an improper interpretation of the wave-packet tunneling and experimental data, but not a real physical effect accompanying the tunneling phenomenon.
Tunneling phenomenon is one of the wondrous events that take place particularly in quantum mechanics and it has no counterpart in classical physics. We focus on its two important implications : tunneling time and tunneling probability. There has been no consensus on tunneling times among the tunneling time community for last seven decades. So far, most of the attempts to calculate tunneling times have apparently been centered on a simple rectangular barrier example. We implement the tunneling times with the most widely accepted approach, which is using physical Larmor clock, into a nonsymmetrical barrier and a double barrier. It results in different and unique insights into the usual understanding of tunneling time. Three characteristics tunneling time expressions are thoroughly studied in these quantum structures. The difference between dwell time and phase-delay time is once again approved. The unique characteristics of the phase-delay time, which was thought to be exactly similar to dwell time when we assign higher energies to the particles, is also revealed in the nonsymmetrical barrier that it should actually be regarded as a distinct feature from the dwell time. The singularity of reflection time in rectangular barrier is no longer valid in nonsymmetrical barrier. The finiteness of reflection time fairly shows that different quantum structures might bear some taboo breaking results. Since a double barrier involves much more complicated features and calculations, the calculation of tunneling times under this barrier is praiseworthy. The dwell time is the most crucial, troublesome and distinctive tunneling time element to look into for a double barrier. Besides the dwell and phase-delay time, we also calculate the three characteristic times as well as the reflection time for a double barrier. Another area that we study within the context of tunneling phenomenon is the tunneling probability. Recently, there has been ground shaking attempts in modifying the conventional WKB formulae. We follow one of the modified versions of WKB formulae in order to be able to calculate the tunneling probability through different quantum structures where conventional WKB formulae fail. These generalized WKB connection formulae are applied to some complex quantum potentials by seeking aid from the Green's function and the path integral approach. One of the biggest problems in quantum mechanics (visualization) is partially overcome through implementing the method of diagrammatic representations that make the life easier in understanding the events happening in the quantum world when we attempt to calculate the transmission and reflection probabilities. This part therefore has a pedagogical sense too in terms of interpreting the results in a diagrammatic way with the help of the path integral approach.
For a one-dimensional barrier of arbitrary shape, an expression of the dwell time of tunneling particles has been obtained by subdividing it into infinitesimal rectangular barrier elements and then summing the dwell times spent by particles inside the barrier elements, and the dwell times in a parabolic potential barrier were calculated. The results show that the dwell times are almost independent of the barrier width as it is greater than a characteristic width W c . This can be ascribed that in the width range beyond W c , the tunneling probabilities through the barrier elements are very small, thus the times taken to tunnel through them make little contribution to the summation of the dwell times in all the barrier elements.
Expressions for the transmission, reflection, and interference contributions to the dwell time for wave packet scattering are obtained by means of scattering theory projector operators that provide a unifying framework for various dwell time decompositions. These different dwell time resolutions are illustrated with the aid of a formal analogy between the two-level system and one-dimensional scattering. The complex "times" and Larmor times proposed by other authors are found as particular cases of the theory. Numerical calculations for a rectangular barrier are provided.
The desire to minimize the response time of tunneling devices has led to renewed interest in the unresolved fundamental question How much time, on average, does an incident electron that is ultimately transmitted spend inside a one-dimensional potential barrier? In addition to this transmission time ⊤t ·,there are two other times of interest: the reflection time ⊤r for electrons that are ultimately reflected and the dwell time TD, which is the average time spent inside the barrier regardless of the ultimate fate of the incident particle. An exact expression for TD and the time-modulated barrier, Feynman path, Larmor clock, and wave-packet evolution approaches for calculating ⊤t and ⊤r arc discussed. It is argued that none of these approaches is completely satisfactory because they all lead to transmission and reflection times that either are complex or, if real, do not satisfy: all of a few reasonable (but not universally accepted) criteria, such as consistency with the exact expression for the dwell time.
The proper basis for the calculation of transmission and reflection times for wave packets scattered off arbitrary tunneling structures in one dimension is considered. With packets narrow in wave-number space, we demonstrate that the classic phase times are indeed correct to lowest order. Explicit, general expressions for the leading correction terms for finite wave packets are given. The physics associated with these corrections is discussed. We also consider the dwell time, as it is currently defined, and derive a general relation between this dwell time and the phase times. This relation shows when the dwell time can and cannot be used. Finally, we discuss wave packets transmitted from narrow resonances, and derive an explicit, exact formula for the tunneling time with resonance transmission from a symmetric double barrier. Comparison with earlier approximate results is made.