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Light Cone Conference (LC)

Quantum Tunneling Time in the Light Front

Alfredo Takashi Suzuki, Jorge Henrique de Oliveira Sales, Daykson N. Possidonio

Published May 25, 2020• 0 citations
Publisher

Abstract

In this paper we study the quantum tunneling time in the light-front using the technique of dwell time (or delay time). We introduce this method of calculation considering the delay time for the Schr\"odinger equation. Then, as we consider the relativistic Klein-Gordon equation, we face a problem, because the probability density for this case is not positive definite, and the delay time technique becomes inapplicable here. Other techniques have been devised to tackle this issue, but comparison of results becomes difficult because methods used are different. Therefore, there is a discrepancy in the tunneling time assessment for the low-energy non relativistic and relativistic case. By treating the Klein-Gordon equation in the light-front, we show that this discrepancy disappears.

DOI
10.22323/1.374.0091

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In this paper we study the quantum tunneling time in the light-front using the technique of dwell time (or delay time). We introduce this method of calculation considering the delay time for the Schr"odinger equation. Then, as we consider the relativistic Klein-Gordon equation, we face a problem, because the probability density for this case is not positive definite, and the delay time technique becomes inapplicable here. Other techniques have been devised to tackle this issue, but comparison of results becomes difficult because methods used are different. Therefore, there is a discrepancy in the tunneling time assessment for the low-energy non relativistic and relativistic case. By treating the Klein-Gordon equation in the light-front, we show that this discrepancy disappears.
A traversal time that has no problem of superluminality was advanced for particles to tunnel through potential barriers in the non-relativistic quantum theory in a previous paper by C.-F. Li and Q. Wang, Physica B 296 (2001) 356. This time is generalized in this paper to Dirac's relativistic quantum theory. Both evanescent and propagating cases are considered. It is shown that the traversal time in the evanescent case has much the same properties as in the non-relativistic quantum theory and thus has no problem of superluminality. It also gets rid of the problem of superluminality in the propagating case. Comparisons with the dwell time, the group delay, and the velocity of monochromatic front are also made.
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.
Using a time-dependent scattering theory for light the authors obtain expressions for the time that light spends in a finite dielectric medium surrounded by vacuum, the so-called dwell time. Given an incoming wavepacket, they focus upon the light scattered in an arbitrary direction. In view of the similarity between Maxwell's equations and the Schrodinger equation, some useful results are derived for the case of Schrodinger potential scattering as well. They show that the dwell time of a Schrodinger particle is a derivative of the phase shifts with respect to the potential, rather than to energy as is the case for the phase-delay time. They indicate the relation to absorption arguments. Because the potential for classical waves is energy dependent, derivatives of the phase shift with respect to potential enter into the phase-delay time for classical waves.
Various approaches have been used to calculate the quantum-mechanical tunneling time through potential barriers including the phase-delay method first introduced by Bohm and Wigner, Buttiker's analysis of the Larmor clock and its generalization, the dwell time of Smith, and numerical studies of wavepacket propagation through potential barriers among others. Most of those previous estimates have only dealt with the tunneling time through simple obstacles, including delta-potential scatterers and simple rectangular barriers under zero bias condition. Even for these simple cases, the agreement among the various estimates is far from being satisfactory. In this manuscript, the transmission line technique is employed to solve the time-dependent Schrodinger equation and an expression of the quantum-mechanical tunneling time is derived for an arbitrary potential profile under non zero bias condition. An exact analytical expression of the tunneling time through a rectangular barrier is derived and shown to be identical to the one obtained recently by Spiller et. al. using the Bohm's quantum potential approach. The tunneling time through resonant tunneling structures under zero bias is also calculated and is shown to be minimum at the quasi-boundstate energy. Finally, the quantum-mechanical tunneling time through the emitter-base junction of a typical heterojunction bipolar transistor is shown to be larger than its semiclassical counterpart.