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arXiv:quant-ph/0004094 · Physical Review A

Estimation of the Büttiker-Landauer traversal time based on the visibility of transmission current

Koh’ichiro Hara, Ichiro Ohba

Published Aug 14, 2000• 10 citations
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Abstract

We present a proposal for the estimation of B\"uttiker-Landauer traversal time based on the visibility of transmission current. We analyze the tunneling phenomena with a time-dependent potential and obtain the time-dependent transmission current. We found that the visibility was directly connected to the traversal time. Furthermore, this result is valid not only for rectangular potential barrier but also for general form of potential to which the WKB approximation is applicable. We compared these results with the numerical values obtained from the simulation of Nelson's quantum mechanics. Both of them fit together and it shows that our method is very effective in measuring experimentally the traversal time.

DOI
10.1103/physreva.62.032104

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•Dec 1, 2017•0 citations
We present a proposal for the estimation of B"uttiker-Landauer traversal time based on the visibility of transmission current. We analyze the tunneling phenomena with a time-dependent potential and obtain the time-dependent transmission current. We found that the visibility was directly connected to the traversal time. Furthermore, this result is valid not only for rectangular potential barrier but also for general form of potential to which the WKB approximation is applicable. We compared these results with the numerical values obtained from the simulation of Nelson's quantum mechanics. Both of them fit together and it shows that our method is very effective in measuring experimentally the traversal time.
The semiclassical expression for barrier traversal time results from a variety of theoretical procedures, but it appears that an opaque barrier is required in each derivation. Following Büttiker and Landauer [IBM J. Res. Dev. 30, 451 (1986)], we use two interfering plane waves as the incident wave and determine the traversal time by attributing differences in the transmitted and incident waves to the ratio of dispersion in the transit time to the modulation period. Calculations are made for rectangular barriers with different sizes. There is essentially agreement with Büttiker and Landauer for opaque barriers which they considered, but the present solution does not diverge for energies near the top of the barrier.
The authors analyse tunnelling through time-modulated barriers. The numerical results for the transmitted sideband intensities, obtained with wave-packet simulations, are in quantitative agreement with the analytical results of Buttiker and Landauer (1982, 1985). For opaque barriers the asymmetry of the sideband intensities can be characterized with a single frequency whose inverse Buttiker and Landauer interpret as the traversal time. This simple behaviour does not hold in general, and the authors discuss some of the difficulties one meets in an attempt to generalise the one-parameter description to arbitrary barriers. Furthermore, they examine analytically the transmitted wavefunction in the adiabatic limit, and show that the related density and current involve both the dwell time and the Buttiker time, defined in the context of a Larmor clock. This suggests the possibility that experiments probing different aspects of the transmitted wavefunction may yield different, but complementary, information about the tunnelling process.
The oscillating-barrier model was used by B\"uttiker and Landauer to determine a ``traversal time for tunneling.'' The model sets a timescale but is not the physically measured flight time of a wave packet scattered on the oscillating-barrier potential. In this paper we show that the flight time in the limit of a narrow-in-momentum wave packet is given by the reflected phase time associated with the various branches of the scattered particle. This is but another example which establishes that tunneling flight times are a reflection of the Wigner phase times. As such, the oscillating-barrier model does not add any new information about tunneling flight times which has not been elucidated previously using static barrier models.
The question of how long a particle takes to pass through a potential barrier is still a controversial topic in quantum mechanics. One of the main theoretical problems in obtaining estimates for measurable times is the fact that several previously defined time operators, which remained within the borders of standard quantum mechanics, present some kind of pathology. Recently, a time operator acting on an additional Hilbert space has been shown to support both Hermiticity and canonical relation with an energy observable. The theory is built in a framework that treats space and time as symmetrically as possible, in the nonrelativistic regime. In this work, we use this formalism to derive a closed analytical expression for the traversal time of a quantum particle impinging on a rectangular potential barrier. To do this, first we discuss the meaning of the expectation values in the additional Hilbert space and, then, we apply it to traversal times. We test our formalism in the specific experimental scenario of a realization by Ranfagni et al. [A. Ranfagni et al., Appl. Phys. Lett. 58, 774 (1991)]. The proposed approach displays a better performance in comparison with the B\"uttiker-Landauer and the phase-time approximations.
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.
The question of how long a particle takes to pass through a potential barrier is still a controversial topic in quantum mechanics. Arguably, the main theoretical problem in obtaining estimates for measurable times is the fact that previously defined time operators, that remained within the borders of standard quantum mechanics, present some kind of pathology. Recently, a time operator acting on an additional Hilbert space has been shown to support both Hermiticity and canonical relation with an energy observable. The theory is built in a framework which treats space and time as symmetrically as possible, in the nonrelativistic regime. In this work, we use this formalism to derive a closed analytic expression for the traversal time of a quantum particle impinging on a constant-potential barrier. We test our theory in the specific experimental scenario of a realization by Rafagni et al [Appl. Phys. Lett. {\bf 58}, 774 (1991)]. The proposed approach displays a much better performance in comparison with the Buttiker-Landauer and the phase-time approximations.