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Physical Review Letters

Unified Derivation of Tunneling Times from Decoherence Functionals

Norifumi Yamada

Published Oct 18, 2004• 71 citations
Publisher

Abstract

The four tunneling times, the Larmor time, the Büttiker-Landauer time, the Bohm-Wigner time, and the Pollak-Miller time, originally obtained from very different physical models, are derived in a unified manner from the Gell-Mann-Hartle decoherence functionals. The origin of the two types of derivatives in the expressions for these tunneling times is clarified at the level of Feynman paths.

DOI
10.1103/physrevlett.93.170401

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The four tunneling times, the Larmor time, the Büttiker-Landauer time, the Bohm-Wigner time, and the Pollak-Miller time, originally obtained from very different physical models, are derived in a unified manner from the Gell-Mann-Hartle decoherence functionals. The origin of the two types of derivatives in the expressions for these tunneling times is clarified at the level of Feynman paths.
In this paper we examine critically and in detail some existing definitions for the tunnelling times, namely: the phase-time; the centroid-based times; the Buttiker and Landauer times; the Larmor times; the complex (path-integral and Bohm) times; the dwell time, and finally the generalized (Olkhovsky and Recami) dwell time, by adding also some numerical evaluations. Then, we pass to examine the equivalence between quantum tunnelling and "photon tunnelling" (evanescent waves propagation), with particular attention to tunnelling with Superluminal group-velocities ("Hartman effect"). At last, in an Appendix, we add a bird-eye view of all the experimental sectors of physics in which Superluminal motions seem to appear.
We derive a general expression for the tunneling time in layered systems based on the Larmor-clock approach. We show that our results are equivalent to those obtained by the Feynman path-integral technique. We establish a relation between functional derivatives of the barrier potential and partial derivatives with respect to the incident energy. In application, we computed the tunneling time and the reflection time for a rectangular barrier, for a double rectangular barrier, and for a Gaussian potential.
In this review (in Italian) we critically and detaily examine various definitions existing in the literature for the tunnelling times: namely, the phase-time; the centroid-based times; the Buttiker and Landauer times; the Larmor times; the complex (path-integral and Bohm) times; the dwell time, and the generalized (Olkhovsky and Recami) dwell time, with some numerical evaluations. Then, we pass to examine the equivalence between quantum tunnelling and "photon tunnelling" (evanescent waves propagation), with particular attention to tunnellings with Superluminal group-velocities ("Hartman effect"). At last, the main (Cologne, Berkeley, Florence, Vienna) experimental data about Superluminal evanescent wave propagation are briefly reviewed.
We show the equivalence of the functions Gp(t){G}_{\mathrm{p}}(t)Gp​(t) and $|\ensuremath{\Psi}(d,t){|}^{2}$ for the ``passage time'' in tunneling. The former, obtained within the framework of the real-time Feynman histories approach to the tunneling time problem, uses the Gell-Mann and Hartle's decoherence functional, and the latter involves an exact analytical solution to the time-dependent Schr\"odinger equation for cutoff initial waves.
A simple solution is given to the tunneling time problem from the viewpoint of interfering alternatives. Using Gell-Mann and Hartle's decoherence functional for real time Feynman histories, it is shown that (i) the probability distribution of tunneling times cannot be defined for ``resident time'' type but (ii) the range of values of the time is definable. The range turns out to be so broad for opaque barriers that a unique tunneling time cannot be considered even in a loose sense.
Exact analytical solutions of the time-dependent Schrödinger equation with the initial condition of an incident cutoff wave are used to investigate the traversal time for tunneling. The probability density starts from a vanishing value along the tunneling and transmitted regions of the potential. At the barrier width it exhibits, at early times, a distribution of traversal times that typically has a peak τpτ_pτp​ and a width ΔτΔτΔτ. Numerical results for other tunneling times, as the phase-delay time, fall within ΔτΔτΔτ. The Büttiker traversal time is the closest to τpτ_pτp​. Our results resemble calculations based on Feynman paths if its noisy behaviour is ignored.