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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Anomalous delay in wave propagation and tunneling: A transition-elements analysis of the traversal time

A. Agresti, Paolo Levi Sandri, C. Ranfagni, Anedio Ranfagni, R. Ruggeri

Published Dec 30, 2002• 12 citations
Publisher

Abstract

An alternative model for near-field propagation and optical tunneling is proposed following the lines of the path-integral method developed by Feynman, and in particular by using a transition-elements analysis. Such a model was able to account for the frequency dependency of delay-time results of an experiment involving microwave propagation in the near field using two horn antennas [A. Ranfagni et al., Phys. Rev. E 66, 036111 (2002)]. Furthermore, this approach is also capable of interpreting delay-time results as a function of the barrier width in a frustrated total internal reflection experiment performed at the microwave scale and in the optical region.

DOI
10.1103/physreve.66.067604

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An alternative model for near-field propagation and optical tunneling is proposed following the lines of the path-integral method developed by Feynman, and in particular by using a transition-elements analysis. Such a model was able to account for the frequency dependency of delay-time results of an experiment involving microwave propagation in the near field using two horn antennas [A. Ranfagni et al., Phys. Rev. E 66, 036111 (2002)]. Furthermore, this approach is also capable of interpreting delay-time results as a function of the barrier width in a frustrated total internal reflection experiment performed at the microwave scale and in the optical region.
The wave-packet simulation (WPS) method for calculating the time a tunneling particle spends inside a one-dimensional potential barrier is reexamined using the Feynman path-integral technique. Following earlier work by Sokolovski and Baskin [Phys. Rev. A 36, 4604 (1987)], the tunneling (or traversal) time tpackT{\mathit{t}}_{\mathrm{pack}}^{\mathit{T}}tpackT​ is defined as a matrix element of a classical nonlocal functional between two states that represent the initial and transmitted wave packets. These states do not lie on the same orbit in Hilbert space; as a result, tpackT{\mathit{t}}_{\mathrm{pack}}^{\mathit{T}}tpackT​ is complex-valued. It is shown that RetpackT{\mathit{t}}_{\mathrm{pack}}^{\mathit{T}}tpackT​ reduces to the standard WPS result, tphaseT{\mathit{t}}_{\mathrm{phase}}^{\mathit{T}}tphaseT​, for conditions similar to those employed in the conventional WPS analysis. Similarly, ImtpackT{\mathit{t}}_{\mathrm{pack}}^{\mathit{T}}tpackT​ is shown to contain information about the energy dependence of the transmission probability. Under semiclassical conditions, ImtpackT{\mathit{t}}_{\mathrm{pack}}^{\mathit{T}}tpackT​ reduces to the well-known Wentzel-Kramers-Brillouin expression for the tunneling time. It is shown there are different definitions for the traversal time of a classical moving object, whose size is comparable to the width of the region of interest. In the quantum case, these different definitions correspond to different ways of analyzing the WPS experiment. The path-integral approach demonstrates that the tunneling-time problem is one of understanding the physical significance of complex-valued off-orbit matrix elements of an operator or functional. The physical content of complex-valued tunneling times is discussed. It is emphasized that the use of complex tunneling times includes real-time approaches as a special case. Nevertheless, there is a limitation in the description of tunneling experiments using tunneling times, whether real or complex. The path-integral approach does not supply a universal traversal time, analogous to a classical time, that can be used in quantum situations. It is demonstrated that the often expressed hope of finding a well-defined and universal real tunneling time is erroneous.
Delay time results obtained in microwave experiments at frequencies above and below the cutoff frequency of different waveguide sections are interpreted on the basis of wave propagation in the presence of dissipative effects. Kac's original suggestion was the starting point for the formulation of a stochastic model, which has now been substantially improved, also in relation to the transition-elements theory of Feynman-Hibbs. In this way, an approach to the problem is provided, which is completely distinct from the ones formulated elsewhere.
A path-integral solution of the telegrapher's equation has been demonstrated to give a plausible description of traversal time, for motions either above or below the top of the barrier, in connection with microwave-simulation experiments [see Mugnai, Ranfagni, Ruggeri, and Agresti, Phys. Rev. Lett. 68, 259 (1992)]. This Brief Report reports an extension of the analysis in order to compare the traversal, or delay, time results relative to a beat envelope signal with those as deduced from the distribution function of the randomized time and its analytical continuation in imaginary time.