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Physical Review E

Real-time analysis of the telegrapher’s equation for tunneling processes

Anedio Ranfagni, Daniela Mugnai, Maria Angela Vitali

Published May 28, 2004• 5 citations
Publisher

Abstract

Based on a close analogy with an RLC circuit, a model for interpreting delay times in forbidden regimes (tunneling) is formulated, avoiding the analytical continuation into imaginary time. In this way, a reasonable description of experimental data, which were previously reported for a waveguide propagation below the cutoff frequency at approximately 9.5 GHz, is obtained.

DOI
10.1103/physreve.69.057603

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arXiv:quant-ph/0210008•Physical Review A•Sep 16, 2002•19 citations
Based on a close analogy with an RLC circuit, a model for interpreting delay times in forbidden regimes (tunneling) is formulated, avoiding the analytical continuation into imaginary time. In this way, a reasonable description of experimental data, which were previously reported for a waveguide propagation below the cutoff frequency at approximately 9.5 GHz, is obtained.
A plausible description of traversal time was given, both in classically allowed and forbidden regions, through a path-integral solution of the telegrapher's equation. This analysis was applied to a simulation based on microwave propagation in a waveguide considered as a one-dimensional system. An extension of the analysis has been performed 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. Subsequently, in tight analogy with a step signal in an electronic circuit (zero-dimensional system), we have searched for a simulation of traversal processes in real time, even for classically forbidden (tunneling) processes. First we have considered a finite series expansion of harmonic functions of the signal in the neighborhood of its rise, and applied the above mentioned procedure to each harmonic, implying analytical continuation in imaginary time and an arbitrary truncation in the range of the signal. Then, in order to avoid these shortcomings, we have considered a waveletlike description of the signal.
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.
Analytic solutions to the time-dependent Schr\"odinger equation for cutoff wave initial conditions are used to investigate the time evolution of the transmitted probability density for tunneling. For a broad range of values of the potential barrier opacity $\ensuremath{\alpha},$ we find that the probability density exhibits two evolving structures. One refers to the propagation of a forerunner related to a time domain resonance [Phys. Rev. A 64, 0121907 (2001)], while the other consists of a semiclassical propagating wave front. We find a regime where the forerunners are absent, corresponding to positive time delays, and show that this regime is characterized by opacities $\ensuremath{\alpha}<{\ensuremath{\alpha}}_{c}.$ The critical opacity ${\ensuremath{\alpha}}_{c}$ is derived from the analytical expression for the delay time, which reflects a link between transient effects in tunneling and the delay time.