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Physical Review Letters

Path-integral solution of the telegrapher equation: An application to the tunneling time determination

Daniela Mugnai, Anedio Ranfagni, R. Ruggeri, A. Agresti

Published Jan 20, 1992• 55 citations
Publisher

Abstract

Delay times relative to a beat envelope are deduced from a path-integral solution of the telegrapher equation, analytically continued to imaginary time when the square of the effective velocity becomes negative. The results are approximately described by a semiclassical model translated in frequency from the nominal to an effective cutoff, thus improving the agreement with experimental results.

DOI
10.1103/physrevlett.68.259

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Delay times relative to a beat envelope are deduced from a path-integral solution of the telegrapher equation, analytically continued to imaginary time when the square of the effective velocity becomes negative. The results are approximately described by a semiclassical model translated in frequency from the nominal to an effective cutoff, thus improving the agreement with experimental results.
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.
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.
It is widely believed that the WKB approximation to quantum barrier penetration is described in the path-integral formalism by a saddle-point approximation based on an expansion about an imaginary-time semiclassical path. This article examines the relationship between tunneling amplitudes and imaginary-time paths. It is shown, using a simple example, that for some tunneling amplitudes the dominant semiclassical path cannot be chosen to have purely imaginary time if all other variables are chosen to be real.
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.