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

Kac’s solution of the telegrapher’s equation for tunneling time analysis: An application of the wavelet formalism

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

Published Aug 1, 1994• 13 citations
Publisher

Abstract

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.

DOI
10.1103/physreve.50.790

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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 ${\mathit{t}}_{\mathrm{pack}}^{\mathit{T}}$ 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, ${\mathit{t}}_{\mathrm{pack}}^{\mathit{T}}$ is complex-valued. It is shown that Re${\mathit{t}}_{\mathrm{pack}}^{\mathit{T}}$ reduces to the standard WPS result, ${\mathit{t}}_{\mathrm{phase}}^{\mathit{T}}$, for conditions similar to those employed in the conventional WPS analysis. Similarly, Im${\mathit{t}}_{\mathrm{pack}}^{\mathit{T}}$ is shown to contain information about the energy dependence of the transmission probability. Under semiclassical conditions, Im${\mathit{t}}_{\mathrm{pack}}^{\mathit{T}}$ 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.

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Path-Integral Resolution of the Tunneling-Time Controversy

Henrique Coelho Barbosa

The time a quantum particle spends traversing a classically forbidden region has resisted un-ambiguous definition since the earliest analyses of wave-packet tunneling, and the proliferation ofmutually incompatible “clocks”—the Wigner phase time, the Büttiker–Landauer traversal time, theLarmor time, and the path-integral complex time among them—has kept the question open fordecades. The controversy has been sharpened, not settled, by the advent of attosecond angular-streaking metrology, in which competing analyses of tunnel ionization continue to disagree over theexistence of a finite delay. Here we show that these rival definitions are not alternative answersto a single question but distinct cumulants of one traversal-time quasi-distribution, generated bythe exact rectangular-barrier transmission amplitude when the latter is promoted to a cumulantgenerating functional through a barrier-height counting field. The first cumulant unifies the Wignerphase time and the Larmor/Büttiker–Landauer time as the real and imaginary parts of a singlecomplex traversal time, recovering the celebrated Hartman saturation as the width-independentvalue ℏ/V0 of its real part. The second cumulant—which, to our knowledge, has not previouslybeen enlisted as a tunneling-time observable—yields a traversal-time variance that entirely evadesthe Hartman saturation and instead grows linearly with the barrier width d. The standard devia-tion of the tunneling time therefore scales as στ ∝ √d: rather than becoming superluminally sharpfor thick barriers, the tunneling time grows progressively more uncertain, in direct opposition tothe naive superluminal reading of the Hartman effect. This parameter-free scaling, independentof the O(1) coefficient that fixes its magnitude, constitutes a falsifiable prediction: attosecond an-gular streaking, which resolves the full arrival-time distribution of tunneled electrons rather thanmerely its centroid, should register a temporal spread broadening as √d even where the mean delaysaturates. We thereby recast a nearly century-old semantic dispute as a concrete experimentalquestion.

•Jul 7, 2026•0 citations
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.
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.
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.
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 time a quantum particle spends traversing a classically forbidden region has resisted un-ambiguous definition since the earliest analyses of wave-packet tunneling, and the proliferation ofmutually incompatible “clocks”—the Wigner phase time, the Büttiker–Landauer traversal time, theLarmor time, and the path-integral complex time among them—has kept the question open fordecades. The controversy has been sharpened, not settled, by the advent of attosecond angular-streaking metrology, in which competing analyses of tunnel ionization continue to disagree over theexistence of a finite delay. Here we show that these rival definitions are not alternative answersto a single question but distinct cumulants of one traversal-time quasi-distribution, generated bythe exact rectangular-barrier transmission amplitude when the latter is promoted to a cumulantgenerating functional through a barrier-height counting field. The first cumulant unifies the Wignerphase time and the Larmor/Büttiker–Landauer time as the real and imaginary parts of a singlecomplex traversal time, recovering the celebrated Hartman saturation as the width-independentvalue ℏ/V0 of its real part. The second cumulant—which, to our knowledge, has not previouslybeen enlisted as a tunneling-time observable—yields a traversal-time variance that entirely evadesthe Hartman saturation and instead grows linearly with the barrier width d. The standard devia-tion of the tunneling time therefore scales as στ ∝ √d: rather than becoming superluminally sharpfor thick barriers, the tunneling time grows progressively more uncertain, in direct opposition tothe naive superluminal reading of the Hartman effect. This parameter-free scaling, independentof the O(1) coefficient that fixes its magnitude, constitutes a falsifiable prediction: attosecond an-gular streaking, which resolves the full arrival-time distribution of tunneled electrons rather thanmerely its centroid, should register a temporal spread broadening as √d even where the mean delaysaturates. We thereby recast a nearly century-old semantic dispute as a concrete experimentalquestion.