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Journal de Physique I

Tunneling with dissipation

G. Nimtz, Harald Spieker, Hanna M. Brodowsky

Published Oct 1, 1994• 28 citations
Publisher

Abstract

Dissipative tunneling was studied with electromagnetic waves. Experiments were carried out with undersized waveguides loaded with a lossy dielectric medium which corresponds to a complex potential barrier. With increasing dissipation the Hartman effect disappears, i.e., the traversal time becomes dependent on barrier length. The results are in agreement with a recent phase time approach on complex tunneling.

DOI
10.1051/jp1:1994194

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Dissipative tunneling was studied with electromagnetic waves. Experiments were carried out with undersized waveguides loaded with a lossy dielectric medium which corresponds to a complex potential barrier. With increasing dissipation the Hartman effect disappears, i.e., the traversal time becomes dependent on barrier length. The results are in agreement with a recent phase time approach on complex tunneling.
We study the phenomenon of one-dimensional non-resonant tunneling through two successive potential barriers, separated by an intermediate free region R, by analyzing the relevant solutions to the Schroedinger equation. We find that the total traversal time does not depend not only on the barrier widths (the so-called 'Hartman effect'), but also on the R width: so that the effective velocity in the region R, between the two barriers, can be regarded as infinite. This agrees with the results known from the corresponding waveguide experiments, which simulated the tunnelling experiment herein considered due to the formal identity between the Schroedinger and the Helmholtz equation.
In this paper we calculate the analytic expression of the phase time for the scattering of an electron off a complex square barrier. As is well known the (negative) imaginary part of the potential takes into account, phenomenologically, the absorption. We investigate the so-called Hartman-Fletcher effect, and find that it is suppressed by the presence of a (not negligible) imaginary potential. In fact, when a sufficiently large absorption is present, the asymptotical transmission speed is finite. Actually, the tunnelling time does increase linearly with the barrier width. A recent optical experiment seems to be in agreement with our theoretical previsions.
We study the phenomenon of one-dimensional non-resonant tunneling through two successive (opaque) potential barriers, separated by an intermediate free region , by analyzing the relevant solutions to the Schroedinger equation. We find that the total traversal time does not depend not only on the barrier widths (the so-called "Hartman effect"), but also on the width: so that the effective velocity in the region , between the two barriers, can be regarded as practically infinite. This agrees with the results known from the corresponding waveguide experiments, which simulated the tunneling experiment herein considered due to the known formal identity between the Schroedinger and the Helmholtz equation.
We consider a scattering process inside a dispersive and absorptive medium, and derive an analytical expression for the scattering-time parameter that characterizes the duration of this process. We consider a microwave wave packet that scatters from a dielectric inhomogeneity inside an otherwise homogeneous waveguide. The waveguide is filled with a dielectric medium with a finite inhomogeneity region in the direction of the wave-packet propagation. The permittivity of the entire medium is described by a complex function of the frequency. Hence both absorption and dispersion affect the propagation of the wave packet. We define the scattering-time parameter to be the difference in the ``average arrival time'' between the free and actual propagation. We show that this time difference (in the limit of a narrow-band wave packet) is given by a linear combination of the real and imaginary parts of the ``complex tunneling time.'' In the special case where there is no dissipation or attenuation outside the scatterer, we recover the well-known result that the scattering-time parameter is identical to the real part of the complex tunneling time. The contribution of the imaginary part of the complex tunneling time to the scattering-time parameter can, therefore, be attributed to absorption and attenuation outside the scatterer. We further show that the real and imaginary-time parameters satisfy dispersion relations. Causality in this process is briefly discussed.
Traversal time in tunneling effect for ultrasonic waves in tapered waveguides is derived considering its analogy with quantum and electromagnetic waves tunneling. If, as traversal time, the so called phase time is considered, the ultrasonic wavepacket shows the equivalent in acoustics of superluminality i.e. the derived velocity , crosses the limit of bulk transverse ultrasonic waves in the medium of the waveguide that is the equivalent of c in the quantum and electromagnetic cases. Graphs clearly illustrating this so called Hartman effect are obtained confirming the experimental results in the three different fields.
Recent papers suggest that tunneling wave packets traverse the region of allowed propagation between two potential barriers with superluminal group velocity and in a time independent of barrier separation. This phenomenon has been termed the "generalized Hartman effect" and extended to multiple barriers. Here we show that this delay time is not a transit time but a cavity lifetime. It does not imply superluminal velocity. Reported experimental verifications of this effect are reinterpreted.