Büttiker–Landauer traversal times in the radial direction of single and double barriers with cylindrical geometry are studied by modulating the incident wave and the barriers, respectively. The traversal times obtained by modulating the incident wave oscillate as a function of the incident wave energy. The oscillation is negligibly small in the traversal times obtained by modulating the barriers. The coherence of particle waves (Hankel functions) in cylindrical geometry is responsible for the oscillation.
Büttiker–Landauer traversal times in the radial direction of single and double barriers with cylindrical geometry are studied by modulating the incident wave and the barriers, respectively. The traversal times obtained by modulating the incident wave oscillate as a function of the incident wave energy. The oscillation is negligibly small in the traversal times obtained by modulating the barriers. The coherence of particle waves (Hankel functions) in cylindrical geometry is responsible for the oscillation.
The semiclassical expression for barrier traversal time results from a variety of theoretical procedures, but it appears that an opaque barrier is required in each derivation. Following Büttiker and Landauer [IBM J. Res. Dev. 30, 451 (1986)], we use two interfering plane waves as the incident wave and determine the traversal time by attributing differences in the transmitted and incident waves to the ratio of dispersion in the transit time to the modulation period. Calculations are made for rectangular barriers with different sizes. There is essentially agreement with Büttiker and Landauer for opaque barriers which they considered, but the present solution does not diverge for energies near the top of the barrier.
The authors analyse tunnelling through time-modulated barriers. The numerical results for the transmitted sideband intensities, obtained with wave-packet simulations, are in quantitative agreement with the analytical results of Buttiker and Landauer (1982, 1985). For opaque barriers the asymmetry of the sideband intensities can be characterized with a single frequency whose inverse Buttiker and Landauer interpret as the traversal time. This simple behaviour does not hold in general, and the authors discuss some of the difficulties one meets in an attempt to generalise the one-parameter description to arbitrary barriers. Furthermore, they examine analytically the transmitted wavefunction in the adiabatic limit, and show that the related density and current involve both the dwell time and the Buttiker time, defined in the context of a Larmor clock. This suggests the possibility that experiments probing different aspects of the transmitted wavefunction may yield different, but complementary, information about the tunnelling process.
The oscillating-barrier model was used by B\"uttiker and Landauer to determine a ``traversal time for tunneling.'' The model sets a timescale but is not the physically measured flight time of a wave packet scattered on the oscillating-barrier potential. In this paper we show that the flight time in the limit of a narrow-in-momentum wave packet is given by the reflected phase time associated with the various branches of the scattered particle. This is but another example which establishes that tunneling flight times are a reflection of the Wigner phase times. As such, the oscillating-barrier model does not add any new information about tunneling flight times which has not been elucidated previously using static barrier models.
The time that a transmitted particle interacts with a tunneling barrier is evaluated in one of the principal existing approaches by considering a small oscillatory perturbation added to the static barrier. The modulation frequency, at which a serious deviation from quasistatic behavior appears, is indicative of the traversal time. This approach is extended here, to a long region including the barrier, and utilizing a modulation potential which is spatially uniform throughout this region, except in the incident region ahead of the actual barrier, where it is made sufficiently smooth to avoid reflections. The resulting traversal time is \ensuremath{\Elzxh}\ensuremath{\partial} ln[\ensuremath{\partial}t/\ensuremath{\partial}E]/\ensuremath{\partial}E, where the complex transmission coefficient t(E) is defined as the ratio of the wave function at the far end to that at the point of incidence. This answer applies to arbitrary potentials and to all energies, including both propagation and tunneling in a single framework. This result is closely related to, but not identical to, that obtained by another earlier approach in which the potential is fixed, but the incident amplitude is modulated.
In the Büttiker-Landauer perturbation approach to electron tunnelling, through a time-modulated rectilinear potential barrier, the Tien-Gordon identity was invoked, together with its infinite energy spectrum. Here, an exact treatment is presented which is based on the temporal wave-function matching procedure, that led to a finite energy spectrum. In seeking the condition governing the time evolution of the tunnelling process, the Euler formula provided the crucial ingredient for time quantization, which discretised the continuous time in the oscillating barrier potential and energy harmonic equations. As a result, a finite system of inelastic scattering channels was created. When an electron entered the elastic channel, it was scattered, instantaneously, into finite neighbouring energy-level scattering channels, by absorption (emission) of photon energy from (to) the oscillating field, during the transit period across the dynamic barrier. The absorption and emission times of barrier traversal, T_+ and T_-, respectively, were derived for the low and high frequency regimes of the barrier oscillations. Calculations revealed that in the low (high) frequency situation T_+= T_- (T_+< T_-).
We present a proposal for the estimation of B\"uttiker-Landauer traversal time based on the visibility of transmission current. We analyze the tunneling phenomena with a time-dependent potential and obtain the time-dependent transmission current. We found that the visibility was directly connected to the traversal time. Furthermore, this result is valid not only for rectangular potential barrier but also for general form of potential to which the WKB approximation is applicable. We compared these results with the numerical values obtained from the simulation of Nelson's quantum mechanics. Both of them fit together and it shows that our method is very effective in measuring experimentally the traversal time.