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Physical Review Letters

Traversal Time for Tunneling

Μ. Büttiker, Rolf W. Landauer

Published Dec 6, 1982• 1,117 citations
Publisher

Abstract

One of several contradictory existing results for the time a tunneling particle interacts with its barrier is confirmed, by considering tunneling through a time-modulated barrier. At low modulation frequencies the traversing particle sees a static barrier. At high frequencies the particle tunnels through the time-averaged potential, but can do it inelastically, losing or gaining modulation quanta. The transition between the two regimes yields $\ensuremath{\int}\mathrm{dx}{[\frac{m}{2(V\ensuremath{-}E)}]}^{\frac{1}{2}}$ for the traversal time.

DOI
10.1103/physrevlett.49.1739

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One of several contradictory existing results for the time a tunneling particle interacts with its barrier is confirmed, by considering tunneling through a time-modulated barrier. At low modulation frequencies the traversing particle sees a static barrier. At high frequencies the particle tunnels through the time-averaged potential, but can do it inelastically, losing or gaining modulation quanta. The transition between the two regimes yields \ensuremath∫dx[m2(V\ensuremath−E)]12\ensuremath{\int}\mathrm{dx}{[\frac{m}{2(V\ensuremath{-}E)}]}^{\frac{1}{2}}\ensuremath∫dx[2(V\ensuremath−E)m​]21​ for the traversal time.
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.
The time required for decay of a metastable state, through tunneling, is a well settled subject. During this lifetime of the initial state, the particle oscillates within its initial well. By contrast, questions about the duration of the actual tunneling process, when it finally occurs, have led to widely divergent answers. In recent years, Jonson, Stevens, and we, have, through independent approaches, pointed out that there is an effective barrier traversal velocity obtained by dividing the magnitude of the imaginary momentum, under the barrier, by the particle mass. Here we present a fourth approach to this earlier answer, by considering a time modulated stream incident on the barrier. At low modulation frequencies the transmitted beam reproduces the incident modulation without lag, and without change in modulation depth. As the modulation frequency is increased we eventually depart from this simple behavior, and that is taken as a measure of the transversal time. We also provide some of the details omitted in our earlier analysis.
The frequency-dependent transmission coefficient for a particle tunneling through a time-dependent barrier is calculated taking into account inelastic and virtual processes. We consider a barrier of arbitrary shape with a time-dependent modulation of single frequency ω. For small modulation amplitudes V 1 , our result is exact up to order V 1 2 . The result is obtained in the low transmission limit and is consistent with the exact calculation for a rectangular barrier. We find that the virtual processes can make a significant contribution to the tunneling current, especially at low ω. In the static approximation ωτ << 1, where τ is the barrier transit time, the contribution of the second order virtual process is equal to one half of the tunneling current from the first order inelastic processes.
The tunneling time of a particle through a given barrier is commonly defined in terms of “internal clocks” that effectively measure the interaction time with internal degrees of freedom of the barrier. It is known that this definition of the time scale for tunneling is not unique in the sense that it depends on the clock used to define it. For the case of resonance tunneling, a particular choice that in the limit of a high/broad square barrier yields the original result of Büttiker and Landauer ( Phys. Rev. Lett. 1982, 49, 1739) is correlated to the lifetime of the resonance state. This is illustrated for analytically solvable one-dimensional double barrier models and for a realistic model of electron tunneling through a static water barrier. The latter calculation constitutes a novel application of this concept to a 3-dimensional model, and the observed structure in the energy dependence of the computed traversal time reflects the existence of transient tunneling resonances associated with instantaneous water structures. These models, characterized by the existence of shape resonances in the barrier, make it possible to examine different internal clocks that were proposed for measuring tunneling times in situations where a “clock independent” intrinsic time scale (the resonance lifetime) for the tunneling time exists. It is argued that this time may be used in order to estimate the relative importance of dynamical barrier processes that affect the tunneling probability.
Numerical simulations of quantum tunneling with time-dependent barriers show that there is a resonance, with a marked increase in the transmission coefficient. For a raised cosine potential, and for low energies with square barriers, the resonance occurs when a modulation quantum can take a tunneling particle to the top of the barrier. For energies near the top of a square barrier the resonance may be understood by hypothesizing that a tunneling particle may travel from end to end of the barrier until it is ultimately either transmitted or reflected.
Quantum particles interacting with potential barriers are ubiquitous in physics, and the question of how much time they spend inside classically forbidden regions has attracted interest for many decades. Recent developments of new experimental techniques revived the issue and ignited a debate with often contradictory results. This motivates the present study of an exactly solvable model for quantum tunneling induced by a strong field. We show that the tunneling dynamics can depart significantly from the scenario in which the barrier-traversal time is zero or very small. However, our findings do not support the idea of a well-defined tunneling time either. Our numerically exact results should help in finding a consensus about this fundamental problem.