Tunnelling through Two Barriers

Recent studies of the tunnelling through two opaque barriers claim that the transit time is independent of the barrier widths and of the separation distance between the barriers. We observe, in contrast, that if multiple reflections are allowed for correctly (infinite peaks) the transit time between the barriers appears exactly as expected. PACS. 03.65.Xp It is a well known result that the tunnelling time, calculated by using the stationary phase method (SPM) approximation in the limit of an opaque barrier, is independent of the barrier width [1]. Such a phenomenon, called the Hartman effect [2] implies arbitrarily large velocities inside the barrier. When this is obtained with the use of the non-relativistic Schrödinger equation, which can be considered exact only when the velocity of light is infinite, there is no paradox involved, only perhaps some doubt about the relevance. One can in particular question, in this context, the use of the terminology “super-luminal velocities”. However, since similar results have more recently been obtained with the Dirac equation [3, 4], there is good reason to be perplexed and to invoke further analysis. We address in this paper a variant of this subject which connects closely to a recent paper upon above-barrier diffusion [5]. In recent years, the Hartman analysis has been extended to a potential model with two successive barriers separated by a free propagation region. Again in the opaque limit for both barriers, it has been observed that, far from resonances, the tunnelling phase time depends neither upon the barrier widths nor upon the distance between the barriers [6, 7]. Thus, this result predicts, contrary to common sense, unbounded group velocities even in the free region between the two barriers. This phenomenon, also valid for an arbitrary number of barriers [8, 9], is known as the generalized Hartman effect. We shall demonstrate that a different analysis, which allows for multiple scattering between the barriers and, consequently, the existence of multiple peaks, alters this result. Indeed the generalized Hartman effect represents an example of an ambiguity in the use of the stationary phase method. The starting point of the analysis is the one-dimensional Schrödinger equation Ψxx(k, x) = 2m ~2 [V (x)− E ] Ψ(k, x) , (1) for a particle of mass m in a double barrier potential, 2 Stefano De Leo, Pietro P. Rotelli: tunnelling through two barriers