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American Journal of Physics

Three-dimensional tunneling in quantum ballistic motion

Christian Bracher, Wilhelm Becker, S. A. Gurvitz, M. Kleber, M. S. Marinov

Published Jan 1, 1998• 60 citations
Publisher

Abstract

We provide an elementary case study of field-induced quantum tunneling in a realistic three-dimensional environment which exhibits both tunneling and classically allowed motion. An analytic description is presented for ballistic motion in a uniform electric field. From there we derive simple expressions concerning the wave function for field emission from an ultrathin quantum well in the tunneling limit. At the same time a wave-packet description of the tunneling process is obtained. The tunneling time can be read off from the lateral spread of the tunneling current distribution.

DOI
10.1119/1.18806

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We provide an elementary case study of field-induced quantum tunneling in a realistic three-dimensional environment which exhibits both tunneling and classically allowed motion. An analytic description is presented for ballistic motion in a uniform electric field. From there we derive simple expressions concerning the wave function for field emission from an ultrathin quantum well in the tunneling limit. At the same time a wave-packet description of the tunneling process is obtained. The tunneling time can be read off from the lateral spread of the tunneling current distribution.
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The dynamics of particle tunneling is characterized by the concept of a tunneling time. The various times suggested in the literature are considered in order to characterize the complementary features of the tunneling process. Explicit calculations show how to find the particle and the wave aspects of different tunneling times. We also expose the physics behind the various definitions and their interrelations. We show that in the classical limit the particle properties dominate and in the opaque limit the wave aspects do. The complementary analysis helps in the interpretation of the tunneling process and eliminates some of the controversies found in the literature. All tunneling times are found to be special cases of a universal tunneling time definition relating to parameter derivatives of the phases of the wave functions. We propose that these relations can be used to extract the tunneling times for any general potential by integrating the wave packet through the tunneling process and extracting the derivatives numerically from these results. Furthermore, we investigate the particle properties of resonant tunneling. They are shown to dominate the dynamics inside the well regions.
We derive a useful expression for the time-development operator for quantum wave functions, exp(−iĤt/ℏ), for the case of a particle subject to a constant force. We apply it, in both position and momentum space, to construct wave packets that exhibit uniform acceleration. We use these results to generate quantum wave packets corresponding to the motion of a charged particle in a uniform electric field in two different gauges, explicitly demonstrating the change in phase of the wave function under a gauge transformation. The more standard description of the time dependence of accelerating wave packets using the formalism of propagators is briefly reviewed in an Appendix for comparison.