Effects of relativity on the time-resolved tunneling of electron wave packets

The phenomenon of tunneling in which a quantum- mechanical particle can penetrate a repulsive barrier with a height that exceeds the total energy of the particle is coun- terintuitive. Any explanation or intuition for this process based on classical mechanics fails. At the same time, this effect is extremely important and has been studied widely. The Josephson effect in high-speed semiconductors @1#, b decay in nuclear physics, and instantons in high-energy physics are just a few examples. In the early 1930s it was already recognized that there was no appreciable temporal delay in the transmission of wave packets through barriers @2#. Wigner discussed the possibility that a particle can ef- fectively travel faster than the speed of light when passing through the barrier. Chiao and co-workers have more re- cently addressed the realization of superluminal speeds in a more systematic way. They used a periodic potential barrier to demonstrate experimentally that superluminal velocities can indeed be obtained, and showed that this result does not violate causality. In this article we intend to address the following ques- tions: Can one trust the predictions of a nonrelativistic theory at all if superluminal effects are being investigated? How accurate are these predictions? Does the relativistic quantum theory predict superluminal speeds? Does a fully relativistic treatment of tunneling increase or reduce the tunneling prob- ability? Does the existence of superluminal velocities imply the violation of Einstein's causality when they are computed in the framework of the Schrodinger equation? Can causality be restored in the Dirac theory? Can one define a physical quantity that describes the time evolution of a wave packet inside the barrier which reduces to the regular peak velocity when calculated from a wave packet that is outside the bar- rier? Due to its lacking a counterpart in classical mechanics, it is not obvious how to apply any intuition to relativistic quantum-mechanical tunneling and to predict any answers to these questions. A full Dirac theory calculation seems nec- essary. Quite remarkably, despite the large amount of literature on nonrelativistic tunneling, we are aware of only two works @3# that have addressed some of these questions. Leavens and Aers @3# used the stationary-state approach to analyze Larmor-clock transmission times for single and double rect- angular barriers. In some special cases the problem of quantum-mechanical tunneling can be mapped onto the fully relativistic problem of evanescent electromagnetic radiation @4-7#. For the special case of nonrelativistic tunneling, the ques- tion of how much time it takes a particle to pass the barrier has triggered considerable controversial debate to the present day. Even though by 1993 the community had largely ac- cepted the fact that there actually is a time scale associated with the duration of tunneling, there is still a lack of consen- sus with regard to the existence of a unique expression for this time scale and on the exact implications of this expres- sion @8#. In fact, Hauge and Stovneng @9# stated that with the exception of two candidates all expressions for tunneling times have logical flaws sufficiently serious that they must be rejected. The only two survivors are the dwell time @10# and the asymptotic phase time @8,9#, which have complementary weaknesses. In this article, we stay away from most of the controver- sial issues and focus on investigating the effect of relativity on the tunneling process. Our model system is an electron that tunnels through a one-dimensional repulsive barrier. The time evolution of this system is given by the solution of the Dirac equation