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Physical Review A

Relativistic approach to the tunneling-time problem

Igor A. Ivanov, Kyung Taec Kim

Published Nov 12, 2015• 10 citations
Publisher

Abstract

We report a study of the effect of relativity on the tunneling time for the process of the tunneling ionization. Our results indicate that the relativistic effects make the tunneling time nonzero. A time interval of a few attoseconds is required for the electron to traverse the barrier. This finding agrees with the basic postulate of the special relativity that two events which are in causal relationship cannot occur simultaneously.

DOI
10.1103/physreva.92.053418

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We report a study of the effect of relativity on the tunneling time for the process of the tunneling ionization. Our results indicate that the relativistic effects make the tunneling time nonzero. A time interval of a few attoseconds is required for the electron to traverse the barrier. This finding agrees with the basic postulate of the special relativity that two events which are in causal relationship cannot occur simultaneously.
Attosecond ionization experiments have not resolved the question "What is the tunneling time?". Different definitions of tunneling time lead to different results. Second, a zero tunneling time for a material particle suggests that the nonrelativistic theory includes speeds greater than the speed of light. Chemical reactions, occurring via tunneling, should then not be considered in terms of a nonrelativistic quantum theory calling into question quantum dynamics computations on tunneling reactions. To answer these questions, we define a new experimentally measurable paradigm, the tunneling flight time, and show that it vanishes for scattering through an Eckart or a square barrier, irrespective of barrier length or height, generalizing the Hartman effect. We explain why this result does not lead to experimental measurement of speeds greater than the speed of light. We show that this tunneling is an incoherent process by comparing a classical Wigner theory with exact quantum mechanical computations.
How much time does a tunneling wave packet spent in traversing a barrier? Quantum mechanical calculations result in zero time inside a barrier . In the nineties analogous tunneling experiments with microwaves were carried out. The results agreed with quantum mechanical calculations. Electron tunneling time is hard to measure being extremely short and parasitic effects due to the electric charge of electrons may be dominant. However, quite recently the atomic ionization tunneling time has been measured. Experimental data of photonic, phononic, and electronic tunneling time is available now and will be presented. It appears that the tunneling time is a universal property independent of the field in question.
The conundrum parameter-operator of time in quantum mechanics, as well as the time-energy uncertainty relation and the tunneling delay time, have recently been addressed in attosecond optical ionization experiments. Dirac's formulation of the electron's relativistic quantum mechanics (RQM) allows the introduction of a dynamical self-adjoint time operator, representing an internal time of the system. Its relation to the parametric (laboratory) time is derived. Its relevance to the tunneling measurements in these experiments is exhibited, namely, as the time it takes the wave packet to cross the exit point of the barrier, thus justifying considering this time operator an additional observable in RQM.
We have analyzed the influence of the wave packet size in the relativistic tunnelling time τ and its uncertainty �τ when it traverses a given potential barrier. The analytical expressions obtained for both magnitudes confirm that the size of the incident pulse has a significant effect on the tunnelling process. This effect is greater for short pulses, compared with the length of the barrier. For the evanescent zone, we have derived an analytical expression for τ with a good limit of validity. This expression constitutes a value tool to calculate the relativistic tunnelling time as a function of the incident wave packet with a good limit of validity. Superluminal propagation is found in this region but with a large value of the uncertainty �τ compared with the tunnelling time itself. We can conclude that the probability of superluminal propagation is practically negligible in the evanescent region. In respect to the Klein zone, we have derived an analytical expression for τ that depends on the size of the incident
Recent results on electron tunneling across a potential barrier, inferred from observations or obtained from theoretical models, have suggested superluminal or instantaneous barrier traversal times. In this work we investigate relativistic wave-packet dynamics for an electron tunneling through a potential barrier employing space-time resolved solutions to relativistic quantum field theory (QFT) equations. We prove by linking the QFT property of microcausality to the wave-packet behavior that the tunneling dynamics is fully causal, precluding instantaneous or superluminal effects. We illustrate these results by performing numerical computations for an electron tunneling through a potential barrier for standard tunneling as well for Klein tunneling. In all cases (Klein tunneling or regular tunneling across a standard or a supercritical potential) the transmitted wave packet remains in the causal envelope of the propagator, even when its average position lies ahead of the average position of the corresponding freely propagated wave packet.
We study the relativistic quantum mechanical problem of a Dirac particle tunneling through two successive electrostatic barriers. Our aim is to study the emergence of the so-called generalized Hartman effect, an effect observed in the context of nonrelativistic tunneling as well as in its electromagnetic counterparts and which is often associated with the possibility of superluminal velocities in the tunneling process. We discuss the behavior of both the phase (or group) tunneling time and the dwell time, and show that in the limit of opaque barriers the relativistic theory also allows the emergence of the generalized Hartman effect. We compare our results with the nonrelativistic ones and discuss their interpretation.