Tunneling time, the Hartman effect, and superluminality: A proposed resolution of an old paradox

The issue of tunneling time is replete with controversy and paradoxes. The controversy stems from the fact that many tunneling time definitions seem to predict superluminal tunneling velocities. One prediction, termed the Hartman effect, states that the tunneling time becomes independent of barrier length for thick enough barriers, ultimately resulting in unbounded tunneling velocities. Experiments done with 'single photons', classical light waves, and microwaves all show this apparent superluminality. The origin of these paradoxical effects has been a mystery for decades. In this article, we review the history of tunneling times starting with the early work of MacColl, Hartman, and Wigner. We discuss some of the tunneling time definitions, with particular emphasis on the phase time (also known as the group delay or Wigner time) and the dwell time. The key experiments are reviewed. We then discuss our recent work, which suggests that the group delay in tunneling is not a transit time as has been assumed for decades. It is, in reality, a lifetime and hence should not be used to assign a speed of barrier traversal. We show how this new understanding along with the concept of energy storage and release resolves all the outstanding tunneling time paradoxes