Quantum traversal time through a double barrier

The generalized Hartmann effect (GHE) predicts a strict inequality between the traversal times across a contiguous and a separated double-barrier system. This is compared to the implications of the time-of-arrival (TOA) operator approach to barrier traversal time [E. A. Galapon, Phys. Rev. Lett. 108, 170402 (2012)]. It is shown that, for initial wave packets with compact supports in the far incident side of the barrier system, the expectation value of the traversal time is independent of the separation between the barriers. On the other hand, for wave packets with supports extending inside the first barrier, the contribution of the barrier separation to the traversal time exponentially increases with the barrier height. Our result shows that if the support of the incident wave packet is far from the barrier region, the GHE inequality is violated. However, if the support of the wave packet extends inside the barrier region, the GHE inequality is consistent with the TOA operator approach, but only when the particle's incident energy is very small.