Only Above Barrier Energy Components Contribute to Barrier Traversal Time

Abstract

A time of arrival operator across a square potential barrier is constructed. The expectation value of the barrier time of arrival operator for a sufficiently localized incident wave packet is compared with the expectation value of the free particle time of arrival operator for the same wave packet. The comparison yields an expression for the expected traversal time across the barrier. It is shown that only the above barrier components of the momentum distribution of the incident wave packet contribute to the barrier traversal time, implying that below the barrier components are transmitted without delay. This is consistent with the recent experiment in attosecond ionization in helium indicating that there is no real tunneling delay time [P. Eckle et al., Science 322, 1525 (2008)].