Path-integral analysis of the time delay for wave-packet scattering and the status of complex tunneling times

Abstract

The wave-packet simulation (WPS) method for calculating the time a tunneling particle spends inside a one-dimensional potential barrier is reexamined using the Feynman path-integral technique. Following earlier work by Sokolovski and Baskin [Phys. Rev. A 36, 4604 (1987)], the tunneling (or traversal) time ${\mathit{t}}_{\mathrm{pack}}^{\mathit{T}}$ is defined as a matrix element of a classical nonlocal functional between two states that represent the initial and transmitted wave packets. These states do not lie on the same orbit in Hilbert space; as a result, ${\mathit{t}}_{\mathrm{pack}}^{\mathit{T}}$ is complex-valued. It is shown that Re${\mathit{t}}_{\mathrm{pack}}^{\mathit{T}}$ reduces to the standard WPS result, ${\mathit{t}}_{\mathrm{phase}}^{\mathit{T}}$, for conditions similar to those employed in the conventional WPS analysis. Similarly, Im${\mathit{t}}_{\mathrm{pack}}^{\mathit{T}}$ is shown to contain information about the energy dependence of the transmission probability. Under semiclassical conditions, Im${\mathit{t}}_{\mathrm{pack}}^{\mathit{T}}$ reduces to the well-known Wentzel-Kramers-Brillouin expression for the tunneling time. It is shown there are different definitions for the traversal time of a classical moving object, whose size is comparable to the width of the region of interest. In the quantum case, these different definitions correspond to different ways of analyzing the WPS experiment. The path-integral approach demonstrates that the tunneling-time problem is one of understanding the physical significance of complex-valued off-orbit matrix elements of an operator or functional. The physical content of complex-valued tunneling times is discussed. It is emphasized that the use of complex tunneling times includes real-time approaches as a special case. Nevertheless, there is a limitation in the description of tunneling experiments using tunneling times, whether real or complex. The path-integral approach does not supply a universal traversal time, analogous to a classical time, that can be used in quantum situations. It is demonstrated that the often expressed hope of finding a well-defined and universal real tunneling time is erroneous.