On the generalized Hartman effect and transmission time for a particle tunneling through two identical rectangular potential barriers

TL;DR

This paper introduces a new quantum-mechanical approach to analyze tunneling times in a two-barrier system, confirming the Hartman effect and clarifying the validity of various time concepts in quantum tunneling.

Contribution

It develops a comprehensive method to model transmission and reflection subprocesses, defining accurate dwell and group times, and clarifies the limitations of existing tunneling time concepts.

Findings

Asymptotic transmission group time confirms the Hartman effect.

Dwell time at resonance points matches B"uttiker's dwell time.

Tunneling time increases monotonically with barrier separation.

Abstract

We develop a new quantum-mechanical approach to scattering a particle on a one-dimensional (1D) system of two identical rectangular potential barriers, which implies modelling the dynamics of its subprocesses -- transmission and reflection -- at all stages of scattering. On its basis we define, for each subprocess, the dwell time as well as the local (exact) and asymptotic (extrapolated) group times. Our concept of the asymptotic transmission group time confirms the validity of the Wigner phase time in the opaque limit, as well as the existence of the usual and generalized Hartman effects predicted on its basis. On the energy scale, this concept is valid everywhere in the high energy region as well as in the low energy region, excepting resonance points and their neighborhoods. On the contrary, the Buttiker dwell time is valid, as the transmission time, just only at the resonance points. Our concept of the transmission dwell time predicts monotonous growth of the tunneling time when the distance between the opaque barriers increases. By our approach only this time scale yields the true time spent, on average, by transmitted particles in the region occupied by the system. We explain why the asymptotic and local transmission group times cannot play this role and why the concept of {\it transmission group} velocity lies beyond the scope of special relativity. And else, all the transmission times admit only indirect measurements. Hence the unambiguous interpretation of all tunneling-time experiments is impossible when the transmission dynamics at all stages of scattering is unknown.