Average clock times for scattering through asymmetric barriers

Abstract

The reflection and transmission Salecker-Wigner-Peres clock times averaged over the postselected reflected and transmitted sub-ensembles, respectively, are investigated for the one dimensional scattering of a localized wave packet through an asymmetric barrier. The dwell time averaged over the same post-selected sub-ensembles is also considered. The emergence of negative average reflection times is examined and we show that while the average over the reflected sub-ensemble eliminates the negative peaks at resonance for the clock time, it still allows negative values for transparent barriers. The saturation of the average times with the barrier width (Hartman effect) is also addressed.PACS. 03.65.Xp Tunneling, traversal time, quantum Zeno dynamics -03.65.Ta Foundations of quantum mechanics; measurement theory -03.65.Nk Scattering theory IntroductionThe search for a sensible definition of quantum tunneling times is one of the most enduring problems in quantum mechanics, despite numerous efforts in the last few decades (see, e.g., [1,2,3,4,5,6,7,8,9,10] and references therein). The difficulty in obtaining a time scale for the tunneling problem lies in the well-known impossibility of defining a selfadjoint time operator canonically conjugated to the (bounded from below) Hamiltonian. Attempts to use a "tempus" operator that is canonically conjugated to the Hamiltonian but not given by the time evolution of the system, have lead to complex stationary times [11,12] (also see [13] for a review of methods attempting to define time operators).A more common approach to this problem is to obtain operational definitions of a parameter with the dimensions of time and investigate its properties. These attempts have rendered several definitions of time that, while useful in specific situations, are generally not considered the definitive answer to the question of how long it takes for a particle to tunnel through a potential barrier. Among the most common (and useful) stationary time scales considered in the literature are the much studied phase time [14,15], dwell time [16,17], Larmor time [17,18,19] (also see [20,21]) and the Salecker-Wigner-Peres (SWP) clock times [22,23,24,25].Recently, the SWP clock has been reconsidered and it was shown that, contrary to previous claims in the literature, it can be directly applied to interacting particles (i.e., without the need for "calibration" [26,27]) provided that one follows Peres' [23] approach to associate the clock's hand with the peak of the clock's wave function [24]. In particular, this approach allowed the derivation, directly from Schrödinger's equation, of a relationship between the dwell time and the SWP clock reflection and transmission times without an interference term [24], thus ending the controversy about the compatibility of such relationship with standard quantum mechanics (see, e.g., [1,2,3,4] and references therein).The SWP clock has also been used to treat the scattering of a wave packet by a potential and to obtain, by making use of a post-selection of the fina...

A Probability Distribution for Quantum Tunneling Times

“…A similar definition of the distribution of tunneling times given in ( 10)-( 11) can be obtained for any time scale which is probabilistic in nature, that is, of the form (8). Although several other probabilistic tunneling times exist in the literature (e.g., [23,31,32,35]), the SWP clock has proven to yield well-behaved real times both in the time-independent [17,37,39] and time-dependent approaches [28,34,40] and it provides a simple procedure to derive the probabilistic expression (8). In addition, the role exerted by circularly polarized light in attoclock experiments [3,25] seems to provide a natural possibility for interpretation in terms of the SWP clock.…”

“…A few works (e.g., [23,28,31,32]) address the issue of localizability and, consequently, arrive at a probabilistic definition of tunneling times (that is, an average time). In particular, in [28] the SWP clock was used to obtain an average tunneling time of transmission (reflection) for an incident wave packet, and such time was employed to investigate the Hartman effect [33] for a particle scattered off a square barrier and it was shown that it does not saturates in the opaque regime [28,34].…”

Double General Point Interactions: Symmetry and Tunneling Times

“…where we have defined the auxiliary phase ϕ 0 (k, q), which allow us to introduce the auxiliary clock and phase times [62,63] given, respectively, by…”

“…With the help of these auxiliary times, the transmission and reflection times are now related by It should be noticed that, contrary to the case of even potentials, for odd potentials (and potentials without a defined parity) the clock times in general do not coincide with the dwell time [63]. Below we consider two cases of odd arrangements of point interactions, the first built from two δ and the second built from two δ ′ interactions.…”