Stationary phase method and delay times for relativistic and non-relativistic tunneling particles

Abstract

This report deals with the basic concepts on deducing transit times for quantum scattering: the stationary phase method and its relation with delay times for relativistic and non-relativistic tunneling particles. After reexamining the above-barrier diffusion problem, we notice that the applicability of this method is constrained by several subtleties in deriving the phase time that describes the localization of scattered wave packets. Using a recently developed procedure -multiple wave packet decomposition -for some specifical colliding configurations, we demonstrate that the analytical difficulties arising when the stationary phase method is applied for obtaining phase (traversal) times are all overcome. In this case, we also investigate the general relation between phase times and dwell times for quantum tunneling/scattering. Considering a symmetrical collision of two identical wave packets with an one-dimensional barrier, we demonstrate that these two distinct transit time definitions are explicitly connected. The traversal times are obtained for a symmetrized (two identical bosons) and an antisymmetrized (two identical fermions) quantum colliding configuration. Multiple wave packet decomposition shows us that the phase time (group delay) describes the exact position of the scattered particles and, in addition to the exact relation with the dwell time, leads to correct conceptual understanding of both transit time definitions.At last, we extend the non-relativistic formalism to the solutions for the tunneling zone of a onedimensional electrostatic potential in the relativistic (Dirac to Klein-Gordon) wave equation where the incoming wave packet exhibits the possibility of being almost totally transmitted through the potential barrier. The conditions for the occurrence of accelerated and, eventually, superluminal tunneling transmission probabilities are all quantified and the problematic superluminal interpretation based on the non-relativistic tunneling dynamics is revisited. Lessons concerning the dynamics of relativistic tunneling and the mathematical structure of its solutions suggest revealing insights into mathematically analogous condensed-matter experiments using electrostatic barriers in single-and bi-layer graphene, for which the accelerated tunneling effect deserves a more careful investigation.

Statistical approach to tunneling time in attosecond experiments

“…They have obviously nothing in common with the ETT, which is based on the potential barrier V (x) alone. On the other hand, two well-known time definitions, the phase time [28,29,30,31,40,41,42] and the dwell time [28,29,30,31,36], which, just like the ETT, are based on the potential energy V (x). They do not involve any external agents like the magnetic field used in Larmor time.…”

“…is the traversing time −iτ of the particle obeying classical motion laws as follows from its definition in (2), and L = x R − x L is the barrier width. The phase time (designated by ϕ), deriving from the stationarity of the phase of the transmission amplitude [28,29,30,31,40,41,42], takes the form…”

“…Tunneling time depends on what kinetic theory is set forth for tunneling process, and therefore, the literature consists of various time definitions [28,29,30,31]. They include traversal time through modulated barriers [32,33,34,35], spin precession time [36,37,38], flux-flux correlation duration [39], phase stationary time [40,41,42], polymer approach [43], and Feynman path integral (FPI) averaging of the classical time [44,45,46]. Some of them are complex, some are difficult to associate with tunneling and some suffer from superluminality.…”

SU(2)⊗SU(2) bi-spinor structure entanglement induced by a step potential barrier scattering in two-dimensions

“…Some preliminary issues have already investigated the relativistic linear transmission [14][15][16][17] and the phenomena of planar diffusion of Dirac particles [18][19][20] through and above potential barriers and steps. In particular, the effects of relative phases between incoming and reflected or transmitted amplitude components of diffused waves have been accurately quantified [14,21,22].…”

Ballistic collective group delay and its Goos–Hänchen component in graphene

“…They describe the tunneling speed through a barrier in different aspects, and are both of paramount importance for solid-state devices working at high frequencies [6]. Recently, as graphene rises as a star material in condensed matter physics, extensive efforts [7][8][9][10][11][12][13] have been devoted to the investigations of group delay and/or dwell time in it.…”