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\begin{document}

\title{TunnelingtimeandtimeinQM}



\author[1]{Kullie}%
\affil[1]{International Winter University}%


\vspace{-1em}



  
  \date{May 08, 2024}


\begingroup
\let\center\flushleft
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\maketitle
\endgroup





\selectlanguage{english}
\begin{abstract}
\scriptsize
%%%%%%%%%%%%%%%%%%%%%%
 %Tunneling and tunneling time are  hot debated and very interesting
 %issues because of their fundamental role in the quantum mechanics.
 {\bf Abstract}\\
 {The measurement of the tunneling time in today's attosecond and strong
 field (low-frequency) experiments, despite its controversial
 discussion offers a fruitful opportunity to understand the time
 measurement and the role of time and the time operator in quantum
 mechanics.
 %In previous work \cite{Kullie:2015}, we suggested a model and
 %derived a simple relation to calculate the tunneling time, which
 %showed a good agreement with the experimental result for He-atom.
 In the present work we discuss our tunneling time model in relation
 to two time operators definitions introduced by Bauer
 \cite{BauerM:2017} and Aharonov-Bohm \cite{Aharonov:1961}.
 We found that both definitions can be generalized to the same type
 of time operator.
 Moreover, we found that the introduction of a phenomenological
 parameter by Bauer to fit the experimental data is unnecessary.
 The issue is resolved with our tunneling model by considering
 the correct barrier width, which avoids a misleading interpretation
 of the experimental data.
 Our analysis shows that the use of the so-called classical barrier
 width, precisely speaking is incorrect.}     %
\end{abstract}%



\sloppy


\title{Tunneling time in attosecond experiments and time operator in 
 Quantum Mechanics}

\author{Ossama Kullie\\\scriptsize
Theoretical Physics, Institute for Physics, 
 Department of Mathematics and Natural Science, University of Kassel, 
 Germany; email:kulliel@uni-kassel.de}

{Correspondence: kulliel@uni-kassel.de}

{keywords: Ultrafast science,  attosecond physics,
 tunneling time, time-energy uncertainty relation, time and time-operator
 in quantum mechanics.}

\section{Introduction}\label{ssec:int}  
       Attosecond science ($attosecond= 10^{-18}$ sec) concerns primarily 
 electronic motion and energy transport on atomic scales and is of  
 fundamental interest to the physics in general.
 In previous work \cite{Kullie:2015,Kullie:2016,Kullie:2018} we 
 presented a tunneling model and a formula to calculate the tunneling  
 time (T-time), by exploiting the time-energy uncertainty relation 
 (TEUR), precisely that time and energy are (Heisenberg) conjugate pair. 
 Our tunneling time is in a good agreement with the experimental 
 finding attosecond (angular streaking) experiment for He-atom 
 \cite{Landsman:2014II,Eckle:2008s,Eckle:2008}, which will be referred 
 to as Keller Attosecond Experiment (KASE). 
 Our model offers a real T-time picture and represents a delay time 
 with respect to the ionization at atomic field strength $F_a$ (compare 
 fig \ref{fig:ptc}). 
 It is also interesting for the tunneling theory in general, because in 
 this model the T-time is related to the height of the barrier 
 \cite{Kullie:2015,Kullie:2016}.  
 Indeed, since the appearance of quantum mechanics (QM) time was  
 controversial, the famous example is the Bohr-Einstein weighing 
 {\it photon box Gedanken experiment (BE-photon-box-GE)}.
 In \cite{Kullie:2015} it was shown that our tunneling model (see sec 
 \ref{ssec:tt}) in the  attosecond experiments (ASEs) is intriguingly 
 similar to the {\it BE-photon-box-GE} (see for example 
 \cite{Auletta:2009} p. 132), where the former can be seen as 
 a realization to the later.    
 And as mentioned, the agreement of the model with the result of KASE 
 \cite{Landsman:2014II,Kullie:2018} is impressively good.

It is worthwhile to mention that Galapon 
 \cite{Galapon:2002I,Galapon:2002II,Galapon:2003} showed in a skillful 
 mathematical way (the consistency theorems) that,  there is no 
 a priori reason to exclude the existence of a self-adjoint time 
 operator canonically conjugate to a semibounded Hamiltonian, contrary 
 to the (famous) claim of Pauli.
 Roughly speaking, see Garrison \cite{Garrison:1970}, for a canonically 
 conjugate pair of operators of a Heisenberg type (i.e. uncertainty 
 relation), Pauli theorem did not apply, unlike a pair of operators 
 that form a Weyl pair (or Weyl system.) 
     Our simple tunneling model was introduced in \cite{Kullie:2015} 
 (see fig 1), in this model an electron can be ionized by laser pulse 
 with a field strength $F$. 
 Ionization happen directly when the field strength is larger than a 
 threshold called atomic field strength $F_a$. 
 However, for field strengths $F<F_a$ ionization can happen by 
 tunneling  mechanism, throw a barrier built by the effective potential  
 due to the Coulomb potential of the nucleus and the electric field, 
 which can be expressed in a one-dimensional form

\begin{equation}
  V_{eff}(x)=V(x)-x F =-\frac{Z_{eff}}{x}-x F,
 
\end{equation}
 compare fig \ref{fig:ptc}. Where $Z_{eff}$ is the effective nuclear 
 charge. In this model all the interesting quantities of the process 
 \cite{Kullie:2015} can be expressed as a function of the ionization 
 potential $I_p$ of the valence or the interacting electron, the peak 
 electric field strength $F$ and  the quantity  
 $\delta_z=\sqrt{I_p^{2}-4 Z_{eff} F}$. 
 In the following $F$ stands (unless it is clear) for {\it the peak 
 electric field strength at maximum}. 
 From fig  \ref{fig:ptc}, the barrier width $d_W=\delta_z/F$,  
 the inner (entrance $x_{e,-}$)  and outer (exit $x_{e,+}$) point 
 $x_{e,\pm} =(I_p\pm\delta)/(2F)$, and the barrier height 
 (at $x=x_m(F)=\sqrt{Z_{eff}/F}$) is 
 $h_B(x_m)=\mid\!-I_p-V_{eff}(x_m)\!\mid=
 \mid\!-I_p+\sqrt{4 Z_{eff}F}\!\mid$.  
     The barrier disappear for $\delta_z=0$ ($d_W=0,\, h_B(x_m)=0$) at 
 $F_a=I_p^{2}/4 Z_{eff}$ where  the direct ionization starts, the 
 barrier-suppression ionization.

The T-time and the tunneling process itself in the ASEs are hot 
 debated and rather still not resolved completely.
 In the (low-frequency) ASEs, the idea is to control the 
 electronic motion by laser fields which are comparable in strength 
 to the electric field in the atom. In today's experiments, usually 
 used intensities are $\sim 10^{14}\, W cm^{-2}$. 
    A key quantity is the Keldysh parameter \cite{Keldysh:1964},

\begin{equation}
\label{gamK} 
 \gamma_{_K} = \frac{\sqrt{2I_p}}{F} \omega_0=\tau_{K} \omega_0, 
 
\end{equation}
   where $\omega_0$ is the central circular frequency of the laser pulse 
 and $\tau_{K}$ denotes the Keldysh time.  
 In eq (\ref{gamK}) and hereafter in this work the atomic units (au) 
 are used, in which the electron's mass and charge and the Planck 
 constant are set to unity, $\hbar = m = e = 1$.
 At values $\gamma_{_K}>1$ one expects predominantly photo-ionization 
 or multiphoton ionization (MPI), while at $\gamma_{_K}<1$ 
 (field-) ionization happens by a tunneling process, which occurs for 
 $F<F_{a}$.
   Tunneling happens for $F<F_{a}$, because the electron does not have 
 enough energy to ionize directly. 
 The electron tunnels (or tunnel-ionizes) through a barrier made by the  
 Coulomb potential and the electric field of the laser pulse,  and 
 escapes at the exit point $x_{e,+}$ to the continuum as shown in fig 
 \ref{fig:ptc} (a sketch for He-atom.)
  In the KASE experiment (see \cite{Landsman:2014II}) an elliptically 
 polarized laser pulse is used with $\omega_0=0.0619\,au\,
 (\lambda=736\, nm)$, the ellipticity parameter $\epsilon=0.87$, while 
 the electric field strengths are in the range $F=0.04-0.11$ and for 
 He atom $I_p=0.90357\, au$.

\section{The tunneling time}\label{ssec:tt}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/OKullie-KASE-gr1/OKullie-TT-fg1-N}
\caption{{\label{fig:ptc}\scriptsize (Color online)
 Graphic display of the potential and the effective potential curves,  
 the two inner and outer points $x_{e,\pm}={(I_p\pm\delta_z)/2F}$ 
 and the barrier width $d_W=x_{e,+}-x_{e,-}$, the ``classical exit'' 
 point $x_{e,c}=I_p/F$ and $x_m(F)=(Z_{_{eff}/F})^{1/2}$  the 
 position at the maximum of the barrier height $h_B(x_m)$, 
 $x_a=x_m(F=F_a)$.%
}}
\end{center}
\end{figure}

Usually the tunneling process in the ASEs is explained by a simple 
 picture, like the one shown in fig \ref{fig:ptc} for the He-atom. 
 It is based on the strong field approximation (SFA) or 
 Keldysh-Faisal-Reiss approximation 
 \cite{Keldysh:1964,Faisal:1973,Reiss:1980}; for introductory reviews  
 see \cite{Zheltikov:2017,Popruzhenko:2014,Ivanov:2005}.    
 This simple picture is very useful in explaining the experiment, 
 although it is strictly true only in the length gauge (see  
 \cite{Bauer:2005,Faisal:2007I,Faisal:2007II,Reiss:2014}.) 
 In the ASEs, according to the SFA (see also \cite{Perelomov:1966}), 
 tunneling means that the electron tunnels and escapes the barrier 
 region at the exit point $x_{e,+}$ (see fig \ref{fig:ptc}) with 
 approximately zero kinetic energy. More precisely, the electron 
 velocity along the (opposite) field direction is zero and negligible 
 in other directions, also called a longitudinal fields approximation 
 because the transverse fields are neglected \cite{Reiss:2014}. 
 In \cite{Kullie:2015} we showed that the uncertainty in the energy, 
 which is related to the height of the barrier $h_B(x_m)$, can be 
 quantitatively discerned from the atomic potential energy at the exit 
 point, $\Delta E \sim \mid\! V(x_e)\!\mid=\mid\!\!-{Z_{eff}}/{x_e}
 \!\!\mid$ for arbitrary strengths $F\le F_a$, where $Z_{eff}$ is  the 
 nuclear effective charge and the atomic field strength is given by 
 $F_a=I_p^2/(4 Z_{eff})$ \cite{Augst:1989,Augst:1991}.   
 With the TEUR, $\Delta E \cdot \Delta T \ge 1/2$, one obtains 
 the symmetrical (or total) T-time \cite{Kullie:2015}:

\begin{equation}
 \label{Tsym} 
 %\hspace*{-1.cm}
 \tau_{_T,sym}=
 \frac{1}{2}\left(\frac{1}{(Ip+\delta_z)}
 +\frac{1}{(Ip-\delta_z)}\right)= \frac{I_p}{4 Z_{eff} F}
 
\end{equation}
 where $\delta_z = \sqrt{I_p^2-4 Z_{eff} F}$ for a single active 
 electron model, and for the Hydrogen atom $Z_{eff}\!=\!1$.    
  We call eq \ref{Tsym} the symmetrical T-time or the total 
 time, because it was obtained by a symmetrization procedure  
 (similar to the Aharonov time operator 
 \cite{Aharonov:1961,Aharonov:2000} see below sec \ref{TBA})) from 
 the unsympathized T-time \cite{Kullie:2015}

\begin{equation}
\label{Tunsy} 
 \tau_{_T,unsy}=\frac{1}{(Ip-\delta_z)}
\end{equation}
  The relation in eq \ref{Tsym}, besides the mathematical simplicity, 
 aids a conceptual reasoning \cite{Kullie:2015,Kullie:2016}.
 The physical reasoning of this relation is the following: 
 the barrier itself causes a delaying time $\tau_{_{T,d}}$, where

\begin{equation}
\label{Tdi} 
 %\hspace{-0.5cm}
 \tau_{_{T,d}}=\frac{1}{2(Ip-\delta_z)}, \, {\rm and}\,\,  
 \tau_{_{T,i}}=\frac{1}{2(I_p+\delta_z)}                                         
\end{equation}
  $\tau_{_{T,d}}$ is the time delay with respect to the ionization at 
 atomic field strength $F_a$, where the barrier is absent (the barrier 
 height, the barrier width $d_W=\delta_z/F$ and $\delta_z$ are zero).
 It is the time duration for a particle to pass the barrier region 
 (between $x_{{_e,-}}, x_{_{e,+}}$) and escapes at the exit point 
 $x_{e,+}$ to the continuum \cite{Kullie:2015}.
 The first term $\tau_{_{T,i}}$ in eq \ref{Tsym} is the time needed 
 to reach the entrance point $x_{_{e,-}}$ from the initial point $x_i$, 
 compare fig \ref{fig:ptc}. 
 The two steps of the model coincide at the limit  $F\rightarrow F_a\,
 (\delta_z\rightarrow 0)$, and the total time becomes the ionization 
 time $\tau_{_T,sym}=\frac{1}{Ip}$, or $\tau_{_T,d}=\tau_{_{T,i}}=
 \frac{1}{2Ip}$ at the atomic field strength $F_a$. 
 For $F> F_a$ the barrier-suppression ionization sets up  
 \cite{Delone:1998,Kiyan:1991}. 
 At the opposite side of the limit $F\rightarrow 0$,   
 $\delta_z\rightarrow I_p$ and $\tau_{_{T,d}}\rightarrow \infty$, hence 
 nothing happens, i.e. the electron remains in its ground state 
 undisturbed, which shows that our model is consistent, 
 for details see \cite{Kullie:2015,Kullie:2016,Kullie:2018}.

\section{Time operator}\label{ssec:Hat}
 In the early days of the QM a time-energy uncertainty relation (analog  
 to the position-momentum relation) and the existence of time operator, 
 faced the well-known objection of Pauli.  
 According to the Pauli theorem the introduction of time operator is 
 basically forbidden and the time $t$ in QM must necessarily be 
 considered as an ordinary number (`$c$' number). 
 On the other side the famous example and its debate is the 
 {\it BE-photon-box-GE}.
 The crucial point is that, to date, no general time operator has been  
 found; thus the uncertainty relation is used dependent upon study 
 cases \cite{Muga:2007}. 
 And there is still a common opinion that time plays a role essentially 
 different from the role of the position in quantum mechanics (although 
 it is not in line with special relativity). 
 Hilgevoord concluded in his work \cite{Hilgevoord:2002}, that when 
 looking to a time operator a distinction must be made between 
 universal time coordinate, t, a c-number like a space coordinate, and 
 the dynamical time variable of a physical system situated in 
 space-time; i.e. clocks. 
 The search for a time operator has a long history 
 \cite{Razavy:1967,Goto:1981,Wang:2007}.
 This led Busch \cite{Busch:1990I,Busch:1990II} to classify three types   
 of time in quantum mechanics: external time (parametric or laboratory 
 time), intrinsic or dynamical time and observable time.

\subsection{Bauer's time operator}\label{TofB}
 Bauer \cite{BauerM:2014,BauerM:2017} introduced a dynamical 
 self-adjoint time operator in the framework of Dirac's relativistic 
 quantum (DRQM).
 The Bauer's time operator (BTO) is defined  as the following,

\begin{equation}
\label{BTO1} 
 \hat{T}={\alpha} \cdot {\hat{r}}/c + \beta \tau_0

\end{equation}
 where $  \alpha,  \beta$ the well-known Dirac matrices,  
 $c$ the speed of light, and $\bf\hat{r}$ is the three-dimensional 
 space vector. 
 The operator defined in eq (\ref{BTO1}) has been shown to commute with 
 the Dirac free particle Hamiltonian $H_D= c  \,\alpha \cdot
 \hat{p} + \beta m_0 c^{2}\!$. 
 Bauer proved the Heisenberg commutation relation and analyzed the 
 dynamical character of $\hat{T}$ and found for  a free  particle

\begin{eqnarray}
 \label{BTO2} 
 \hat{T}(t)&=&\left(\frac{v_{gp}}{c}\right)^{2} t + 
 \left(\frac{m_0 c^{2}}{H_D}\right) \tau_0 \\
  {r}(t)&=& v_{gp}\, t \\
  {r}(t)&=& v_{ph}\, T  
\end{eqnarray}
 where $v_{gp}$ is the group,  $v_{ph}$\footnote[1]{Eq (\ref{BTO1}), 
 Bauer private communication} the phase velocity of the particle. 
 According to Bauer $T(t)$ is the internal time and $t$ the external 
 (laboratory) time, where $T(t)<t$. 
 In the limit $m_0=0$ then $v_{gp}=c$  and $T(t)=t$.
 The parametric time $t$ is according to Bauer can be interpreted as 
 the laboratory time, which is the time variable appears in the 
 time-dependent Schr\"odinger equation and characterize the dynamical 
 evolution of microscopical systems

 Bauer argued that DRQM allows the introduction of a dynamical time 
 operator that is self-adjoint, unlike the parametric time entering in 
 the Schr\"odinger equation.
 While Galapon \cite{Galapon:2002I,Galapon:2002II,Galapon:2003} showed 
 that in nonrelativstic QM (NRQM), for Heisenberg pairs there is no 
 reason to exclude the existence of a time operator and consequently an  
 observable of time and a dynamical time, as already mentioned.

For a non-relativistic particle Bauer found \cite{BauerM:2017} 
 a relation between the parametric (external in Bauer's notation) time 
 intervals and the dynamical (internal in Bauer's notation) intervals,

\[
\Delta T_B=<T_B(t_{2})>-<T_B(t_{1})>=(t_{2}-t_{1})=\Delta t
\]

\begin{eqnarray}
\label{BT1} 
 \Delta T_B&=&<T_B(t_2)>-<T_B(t_1)> \\ \nonumber
 &=&\frac{1}{2}\left(\frac{v_{gp}}{c}\right)^{2} 
 (t_{2}-t_{1}) << (t_{2}-t_{1})=\Delta t_B

\end{eqnarray}
 The parametric time interval $\Delta t$ are enhanced relative to 
 the internal time interval $\Delta T$, which by the virtue of eq 
 (\ref{BTO1}) are related to the time, which the light would take to 
 travel the same distance. 
 On the other hand, for high ("relativistic") energies on obtain,
 
 In other words, in the  relativistic case internal time intervals 
 coincides with external time intervals, whereas in the non-relativistic 
 case the later is enhanced relative to the internal time intervals.   
 At this point it is important to note that in the presence of 
 a potential dependent only on position, e.g. Coulomb type potentials,

\begin{equation}
\label{THCR} 
 [\hat{T},\hat{H}_D+V(\tilde{r})]=[\hat{T},\hat{H}_D]  

\end{equation}
 hence the commutation relation of the time operator is reduced to 
 the commutation relation with the relativistic free  particle of 
 Dirac operator $\hat{H}_D$, since the later is a linear function of  
 relativistic momentum $\hat p$, from eq (\ref{BTO1}) one find that eq  
 (\ref{THCR}) is reduced to the position momentum commutation relation 
 $[\hat{r},\hat{p}]$.

For the tunneling in attosecond experiment Bauer uses the argument of 
 Kullie \cite{Kullie:2015}, that the potential energy at the exit point  
 define the uncertainty in the energy and leads to the time of passage 
 of the barrier or the time needed to cross through the exit point and 
 represents a tunneling internal time of the system $\tau_{T}$.  
 With his view of $\Delta T_B \simeq\tau_{T}$ Bauer obtained a relation    
 for the laboratory time lapse to cross the barrier,

\begin{eqnarray}
\label{BT2} 
 \Upsilon_T  =\frac{1}{4 \pi}\, \xi \,
 \frac{1}{\frac{1}{2}(\frac{v_{gp}}{c})^{2}}\, \tau_T = \frac{1}{4 \pi}
 \,\,2\,\xi \cdot \Gamma \cdot\tau_T

\end{eqnarray}
 where $\Gamma=(\frac{v_{gp}}{c})^{-2}$, is called the enhancement 
 factor, $\xi$ is a phenomenological parameter (see below), and 
 $\tau_T$ the internal time interval, compare eq (\ref{BT1}). 
\vspace{0.2cm}
\paragraph{Properties of BTO}\label{PBTO} 
 Despite that BTO is interesting and satisfy the property and the 
 conditions of an ordinary time operator in the relativistic frame 
 work of the QM. 
 There is some unexpected of the relation (\ref{BT2}) given by Bauer 
 as a consequences of the time operator definition in (\ref{BTO1}). 
  The laboratory time interval $\Delta t$ is connected the internal 
 time interval $\Delta T\equiv \Delta T_B$ by eq (\ref{BT2}). 
 However, with eqs (\ref{BTO1}), (\ref{BT1}) and similar  
 to eq (\ref{BT2})

\begin{eqnarray}
\label{BTP1} \nonumber
 \Delta T_B &=& \Delta \hat{r}/c = 4\pi\,
 \Delta \hat{r} /c = 4\pi \, \Delta r/c \\  
 
 \Delta t&=& \frac{1}{4 \pi}\, \xi \,
\frac{1}{\frac{1}{2}(\frac{v_{gp}}{c})^{2}}\Delta T

\end{eqnarray}
 Consequently one finds

\begin{eqnarray}
\label{BTP2} 
 \Delta t&=& \frac{1}{4 \pi} \kappa\,
 \frac{c^{2}}{\frac{1}{2}(\frac{\Delta r}{\Delta t}) v_{gp}} 
 (4\pi \frac{\Delta r}{c})\\ 
   &=&\, \xi\,  \frac{c}{\frac{1}{2} v_{gp}} \Delta t \Rightarrow
   v_{gb} = 2\, \xi\, c = const

\end{eqnarray}
 Bauer introduced the phenomenological parameter $\xi<1$ so that 
 $\Upsilon_T$ in eq (\ref{BT2}) somehow fits the experimental data 
 and the Feynman path integral calculation (FPI), presented by Landsman 
 \cite{Landsman:2014II}, compare fig 1 of \cite{BauerM:2017} (the same 
 plot is given in fig \ref{fig:tundB1}, sec \ref{EAF}).  
 He concluded that $\xi=0.45$ gives the best fit to the experiment. 
 The result of Bauer fit well the FPI result of Landsman 
 \cite{Landsman:2014II}, figure 1 in \cite{BauerM:2017} (as in fig 
 \ref{fig:tundB1}, sec \ref{EAF}).  
 However there is no theoretical justification for the choice of 
 $\xi$ as a parameter or its value $\xi=0.45 \Rightarrow v_{gb} >c $ in 
 eq (\ref{BTP2}).  
 This is rather unexpected result, then without the parameter $\xi$ 
 (i.e. $\xi=1$) one obtain $v_{gb} =2 c$, which is a consequence of the 
 definition in eq (\ref{BT2}) and eq (\ref{BTO1}), where the speed of 
 light is used in the denominator.   
 In other words, $\xi=0.5$ gives $v_{gb} =c$ as it should be in 
 accordance with the definition in eq (\ref{BTO1}).   
 The idea of Bauer to set an universal internal time eq (\ref{BTO1}) is 
 reasonable, however to use it to  measure external time intervals,  
 i.e. the relation between internal time intervals and external time 
 intervals eq (\ref{BT2}), leads to unexpected implications.    
 We will see with our tunneling model and our definition, or the 
 generalization of time operators of Bauer and Aharonov (see below), 
 that there is no need to introduce a phenomenological parameter.

\subsection{Time operator of the type Aharonov-Bauer}\label{TBA}
 The recent interesting work of Bauer and the introduction of time 
 operator in the frame work of the DRQM, together with the well known 
 Aharonov time operator (eq (\ref{KOPA}) below) has stimulate us to 
 a generalized form of a time operator of the same types.     
 We suggest that the time operator definition of Aharonov and Bauer 
 can be extended and combined as the following. For one particle 
 (in atomic units)

\begin{equation}
\label{KOP0} 
 \hat{t}_{gp}= {  \alpha} \, \frac{\hat{\bf r}}{v_{gp}},\quad
 \hat{t}_{ph}= {  \alpha} \, \frac{\hat{\bf r}}{v_{ph}}  

\end{equation}
 where ${v_{ph}}$, ${v_{gp}}$ are the phase and the group velocity, 
 respectively.  
 In the following we look to 1-dimensional case with the radial 
 coordinate $r$, i.e. we neglect the factor $1/(4 \pi)$ used by Bauer 
 of the 3-dimensional case, where $\mathbf{r}=4\pi\, {r}$. 
 And the Dirac matrices $\alpha$ will be set to unity 
 ${\alpha}=1$. 
 Under the definition in eq (\ref{KOP0}) the notation internal,
 external time operator is misleading.   
 We denote $\hat{t}_{gp}$ the dynamical time operator (Dynamical-TO) 
 and $\hat{t}_{ph}$ the phase (or phase-velocity) time operator 
 (Phase-TO), without any connection to the notation external, internal 
 time classification of Busch \cite{Busch:2008}. 
 The relation between the dynamical and phase times,  follows 
 immediately from eq (\ref{KOP0}), using an interval 
 $\Delta r=\Delta{t_{ph}}\cdot v_{ph}=
 \Delta{t_{gp}}\cdot v_{gp}$

\begin{equation}
\label{KOTt} 
 \Delta{t_{gp}} \cdot v_{gp} =\Delta{t_{ph}} \cdot\, v_{ph} 
 \Rightarrow \Delta{t_{gp}} = \frac{v_{ph}}{v_{gp}}\, \,
 \Delta{t_{ph}}\, \ge\, \Delta{t_{ph}}  

\end{equation}
 Because of the relativistic relation $v_{ph}\,v_{gp}=c^{2}$, for 
 a matter particle with a mass $m$, $E= m c^{2}$, $v_{ph}=E/p$ one 
 obtains $\hat{t}_{gp} \ge \hat{t}_{ph}$, where $p$ the momentum of the  
 particle.   
 For this reason it better to adopt the notations, $ \hat{t}_{ph}$ the 
 phase and $\hat{t}_{gp}$ the dynamical time operator. For the light 
 (photon) particle $v_{gp}=v_{ph}=c$ one obtain the BTO

\begin{equation}
\label{KOPB} 
 \hat{t}_{gp}= \hat{t}_{ph}= \alpha\, \frac{\hat r}{c}  

\end{equation}
 For the non-relativistic one-dimensional case ($\alpha=1$), it is  
 straightforward to obtain the well-known Aharonov-Bohm  
 \cite{Aharonov:1961} time operator (ABTO) ($\hat{t}_{gp}=  
 \frac{\hat r}{v_{gp}}$) or the quantum mechanical symmetric operator

\begin{equation}
\label{KOPA}  
  \hat{t}_{gp}= \frac{1}{2}\left({\hat r}\,\,{v_{gp}^{-1}} 
  + {v_{gp}^{-1}}\,\,{\hat r}\right)  

\end{equation}
 For the $\hat{t}_{ph}$ the symmetrization has no meaning, 
 because ${v_{ph}}$ is not an observable unlike ${v_{gp}}$. 
 The commutation relations were verified in both cases the 
 non- relativistic Bohm-Aharonov operator 
 \cite{Aharonov:1961,Paul:1962}, and the relativistic, Bauer operator 
 \cite{BauerM:2014,BauerM:2017}. 
 For $\hat{t}_{ph}$ because ${v_{ph}}$ is not an observable, 
 the commutation relation is reduced to the known commutation relation 
 $[\hat{r},\hat{p}]$.  
 
 The first consequence our definition is the equivalence to the BTO, 
 as given in eq (\ref{KOPB}) and to the ABTO eq (\ref{KOPA}).
 There is  also an equivalence between BTO and ABTO, then from  eq 
 (\ref{KOPA}) for a light  particle with ${v_{ph}}={v_{gp}}=c$

\begin{eqnarray}
  \hat{t}_{gp}= \frac{1}{2}\left(\frac{\hat r}{c}\,+\,\frac{\hat r}{c}\right)
  =\frac{\hat r}{c}   

\end{eqnarray}
 and for a matter  particle 
 ${v_{ph}}\cdot {v_{gp}}=c^{2}$

\begin{eqnarray}
\label{KOtt} 
 \Delta{t_{gp}}&=& \frac{\Delta{r}}{v_{gp}}
  = \frac{v_{ph}\cdot\Delta{t_{ph}}}{v_{gp}}
 =\frac{c^{2}/v_{gp}}{v_{gp}}\,\,\Delta{t_{ph}}\\\nonumber
 &=&\frac{1}{(v_{gp}/c)^{2}}\,\,\Delta{t_{ph}}

\end{eqnarray}
 as already found by Bauer, compare eqs (\ref{BTO2}), (\ref{BT2}), 
 (\ref{BTP1}). Where no approximation is used, but with 
 $\Delta{t_{gp}}$ instead of the parametric time $\Delta t$ and 
 $\Delta t_{ph}$ instead the internal time $\Delta T$ of Bauer's 
 notations.
 
 Bauer obtained the factor $\xi\,(1/2)^{-1}=2\,\xi$ in eq (\ref{BT2}) 
 by going from relativistic to non-relativistic approximation (the 
 factor $(1/2)^{-1}$) and using a phenomenological parameter $\xi$ to 
 fit the experimental data. 
 And $\hat{t}_{ph}$ is the phase time, whereas Bauer refers to it  
 as internal time $T_B$ (using $v_{ph}=c$). 
 In the Bauer case (notation) one has $\Delta T_B \le \Delta t_B$,  
 see eqs (\ref{BT1}), (\ref{BT2}), likewise we have a relation between 
 the dynamical and phase times, $\Delta  t_{ph}\le \Delta  t_{gp}$ eq 
 (\ref{KOtt}).   
 However, we have $T_B=\hat{r}/c \ge t_{ph}=\hat{r}/v_{ph}$  because 
 $v_{ph}\ge c$. 
       Further, we get Bauer procedure for 3-dimensional case by the replacing 
 $\Delta r/v_{ph}$ with $\Delta r/c$, which with $\Delta{\bf r}= 4 \pi
 \Delta r$ gives eq (\ref{BT2})

\begin{eqnarray}
\label{tgK} 
  \Delta \hat{t}_{gp} =(\frac{1}{4 \pi})\,
  \frac{1}{(\frac{v_{gp}}{c})^{2}}\Delta{t_{ph}}

\end{eqnarray}
 The only different is that Bauer used $2 \,\xi=2\cdot 0.45=0.9$, 
 compare eqs (\ref{BT2}), (\ref{BTP1}), where it is equal $2\cdot 0.5$,  
 or exactly $\xi=1/2$ in our case without any approximation, the 
 difference is very small and $\xi=0.45$ does not fit perfectly to the 
 experimental data of \cite{Landsman:2014II}, see below.   
 Thus, to refer to  $\Delta T=\Delta r/c$ as an internal time and 
 introduce a phenomenological parameter $\xi$ has no justification,  
 unless one relates every time interval to its counter part of an 
 unique time interval of the light propagation, which is $\Delta r/c$ 
 by replacing the phase time of a matter particle by $v_{ph}=c$, which 
 is straightforward and a parameter $\xi$ is redundant.
 
 With his approximation, Bauer obtained eq (\ref{BT2}),  
 whereas on the basis of our tunneling model, eq (\ref{Tdi}) can be 
 written, after a small manipulation, in the form ($F$ is the field 
 strength)

\begin{equation}
\label{UpsK} 
 t_{gp}(F)\equiv\Upsilon_K(d_W)= \tau_d  = \frac{1}{4 Z_{eff}}\,
 \left(d_W+ x_{e,-}\right)
 
\end{equation}
 where $d_W=\delta_z/F$ is the barrier width and $x_{e,-}=
 (I_p-\delta_z)/(2F)$ \cite{Kullie:2015}, compare fig \ref{fig:ptc}. 
 In the following to avoid a confusion, we refer in the general case 
 to a barrier width as $D_{BW}$. 
 Whereas in our model we set $D_{BW}=d_W$ and for the classical barrier  
 width we set $D_{BW}=d_C$. 
 On the basis of numerical values from the experimental data 
 in \cite{Landsman:2014II}, with approximate barrier width $d_C$ 
 (compare \cite{Landsman:2014II}, \cite{BauerM:2017}) and with the 
 values $d_C=13\,au$ (in atomic units), $\tau= 40\,as$ (in attosecond),     
 $F\approx 0.069\,au$, $v_{gb}=13/40\,( au/as)=6.88/40$
  {\AA}/as),  Bauer obtained \cite{BauerM:2017},

\begin{eqnarray}
\label{numU} 
  \Upsilon_B (D_{BW}) &=& \frac{1}{4 \pi}\,\,608.44\,\,
  {\rm A^{\circ}}/as)\,\,\left(\selectlanguage{greek}ξ\selectlanguage{english}\,
  \frac{D_{BW}}{c} \right)\\\nonumber
  &=&16.14 \cdot\xi \cdot D_{BW} \, \,as \,\,(D_{BW} \,\, {\rm in } \, \, 
  \,{\rm A^{\circ}}) \\\nonumber
  &=&4.52  \cdot\xi \cdot D_{BW} \, \,as \,\,(D_{BW} \,\,{\rm in }\,\, 
  \, au)
 
\end{eqnarray}
 Where $D_{BW}$ is a barrier width,  \AA is Angestr\"om 
 length unity and $au$ atomic units. 
 The factor $(2/[(v_{gb}/c]^{2})=2/[(6.88/40)/c]^{2}=608.44$ is 
 calculated by using the numerical data above, and the best fit 
 to the experimental data according to Bauer is $\xi=0.45$.    
 ONE notices that Landsman \cite{Landsman:2014II} assumed a classical 
 barrier width, i.e. $D_{BW}=d_C=I_p/F$ which is larger than $d_W$ 
 (compare fig \ref{fig:ptc}), it is usually taken to be approximately 
 valid and is adopted by Bauer.  
 However, as we will see below, this leads to a confusion in the 
 evaluation of the tunneling time against the barrier width.      
 Whereas from our model eq (\ref{UpsK}), (\ref{Tdi}), it follows that  
 $\tau_d =3.6\, (d_W+ x_{e,-}) \,as $,  using $Z_{eff}=1.6875$ of 
 Clementi \cite{Clementi:1963} (for small barrier width), or $\tau_d 
 =4.4\, (d_W+ x_{e,-})\,as$, using $Z_{eff}=1.375$ of Kullie (for 
 large barrier width), see \cite{Kullie:2015}, but no fitting 
 procedure is used and our $\tau_d$ is in good agreement with the 
 experimental data. 
     One can imagine that BTO, eq (\ref{BTO1}), presents an universal time  
 scale or internal clock of a light particle, a photon, but its 
 relation to the external time or clocks is then not presented by eqs  
 (\ref{BT2}), see below.   
 We think that our definition is a generalization, eq (\ref{KOP0}) with  
 a straightforward transition to both the BTO and the non-relativistic 
 ABTO.

\subsection{Experimental affirmation}\label{EAF}
 It is worthwhile to mention that the velocity 
 with which the electron passes through the tunnel vary slowly with  
 the barrier width. In our model and with some manipulation one 
 obtains the mean velocity as a function of $d_W$

\begin{eqnarray}
\label{vgpK} \nonumber
  \overline{v_{gb}(d_W)} &=& d_W/\tau_d = 
  \frac{d_W}{\frac{1}{2(I_p - \delta_z)}}
  =\frac{d_W}{(d_W+x_{e,-})/4Z_{eff}}\\
  &=&  4\,Z_{eff} \frac{d_W}{(d_W+x_{e,-})} < 4\,Z_{eff}
 
\end{eqnarray}
 For $Z_{eff}=1.6875,\,1.375$ we obtain the values 
 $\overline{v_{gp}(d_W)}<4Ze_{eff}=6.75, 5.5$ respectively.
 Using the experimental data at one point, see eq (\ref{numU}), 
 Bauer extracted a value $(2\cdot\xi/(v_{gb}/c)^{2})=\xi \cdot 608.44$   
 (compare eq (\ref{tgK}), or $ \overline{v_{gp}^{C}}=
 (d_C/\tau)\!\!\!\!\mid_{_{F=0.069}}=(13\,au)/(40\,as) = (13/1.66) 
 \,au =7.86\, au$, which was used as a fixed (independent of $F$) mean 
 value in eq (\ref{numU}) independent of the barrier width.  
 It is sightly larger than our values $6.75 (5.5)$, which is caused by 
 the use of the classical barrier width $d_C=I_p/F=d_W+2 x_{e,-}$, 
 compare eqs (\ref{BT2}), (\ref{numU}), and we think it is one of the 
 reasons why Bauer needed to introduce a phenomenological parameter. 
     
 In fig \ref{fig:tundB} we plot our result of tunneling time 
 $\Upsilon_K(d_W)$, eq (\ref{UpsK}), against the barrier width 
 $d_W(F)=\delta_z/F$, where $Z_{eff}=1.6875$ is used, together with   
 Bauer's tunneling time formula $\Upsilon_B(D_{BW}=d_W)$, eq 
 (\ref{numU}), with $\xi=0.45$ used by Bauer, and with our value 
 $\xi=0.5$ and the values $\xi=0.3, 0.6$ used by Bauer.     
  In addition the experimental tunneling time data (plotted against  
 $D_{BW}=d_W$) together with the error bars are shown. 
 The elementary data  or the experimental values of time of 
 \cite{Landsman:2014II} (at the corresponding $F$ values) were sent by 
 Landsman, where $d_W$ easily calculated from $d_W=\delta_z/F$.   
 As seen in the figure eq (\ref{UpsK}) (solid red line) shows the best  
 fit with the experimental data.  
 The dashed (pink) line of eq (\ref{numU}) with $D_{BW}=d_W$ and 
 $\xi=0.5$ is slightly below the red solid line (for small $d_W$), and 
 the dashed dotted (blue) one of eq (\ref{UpsK}) with $D_{BW}=d_W$ and 
 $\xi=0.45$ is slightly below both.     
 
 One notice we used $Z_{eff}=1.6875$ which is suitable for small 
 barrier width, whereas $Z_{eff}=1.375$ is better for large barrier 
 width, with wihch the lines (not shown) will slightly shift towards 
 higher values, and for large barrier widths the red line is then 
 closer to the experimental values, compare fig 5 in 
 \cite{Kullie:2015}, and the pink and blue lines stay below the red 
 line in this region. 
 
 The fig \ref{fig:tundB} has a small difference to the figure shown in 
 ref \cite{Landsman:2014II} (fig 3(d) Feynman path integral) and to  
 Fig 1 of ref \cite{BauerM:2017} (same as eq (\ref{numU}), with 
 $\xi=0.3, 0.45, 0.6$), where in both works $D_{BW}=d_C$ was used, and 
 where the plotted lines are slightly above the experimental data, i.e 
 a less agreement than in our evaluation plotted in fig 
 \ref{fig:tundB}.
  This is because, as mentioned, the classical barrier width $d_C=I_p/F=
 d_W+2x_{e,-}$ was used, compare \cite{Landsman:2014II}. 
 In other words a parameter $\xi$ is not needed, when one uses the 
 correct barrier width $d_W=\delta/F$, compare fig \ref{fig:ptc}.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/OKullie-KASE-gr2/TunndB}
\caption{{\label{fig:tundB}\scriptsize (Color online) Graphic display. 
 $x-$axis is the barrier width $d_W$ in $au$, vertical axis the 
 tunneling time in $as$ of formula eq (\ref{UpsK}) $\Upsilon_K(d_W)$, 
 and of formula eq (\ref{numU}) $\Upsilon_B(D_{BW}=d_W)$ with 
 $\xi=0.3,0.45,0.5,0.6$. 
 Also shown experimental tunneling time data  at the corresponding 
 barrier width $d_W$ with error bars see \protect\cite{Kullie:2015}.  
 Data were kindly sent by Landsman, see \protect\cite{Landsman:2014II}.%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Tunn-dC/TunndB-1}
\caption{{\label{fig:tundB1}\scriptsize (Color online) Graphic display.
 horizontal axis (classical) barrier width $d_C$ in $au$, vertical axis
 the tunneling time in $as$. 
 The figure reproduce Fig 1 of \protect\cite{BauerM:2017} (using figure 
 digitizer), tunneling time of Bauer formula for $\xi=0.45$, the FPI 
 and the experimental data, note these data are extracted from Fig 1 
 of \protect\cite{BauerM:2017}), compare fig 3(d) of \protect\cite{Landsman:2014II}. 
 Together with the formula in eq (\ref{numU}) $\Upsilon_B(d_C)$ 
 (same formula of ref \protect\cite{BauerM:2017}), for $\xi=0.45,\xi=0.5$, 
 showing we reproduce the result of Bauer.%
}}
\end{center}
\end{figure}

For a clarity we plot in fig \ref{fig:tundB1} a copy of fig 1 of ref 
 \cite{BauerM:2017} (extracted data by using web digitizer), i.e. the 
 data of Bauer $\Upsilon_B(D_{BW}=d_C)$ for $\xi=0.45$ (dahsed, black) 
 and also the FPI (solid, light blue) which used by Bauer after it 
 extracted form fig 3(d) of ref \cite{Landsman:2014II}.  
 Additionally we plot $\Upsilon_B(D_{BW}=d_C)$ with $\xi=0.45$ (dashed 
 doted, magenta), which as expected reproduce the line of Bauer (dashed, 
 black), the tiny difference is only because we extract the data of 
 Fig 1 of Bauer \cite{BauerM:2017} by web digitizer. 
 Also $\Upsilon_B(D_{BW}=d_C)$  with $\xi=0.5$ is plotted (dashed 
 dotted dotted, blue), which lies slightly higher.     
 We can reproduce the data of Bauer, which makes our conclusion 
 reliable.  
 Thus, we can see why Bauer found that $\xi=0.45$ fits better to FPI, 
 it is because the use of approximate barrier width $D_{BW}=d_C$, 
 precisely speaking the use of $D_{BW}=d_C$ is incorrect. 
 The small difference was not crucial for the Landsman in the work 
 \cite{Landsman:2014II} (fig(3d)), but Landsman noted that it is 
 an approximate barrier width, unlike our model \cite{Kullie:2015}  
 (published later), where a correct barrier width $D_{BW}=d_W=
 x_{e,+}-x_{e,-}=\delta_z/F$ is obtained, compare fig \ref{fig:ptc}. 
 Our conclusion, although the difference between $d_W$ and $d_C$ seems   
 to be not crucial as thought by Landsman \cite{Landsman:2014II}, but  
 the use of a barrier width $D_{BW}=d_C$ instead of $d_W$ leads to 
 a confusion and is misleading, when plotting the tunneling time data 
 against the barrier width, and led Bauer to introduce a parameter 
 $\xi=0.45$, which in our view is unnecessary. 
 Regardless how to understand Bauer's definition of ``internal'' time 
 operator in eq (\ref{BTO1}). 
 Thus our definition of a time operator eq (\ref{KOP0}) is reasonable,  
 it is a general form with a straightforward transitions to BTO eq 
 (\ref{BTO1}) \cite{BauerM:2014,BauerM:2017} and ABTO 
 \cite{Aharonov:1961}.
                   
 
           A final note to demonstrate the importance of our model to the 
 tunneling theory in general, because it relates the tunneling time to 
 the barrier height. 
 The T-time found in Eq. (\ref{Tsym}) can be also derived in a simple 
 way, when we assume that the barrier height corresponds to a maximally 
 symmetric operator as the following.
 The barrier height $h_B(x_m)\equiv h_M$ can be related to a (real)
 operator $\widehat{h}_B(x_m)$

\begin{equation}
\label{hm1} 
 \widehat{h}_M^{\pm}=-I_p\pm\sqrt{4 Z_{eff}F}= -I_p\pm\Delta
 
\end{equation}
 and the uncertainty in the energy caused by the barrier

\begin{equation}
\label{hm2} 
 \left(\widehat{h}_M^{-} \widehat{h}_M^{+}\right)^{1/2}=
 [(-I_p-\Delta)(-I_p+\Delta)]^{1/2}=\delta_z
 
\end{equation}
 From this we get ${\Delta E^{\pm}}=abs(-Ip\pm\delta_z)$, i.e. we have 
 to add (subtract) the internal energy of the system (the ionization 
 potential).  
 And hence we get $\tau_{_{T,d}}$, $\tau_{_{T,i}}$ and 
 $\tau_{_{T,sym}}$ by the virtue of the time-energy uncertainty 
 relation, where we assumed that the time is intrinsic and has to be 
 considered (a delay time) with respect to the ionization (time) at 
 atomic field strength, i.e. in respect to the internal energy or the 
 ionization potential $I_p$.   
 This suggests to consider $h_M$ as a perturbation (energy) operator, 
 where the full operator of the system can be taken as 
 $H_0 + h_M^{\pm}$. 
   In fact, one can argue that $-Ip$ must be taken to avoid the 
 divergence of the time to infinity for $F=F_a$, because it is 
 physically incorrect, as $\delta_z(F_a)=0\Rightarrow \tau=1/(2\delta_z) 
 \rightarrow \infty$, which in turn can be seen as an initialization 
 of the internal clock, i.e. the T-time is counted as a delay with 
 respect to the ionization at $F_a$ (the limit of the subatomic field  
 strength), after which no tunneling occurs and the ionization is 
 classically an allowed process, the barrier-suppression ionization.
 Here are new references \cite{Kullie:2018II}


{\bf Conclusion} In this work we showed that our
tunneling time model for the tunneling in attosecond angular streaking  
 experiment, enables us to discuss and understand  the time and time 
 operator in quantum mechanics, where we found a generalized form  
 of a time operator with straightforward transitions to the Bauer  
 time operator, which was introduced  in frame work of relativistic 
 Dirac theory and to the Aharonov-Bohm time operator, which was 
 introduced in the non-relativistic quantum mechanics.   
 Moreover, we found that the introduction of a phenomenological 
 parameter as done by Bauer is unnecessary. 
 The issue is resolved by recovering a confusion caused by the use 
 a classical barrier width, and is clarified by using the correct 
 barrier width, as found by our tunneling model. 
\twocolumn\cite{2007}

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