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\begin{document}
\title{TunnelingtimeandtimeinQM}
\author[1]{Kullie}%
\affil[1]{International Winter University}%
\vspace{-1em}
\date{\today}
\begingroup
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\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 $F1$ one expects predominantly photo-ionization
or multiphoton ionization (MPI), while at $\gamma_{_K}<1$
(field-) ionization happens by a tunneling process, which occurs for
$F F_a$ 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_{2}-t_{1})=\Delta t
\]
\begin{eqnarray}
\label{BT1}
\Delta T_B&=&- \\ \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}ΞΎ\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.
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