1. Introduction
Attosecond science (
s) concerns primarily electronic motion and energy transport on atomic scales. In previous works [
1,
2,
3], 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 a (Heisenberg) conjugate pair. Our T-time is in good agreement with the attosecond (angular streaking) experiment for He-atom [
1] with the experimental finding of Eckle et al. [
4,
5,
6], and for hydrogen atoms [
7] with the experimental finding of Sainadh et al. [
8]. Our model presents a real T-time picture or a delay time with respect to the ionization time at atomic field strength
(see below, compare
Figure 1). Our T-time model is also interesting for the tunneling theory in general because it relates T-time to the energy gap or the height of the barrier [
1,
2].
Indeed, the role of time has been controversial since the appearance of quantum mechanics (QM). The best known example is the Bohr–Einstein weighing
photon box Gedanken experiment (BE-pb-GE) [
9] and [
10] (p. 132). Our T-time picture [
1] shows an intriguing similarity to the
BE-pb-GE, where the former can be seen as a realization of the later [
1,
3]. Concerning the time operator in QM, recently Galapon [
11,
12,
13] showed 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 objection of Pauli (known as Pauli theorem). The result is, as noted earlier by Garrison [
14], for a canonically conjugate pair of operators of a Heisenberg type (i.e., uncertainty relation), the Pauli theorem does not apply, unlike a pair of operators that form a Weyl pair (or Weyl system.)
Our tunneling model was introduced in [
1] (see
Figure 1). We take a one-dimensional model along the
x-axis as justified by Klaiber and Yakaboylu et al. [
15,
16]. Hereafter, we adopt the atomic units (au), where the electron’s mass and charge and the Planck constant are set to unity,
. In this model, the effective potential of the atom-laser system is given by
where
F (throughout this work) is
the peak electric field strength (at maximum) of the laser pulse (quasistatic limit), and
is the effective nuclear charge that can be found by treating the (active) electron orbital as hydrogen-like, similar to the well-known single-active-electron (SAE) model [
17,
18]. The choice of
is easily recognized for many-electron systems and well-known in atomic, molecular, and plasma physics [
19,
20,
21,
22].
The active electron can be ionized by a short laser pulse with an electric field strength
F, where ionization happens directly when
F equals a threshold called atomic field strength (see below Equation (
4))
[
19,
23], where
is the ionization potential of the system (atom or molecule). However, for field strengths
, ionization can happen by a tunneling mechanism, through a barrier built by the effective potential and the ionization potential, as seen in
Figure 1. The barrier height at a position
x is given by (see
Figure 1)
that is equal to the difference between the ionization potential and effective potential
of the system (atom+laser) at the position
x. The crossing points
of
with the
-line are given by [
1]
is the barrier width, where
is given below in Equation (
5).
is usually called the “classical” exit point; it is the intersection of the field line
with
-line, which equals what is usually called the “classical” barrier width
.
We obtain the maximum
from the derivative of Equation (
2),
. From
, see
Figure 1 (the lower green curve), we get the atomic field strength
Fortunately, Equation (
4) can be generalized as the following. For a field strength
:
the equality
occurs at
. Indeed,
is a key quantity; it controls the tunneling process and determines the time “delay” due to the barrier (
) as we will see.
In
Figure 1 (for details see [
1]), the inner (entrance
) and outer (exit
) points,
. Note that, for forward, backward tunneling, the entrance/exit, Equation (
3) is exchanged, where the barrier hight (at maximum) is given by (compare Equation (
4)
. The energy gap is given by
[
2]. For
(
), the barrier disappears and
at which the barrier-suppression ionization (BSI) starts.
In the (low-frequency) attosecond experiment, the laser field is comparable in strength to the electric field of the atom. Usually, intensities ∼10
W cm
are used. It is usual to characterize the strong-field approximation (SFA) by the Keldysh parameter [
24],
where
is the central circular frequency of the laser pulse and
denotes the Keldysh time. According to Keldysh or SFA, in Equation (
6), at values
, the dominant process is the multiphoton ionization (MPI). For
(precisely
), the ionization (or field-ionization) happens by a tunneling process, which occurs for
. SFA was developed and refined later by Faisal [
25] and Reiss [
26] and known under the term Keldysh–Faisal–Reiss (KFR) approximation, where the two regimes of multiphoton and tunneling are more or less not strictly defined by
[
27,
28,
29]. In the tunneling regime (for
) at a quasistatic limit, the electron does not ionize directly. It tunnels (tunnel-ionizes) though adiabatically, and escapes the barrier at the exit point
to the continuum as shown in
Figure 1 (a sketch for He-atom.)
With this model, we derived the following relations of the T-time [
1],
As discussed in [
1,
2]
(or
) correspond to the forward, backward tunneling, respectively. An intuitive picture is given by a physical reasoning of these relations [
1] as the following:
is the time delay with respect to the ionization time at atomic field strength
. The latter is undoubtedly real and not zero
.
is the time delay for a particle to pass the barrier region and escapes at the exit point
to the continuum [
1].
is the time needed to reach the entrance point
from the initial point
, compare
Figure 1. The two steps of the model coincide at the limit
, and the total time becomes the ionization time at the atomic field strength
,
,
. For
, the BSI starts [
30,
31]. At the opposite side of the limit
,
and
, hence, nothing happens, i.e., the electron remains in its ground state undisturbed, indicating that our model is consistent; for details, see [
1,
2,
3]. For the first term
, see below
Section 3.
3. Aharonov–Bohm Time Operator
In
Section 2.2, we have seen that classically one obtains the Keldysh time
from
p/
F (or Δ
p/
F) and similarly from
x/
p (or
). In this section, we discuss a derivation of the T-time from the Aharonov–Bohm time operator (ABTO) the counterpart of classical form
or
, as we have done for the counterpart of
in
Section 2.2, the Fujiwara–Kobe time operator
of Equation (
11). The well known ABTO for a free particle with a momentum
p and position
x [
48] is
where the second line is the momentum representation, and the third is the energy (
) representation, see Allcock [
43], where a transformation of the wave function
leads to
We first compare our T-time in Equation (
7) with Equations (
20) and (21). In the limit of
,
(ground states)
It is clear that at this limit (ground state) the electron can not escape the atom
[
1], whereas
corresponds to the second term of ABTO in Equation (21). We will see later that
correspond (forward, backward) to
, the Olkhovsky time operator. Thus, the limit for
reveals a connection between our T-time and ABTO. In the general case
, we use Equation (21) as the following. When the electron interacts with the field, we can assume the quasistatic (adiabatic) dynamics tunneling model (QSTM). The electron moves through the barrier region adiabatically with a mean momentum
, which, as discussed in [
45], can be seen from
The QMal correspondence to
(i.e.,
) leads to
as the following. With
and
, and noting that the QM correspondence to
is
, the time to reach the exit and entrance point, respectively, and their difference is the time due to the barrier itself, we get
The similarity between Equations (21) and (
24) is obvious. It should be understood as a mean value due to the assumption
. However, by comparing Equation (
24) with Equation (
20), we clearly see the difference in the minus sign. In Equations (
24) and (25), the ABTO with the minus sign or
corresponds to the time of arrival time (or transit time) as we have seen in
Section 2.2.2, Equation (
18) (where
), we discuss this later below (see Equation (
29)).
Our total T-time is obtained by taking the plus sign. Then, with a mean momentum
,
and with Equation (
20), we obtain
In fact, it has been already discussed in [
1] that the symmetry of the tunneling process (forward, backward) leads respectively to
and the total time
, see Equation (
7)—hence, the factor
, which implies a relation of
to the ABTO. Equation (
26) might be considered as the symmetric form of ABTO, whereas Equation (
24) is the antisymmetric form. QMly a symmetrization leads to a real quantity, which is obvious in Equation (
26) (or (27)) and it is the time delay
given in Equation (
16) (see also Equation (
33)). However, this is also the case in Equation (25), since
is real which involves a tunnel ionization. Surprisingly, we obtain from ABTO a delay time
(as given Equations (
16) and (
33)) of a process, in which
the tunneling part (due to the barrier itself) cancels out, assigned a self-interference term by Winful [
42]. The importance of this result especially to the nonadiabatic effects is discussed in [
41].
Actually, as seen in Equation (27), the delay time due to the barrier itself is eliminated by the sum of the two process forward and backward in the immediate step (at second line), which invokes the question about the real, imaginary pictures of the T-time, whereas the antisymmetric form in Equation (25) eliminates the self-interference term and performs the delay time due to the barrier itself or the dwell time, compare Equation (
32). A complex T-time picture can be obtained by the substitution
or equivalently an imaginary barrier gap
, which leads to the same result in Equation (
26)
Apart from the fact that
is classically allowed as discussed in [
1] (see the physical reasoning of Equation (
7)), which means no need to an imaginary part (i.e.,
is redundant), only the real part of the delay time
(unlike
, which is real) can be compared with the experimental and the Feynman Path Integral (FPI) results in the adiabatic tunneling picture [
1,
4], see discussion later in
Section 6. Hence, in our view, the complex delay time of Equation (
28) can not satisfactorily describe the delay time in the attoclock,
Section 6.
However, the choice of the sign allows a question opened namely about the symmetric and the antisymmetric forms given by Equation (
26) or Equation (
24). In the former (
), we obtain the self-interference term, which eliminates the tunneling part, or in other words no tunneling is involved is this case. This is in accordance with the well known observation of the numerical methods that the velocity gauge the SFA (see also Reiss [
29]), in which no tunneling time is encountered, as we will discuss in
Section 6. The later (
) corresponds to
and we calculate
,
as done in Equations (
10), (
14) and (
19). In this case, forward and backward processes eliminate the self-interference term and give the net time due to barrier itself, and leads to
with a good agreement with the experimental result [
1] in the adiabatic tunneling [
1,
4], as already mentioned. In terms of a classical point of view, it is nothing but the time to reach the exit point
(forward tunneling
) minus the time to reach the entrance point
(backward tunneling
) (multiplied by the factor
for half scattering, which is of a QMal origin) and this is what makes
equivalent to the time of arrival (
Section 2.2.1 and
Section 2.2.2), the Wigner time (further below
Section 4), and the dwell time as we already noted in
Section 2.1. Note, in all our time relations, that we can drop out the factor
and we get, for example,
, which fits exactly to the experimental data of Sainadh et al. for H-atom and the accompanied result of numerical integration of time-dependent Schrödinger equation (NITDSE). It makes, however, the agreement with the experimental data of Landsman et al. for He-atom lacking this factor.
The question is now which of the experimental data, He-atom [
4], or H-atom [
49] is the right choice to compare with. The factor
is of a QMcal origin as we see from Equations (
24) and (
26); see further below Equation (
29). Thus, we think that the data of Landsman et al. is reliable. On the experimental side, the difference comes from the evaluation of the experimental data, i.e., how the final photoelectron momentum distribution and the streaking offset angle difference transformed to a tunneling delay time. Finally, the same factor was also introduced by Wigner in the scattering or atomic collision to calculate the Eisenbud–Wigner time delay, as we will see in the next section. Note that it is only when
is an imaginary quantity that we end with an imaginary T-time picture, the one supported by Sainadh et al. [
49], see
Section 4 and
Section 6.
In the energy representation, the T-time is given by the form of a bilinear operator of Equation (
22), see Olkhovsky [
50]
where
means to the left, right (or forward, backward) applied forms
,
. The comparison of Equations (
26) and (
29) with Equations (
20)–(
22) (and the transition
) makes it clear that ABTO can be used to derive the (or directly connected to our) T-time, though we take advantage of the QSTM. This is similar to the approach introduced by Allcock in his seminal work [
43] to define the time of arrival (TOA) and obtain the TOA distribution or observable time after the classification of Busch [
51] (chap. 3). In addition, it is again related to the similarity we encountered before with Fujiwara time of arrival
, see
Section 2.2.3. It turns out that a unified picture of the tunneling time is reliable, as we will see in
Section 4.
Note that
is a parameter with
[
52] for a small barrier when the electron escapes closer to the nucleus (large
F).
is better for a large barrier width when the electron moves far from the nucleus (small
F), see Figure 4 of [
1]. In a more reliable approximation,
depends on the barrier width (or the distance to the nucleus). However, there is no such reliable approximation available so far.
4. Relation to Eisenbud–Wigner Delay Time
As we have seen in
Section 3 and Equation (
24), the barrier width can be used to obtain the delay time
. Surprisingly, this is also useful to find a relation to Eisenbud–Wigner time or the more commonly used term
mentioned in
Section 2.1. The
has a long history going back to atomic collision and particles scattering by a Coulombic potential. We find in [
35,
53] (and according to Wigner [
54,
55]) the following definition of the
:
where
are the barrier width and the velocity of the particle or the propagation velocity of the wave packet, respectively, and
k is the wave vector of the monochromatic wave packet. Similar to Equation (
24), we get for a tunneled electron (wave-packet) the time shift [
35] due to the barrier (
)
where again
(
as mentioned above) is the mean velocity of the particle (propagation velocity of the wave-packet) in the tunneling region. Obviously, one has to add the barrier-free term to get the total EWS-time
[
35]. Thus, adding the self-interference term (see Equation (
32)) or
(Equation (
16)) as we have done in Equations (
10), (
14), (
19) and (
26), we get immediately the time delay
. Clearly,
corresponds to the
. Although this result is unexpected, we think it is not surprising. Then, from Equation (
30), we see immediately the similarity to Equations (
22) and (
29) in the energy form
, where the phase
is related to the semiclassical action. We note that such a unified T-time picture was already found by Winful [
42] for the quantum tunneling of a wave packet or a flux of particles scattering on a potential barrier. Winful showed in his work that the group delay or the Wigner time delay can be written in the form
where
is the dwell time which corresponds to our
as mentioned in
Section 2, and
is a self-interference term which corresponds to our
in Equations (
16) and (
33), see further below
Section 6. Recently, Han et al. [
56] tried also to find a unified T-time picture, but they claimed that the concept of T-time depends on the model used to derive it. We think that the unified picture brought by Winful is more reliable, especially since it fits well with our result.
It is worthwhile to note that Bray et al. used a similar point of view with a semiclassical Keldysh–Rutherford (RK) model based on the Rutherford scattering [
33,
57]. They found a Coulombic origin of the attoclock offset angle (ACOA). Nevertheless, they claimed to confirm an imaginary T-time picture, which was justified by NITDSE with a short-range potential, see
Section 6.
In the imaginary time picture as also done by Bray et al., one claims that experimentally measured ACOA is rooted to the tail of the potential, whereas the T-time is imaginary and its real part is zero (instantaneous tunneling) with a decomposition of the form
(
), where
are the saddle point solution, the exit time and the T-time, respectively, see [
49,
58]. According to [
58], the exit time is the time at which the electron leaves the tunneling barrier, or the ionization time that is in the tunneling picture corresponds to the moment at which the electron emerges in the classically allowed region [
58].
As we found in this section, permits a scattering mechanism interpretation for the T-time where the ACOA corresponds to a real because and hence are real quantities.
An important point is that we now clearly see that the factor
in Equation (
30) (and the same in
) enters because the tunneling is half scattering. Taking a second
factor by symmetrization (anti-symmetrization) as done in Equations (
24)–(27) (and (
28)) corresponds to forward, backward scattering. In our view, this again shows that the result of Landsman et al. [
4] for He-atom [
1], is more reliable than the result of Sainadh et [
49] for H-atom [
7] (which lacks a factor
), as already mentioned in
Section 3.
Therefore, the adiabatic tunneling process can also be understood in term of scattering in which the ACOA is due to the scattering by Coulombic potential in the barrier region (not the tail of the potential) and corresponds to the T-time with real quantities (i.e., a real T-time), whereas Bray et al. [
33,
57] as mentioned above claims that their RK model confirms the imaginary time picture or that the ACOA is due to the tail of the Coulombic potential by comparing the model to a numerical result. In [
57] by a comparison to experimental results for H-atom [
49], they found that the latter confirms the
dependence of RK-scattering curve (where
I is the intensity of the laser pulse), which is in accordance with an
dependence of the T-time, e.g., Equation (
15) with an additional nonlinear term
, which is quite smaller than unity (the first term).
Thus far, we see that our real T-time picture is highly consistent and fundamental. However, we have to add in regard to imaginary tunneling picture that it is common (rather fundamental) in QM that two pictures exist for a QMal physical process or phenomena [
7]. Indeed, the debate continues and a complete picture of the T-time is not reached so far.
5. Discussion
In
Section 2.2, we introduced a time operator and calculated the T-time Equations (
14) and (
19). Our approach is supported by the works of Busch et al. [
46], Fujiwara [
37], and Kobe et al. [
38]. We also have shown in
Section 3 that the Aharonov–Bohm time operator leads to the same result Equation (
24), by taking advantage of a QSTM or an adiabatic tunneling with a mean value of the momentum
. A similar conclusion is found in
Section 4 with the Eisenbud–Wigner delay time
. It turns out that the specific form of the time operator is not crucial, and a dynamical variable such
p of the system under consideration maintains the time observable through an operator form, which represents this time observable. This is similar to the classical case or limits
, which lead to the Keldysh time
, as mentioned in
Section 3. Thus, time operators representing a dynamical variable (intrinsic time) are equivalent.
Furthermore, to our knowledge, this is the first time where one encounters a close relation between T-time delay and time of arrival, see
Section 2.2 last paragraph and
Section 3. Time of arrival is widely used to calculate T-time. However, the usual belief is that time delay and time of arrival are two different definitions of the time observable, i.e., they result in two time intervals (different
’s) for the same process in quantum systems. According to our present work, we argue that intrinsic time can be represented equivalently by different well defined (dynamical) operators. The most known types, among others, are the delay time and the time of arrival. It is worthwhile to explain what we mean with the above stated well defined. An example is the definition of the Mandelstam–Tamm time operator (MTTO) [
59,
60,
61] (
are two operators of the system, and
is the wave function)
Recently, Gray et al. [
60] argued that the TEUR-version of MTTO depends on the states that are not eigenvectors of an observable. It means that the Mandlestam–Tamm time (MT-time) is the infimum of the ratio of static uncertainty to dynamic uncertainty per unit time (see discussion in [
2]). Messiah [
62] used MTTO to define “the characteristic time of evolution”, whereas Gray et al. claimed that this formulation recommends itself since it does not promise too much. They also provided what they called a (mathematical) precise definition of MTTO [
60]. However, following Kullie [
2], the MTTO can be, in fact, brought in line with the time of arrival of ABTO and the time delay in Equation (
7), by using
p as a dynamical observable. Furthermore, this contrasts the claim of Orlando et al. [
63] that the MT-time is equal to the Keldysh time. Our conclusion is that, under the same conditions, well-defined time operators by a dynamical observable equivalently represent one and the same intrinsic (dynamical) time or time observable of the system (called tempus by Kobe et al. [
38]), a clock dynamically attached to the system. The remaining question concerns the equivalence between different time operators, which are defined on the basis of different dynamical observables of the system. In the present work, solely
p was used as a dynamical observable.
6. Attoclock and Tunneling Time in Strong Field Interaction
Attosecond angular streaking experiments (termed attoclock) have triggered a hot debate about T-time, whether it is a real or an imaginary quantity. In adiabatic tunneling, for example [
4,
5,
6] with a Helium (He) atom, we have found that T-time is real. It is a delay time with respect to the ionization at atomic field strength [
1,
2,
3]. In an experiment with H-atoms, Sainadh et al. [
49] argued using a short-range potential model and NITDSE that tunneling is instantaneous. Hence, they support the point of view that T-time, i.e., the time to traverse the barrier region, is an imaginary quantity. In [
7], we found that the measured data in the experiment with H-atom of Sainadh et al. and the accompanied NITDSE [
8,
49] fits very well to our real T-time picture (Equation (
7)). Solely the factor of (1/2) is still unclear; it is absent in the experimental data of Sainadh et al. [
8,
49], unlike the data of Landsman et al. [
4] for the He atom, as discussed in
Section 3 and
Section 4. Furthermore, according to the imaginary time picture, one also claims that time is a parameter in QM and not an observable. Hence, no time operator exists in QM in line with the well known Pauli theorem. However, the later point has been clarified and the list of references is too large to be specified; see, for example, [
11,
12,
13,
37,
38,
39,
45,
64,
65,
66,
67] and the references therein. The present work is a step in this direction.
Despite this, Rost et al. [
68] argued that, thanks to the attoclock experiments with He- and H-atoms, one can be sure that a reasonably defined T-time is zero in these cases. Furthermore, Rost et al. concluded that, using classical back propagation of an ionized wave packet (determined QMly by Schrödiger equation), one arrives at the conclusion that the T-time defined at the exit point (as defined in their work) is zero for single-electron dynamics, see Ni et al. [
69]. Tunneling is a QMal manifestation and a classical point of view is not appropriate anyway, a deterministic T-time is not expected. Furthermore, although they found a positive time at the exit point by back propagating with the position-based criterion, they claimed that the T-time is zero and the criterion can be considered as a misinterpretation of the attoclock experimental data based on models that do not take full and consistent account for nonadiabaticity, with the argument that they found a high non-tunneled fraction [
70], we come back later to this important point. In the same work [
69], they found that a static energy-based (again in their notation) adiabatic tunneling criterion fails completely.
The core point in the result (Equation (
7)) of our tunneling model is the real T-time to overcome the barrier region as a delay time with respect to the ionization time at atomic field strength; the latter is undoubtedly real and QMly does not vanish. The issue becomes more evident by rewriting Equation (
15) in the form (see also Equation (
16))
where both terms are time delays and real, the second term
is real because
is a real quantity for
[
1]. The relation to
,
, and
of ABTO that we have discussed in
Section 3 and
Section 4 confirms our point of view, in particular that
provides a similar picture of a scattering particle on a Coulombic potential. However, it is not yet entirely clear whether the origin of the delay time is the Coulombic barrier, due to the similarity with
, or due to the tail of the Coulomb potential, as claimed by th imaginary T-time picture [
49,
58]. The latter enforces a
, which is not reflected in our comparison of the experimental data, especially for
F values near the atomic field strength
, where
shows a good agreement with the experimental result [
1]. Our model uses the length gauge or the dipole approximation of the interaction Hamiltonian [
2] due to the Göppert–Mayer gauge-transformation [
71]. However, to claim that it is a kind of a position criterion seems vague, apart from the fact that Equation (
15) fits well with the experimental results, as shown in [
1]. We think that the work of Ni et al. [
69] indicates that numerical methods, such as back propagation including NITDSE imply a dependence on the applied concepts, e.g., the back propagation with the different criteria in the work of Ni et al. [
69]. Hence, they end with inconclusive results and interpretations. A drawback of their approach is that they only consider backward propagation, which is not sufficient to give a clear tunneling picture (forward propagation is not considered).
Actually, we have already encountered such a matter in the work of Orlando et al. [
63], where they claimed to obtain Keldysh time as a lower limit of the T-time from the NITDSE and the MTTO. However, we found in [
2] and discussed in the present work that applying the MTTO in our tunneling model leads to the T-time relation in Equation (
7), which agrees well with the experimental results. Reiss [
29] divided the strong field approximation methods into two categories, strong electric-field approximation (SEFA) and strong propagating-field approximation (SPFA). He argued that they are critically different. The SEFA employs a tunneling model that trends to an adiabatic limit at low frequencies; the SPFA does not invoke the tunneling concept. This could be one of the reasons why Ni et al. get different results by applying different criteria. It is worth noting that earlier works using NITDSE [
72,
73] have also reached a similar conclusions like Ni et al. As a consequence, the conclusion of Ni et al. of an imaginary T-time, though possible, is not a closing answer and does not tell much concerning the T-time and tunneling theory in general. The imaginary T-time picture is in line with the fact that the tunneling is classically forbidden. This is important; nevertheless,
it is not instructive in the case it obscures an insight and otherwise accessible conceptual understanding (See Carver Mead, the Nature Of Light What Are Photons,
www.cns.caltech.edu). It offers a picture of tunneling mechanisms among other pictures, a situation that is common in QM. A complete picture to the tunneling in strong field science is still not available, where certainly it is crucial to distinguish between adiabatic and nonadiabatic tunneling.
Our T-time concept for adiabatic tunneling opens a new insight for the understanding and of a fundamental importance to the tunneling theory in general, as mentioned in
Section 1. Note that a complex time point of view, i.e., real and imaginary parts, such as the one brought by Torlina et al. [
58], would not change our conclusion. The difference is that Torlina et al. [
58] claim that the measurement of the attoclock, the real part, corresponds to the tail of the atomic Coulomb potential by using a short-range potential model, whereas in our picture the atomic potential energy at the exit point of a single-active electron model determines the uncertainty in the energy and, by the virtue of TEUR, we get the real delay time measured in the attoclock; see our detailed discussion in [
3]. The utilization of TEUR in our model is important to compare with the statistical point of view of the T-time and to the tunneling theory in general, as we will see in the following.
The measurement data in both experiments with He- and H-atom are of a statistical nature. Landsman et al. have shown [
4] a good agreement between the experimental result and the FPI. Demir et al. [
74] came to the same result of the T-time using a statistical approach to the T-time (SATT) based on FPI. Camus et al. [
75] claim that the agreement between theory and experiment in their work with Ar-, Kr-atom provides clear evidence for a nonzero T-time delay, however without clear details about their tunneling picture, i.e., adiabatic or nonadiabatic, which is certainly a crucial point. In our T-time model, we have found a good agreement with FPI of Landsman et al. for He-atoms [
1], and with NITDSER of Sainadh et al. for H-atoms [
7]. Thus, as far as adiabatic tunneling is considered, we think the question is rather how to compare our result and model with the FPI, NITDSE and other statistical and wave packet dynamics model calculations. We will now discuss a possibility to resolve this puzzle.
First, as known, a single electron model and statistical or probabilistic interpretation inherently coexist in the QM. One thinks about a single electron atom such as the H-atom, the Bohr model, the radial Schrödiger equation, and the Born interpretation of its wave function.
Second, our model is not alone or a pure semiclassical single-active electron model because our T-time is derived by the virtue of the TEUR [
1], and, in this sense, it is a QMal model with a direct relation to the statistical or probabilistic point of view, see Landsman et al. [
4,
35]. It is not the time spent by a single electron moving over or through a barrier having an energy less than the barrier height, which is a classical inadequate picture to offer an understanding to our result and the T-time in general. Having such a classical picture in mind does not help to find the correct answer or the right interpretation to the T-time.
In his work, Galapon [
76] constructed a time of arrival operator for a square potential barrier and calculated its expectation value for a wave packet. By a comparison to the result of Eckle et al. [
5,
6] for He-atoms, he showed that only the above-the-barrier components of the momentum distribution of the incident wave packet contribute to the traversal time to cross the barrier region. In statistical methods or wave packet dynamics, one considers the momentum distribution which corresponds to a collective incident of a stream of particles (a statistical ensemble) upon a potential barrier. From this point of view, what Galapon found is that the particles partially tunnel in real time (traversal time). Thus, in terms of FPI [
35], this is the part that corresponds to the amount of the transmitted particles or transmission amplitude. It is the part of the wave function that corresponds to a specific time
that the electron spends inside the potential barrier [
35]. In this sense, one can imagine that our T-time would correspond, in terms of a transmission amplitude, to the time
of the part of the wave function that tunneled (transmitted) the potential barrier. For example, at a field strength
, the ionization rate is the part of ionized atoms (transmission amplitude of the wave function) relative to the amplitude of the total wave function of the system, which corresponds to the total number of the atoms. In this respect, in our model, we quantify tunneling through the energy gap (potential barrier) and T-time using TEUR, which is equivalent to quantifying it through the transmission amplitude of the wave function. This follows in particular from the good agreement of our T-time with FPI [
4,
35] and SATT [
74] for He-atoms. We see now why Ni et al. [
69] have found a high non-tunneled fraction as mentioned above (using a position criterion). The tunneling probability is always less than unity and only at
reaches unity, see Delone [
77] (p. 72). They ignore the fact that tunneling unlike ionization depends on the probability amplitude or transmission and reflection amplitudes and coefficients. The tunneled fraction corresponds to the part of the transmission amplitude, which corresponds to the T-time
and
, which agree with the experimental result in the adiabatic tunneling [
1,
4,
35].
At the atomic field strength
, the T-time is the ionization time
(compare Equations (
15) and (
33)), at which the ionization rate equals unity [
77] (p. 72), where the barrier disappears and the BSI region starts, which is a classically allowed process. Actually, we can express our views with a simple form in terms of relative probability amplitude,
. Hence, we find that
is an enhancement factor similar to Equation (
15). One notices that
is the QMal limit and
for any time measurement through an interaction with a field
[
9], see [
1,
2].
In a way, our approach is tied to the Heisenberg picture of the QM, where the dynamics are performed by the operators acting on the time-independent state vectors, which are represented by a basis in the phase space. The expectation values are then extracted information by virtue of the operators acting on the sates vectors, where the probability character of the expectation value corresponds to the probability amplitude or the factor
. However, certainly the calculation of
is a difficult task, whereas our approach offers a simple procedure to calculate the tunneling time by exploiting the TEUR [
1].
Therefore, we think that our tunneling model is in line with the statistical, probabilistic point of view, e.g., such as FPI, the wave packet dynamics and wave function methods or NITDSE. It might explain the good agreement of our T-time [
1] with FPI of Landsman et al. [
4], and NITDSE of Sainadh et al. [
8,
49] (apart from the factor
as already discussed). This point of view is supported by the similarity between the delay time (interval) due to the barrier itself
and the dwell time
, see
Section 2.1. The latter is defined statically by the transmission and reflection coefficient or amplitudes of the wave function [
35] (in our picture forward, backward scattering). In addition, similarly, it is supported by the agreement with the unified tunneling picture of Winful (Equation (
32)), see
Section 4. This way we showed that our real T-time conciliates with various interpretations and pictures, it is in this sense sturdy, thorough, and fundamental.
Finally, one might argue that the imaginary T-time picture is physically more reliable, see Rost et al. [
68], but why then is the ACOA caused by the tail of the potential [
49,
58] if the energy gap is mainly covered by the nonadiabatic effects? There are several concerns with this picture. 1. It ignores the self-interference term, Equations (
32) and (
33), which also exists for a zero-range potential. 2. We have already seen from
in Equation (
30) that the ACOA is likely caused by a scattering mechanism due to the Coulomb potential and not solely due to the tail of the potential. 3. It ignores that the time to reach the entrance point
(Equation (
7), see
Figure 1), is a classically allowed process. Furthermore, the electron becomes a free particle at the exit point, when it gains the energy through the nonadiabaticity. It is then less affected by the tail of the Coulombic potential, which is screened by the field [
45]. Otherwise, we do have to rely on the adiabatic tunneling picture, where our real T-time model, together with the statistical or probabilistic time interpretation as discussed above, offers a reasonable picture in which the nonadiabatic effects are small and not crucial to the ACOA or the T-time. Where the nonadiabaticity due to the oscillatory field is smaller than the error bars, as already mentioned, and, in our opinion, it is reflected by the spread of the points in the experimental result, compare the figures given in [
4]. Further discussion is given in [
41], and we follow this issue in future works.