# The role of reflections in the generation of a time delay in strong field
ionization

Daniel Bakucz Canário<EMAIL_ADDRESS>Michael Klaiber
Karen Z. Hatsagortsyan<EMAIL_ADDRESS>Christoph H. Keitel
Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg,
Germany

(August 27, 2024)

###### Abstract

The problem of time delay in tunneling ionization is revisited. The origin of
time delay at the tunnel exit is analysed, underlining the two faces of the
concept of the tunnelling time delay: the time delay around the tunnel exit
and the asymptotic time delay at a detector. We show that the former time
delay, in the sense of a delay in the peak of the wavefunction, exists as a
matter of principle and arises due to the sub-barrier interference of the
reflected and transmitted components of the tunneling electronic wavepacket.
We exemplify this by describing the tunnelling ionization of an electron bound
by a short-range potential within the strong field approximation in a “deep
tunnelling” regime. If sub-barrier reflections are extracted from this
wavefunction, then the time delay of the peak is shown to vanish. Thus, we
assert that the disturbance of the tunnelling wavepacket by the reflection
from the surface of the barrier causes a time delay in the neighbourhood of
the tunnel exit.

##  I Introduction

In both classical and quantum mechanics, the passage of time is always defined
with respect to some dynamical variable (for instance, the hands of a clock).
In the absence of a canonical quantum mechanical time operator, the definition
and measurement of time in quantum mechanics is challenging since dynamical
observables are inherently non-deterministic. Despite the conceptual
difficulties involved, operational techniques with which to measure time have
been developed (see e.g. [1, 2, 3, 4, 5, 6, 7, 8, 9] for overviews).

One of the main applications of these time measurement protocols is the study
of the time taken for a particle to tunnel through a barrier, known as
tunnelling time. Several definitions of the tunneling time exist, such as the
Wigner time [10, 11, 12, 13, 14], the Büttiker-Landauer time[15], the Pollak-
Miller time [16], the Larmor time [17, 18, 19, 20, 21], the dwell time [22],
etc. [23, 24, 25, 26, 27]. Each definition corresponds to a specific aspect of
the measurement process and, in general, these do not coincide.

The tunneling phenomenon plays an essential role in the strong field
ionization process of atoms and in the related field of attoscience. State-of-
the-art techniques in attosecond science are now able to provide exceptional
time and space resolution, reaching tens of attoseconds time- and Angström
space-resolution. In particular, the attoclock technique provides time
resolution of tunneling ionization [28], and has inspired researchers to
experimentally address the challenging problem of tunneling time [29, 30, 31,
32, 33], i.e., how much time, if any, elapses during the quantum tunnelling
process.

This question strikes at the fundamentals of quantum mechanics but experiments
have often been followed by controversial discussion [29, 30, 31, 32, 33, 34,
35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 26, 46, 47, 48, 49, 50, 51, 52,
53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66] in the strong field
community arguing either for, or against, the existence of a tunnelling
induced time delay between ionised electron wavepacket and ionising laser
field.

In this respect  we emphasise the distinction between two concepts of the
tunneling time in strong field ionization, namely, the time delay near the
tunnel exit (the classically expected coordinate for the tunneled electron to
emerge), and the asymptotic time delay [37]. While the latter is relevant to
attoclock experiments, the former, known also as the Wigner time delay, can be
calculated theoretically and measured in a Gedanken experiment with a so-
called virtual detector [67, 68]. In both cases, time delay arises due to sub-
barrier dynamics.

The asymptotic time delay is derived from the asymptotic photoelectron
momentum distribution (PMD). It is defined by the classical backpropagation of
the peak of the photoelectron asymptotic wavefunction up to the tunnel exit
[42, 45, 43, 44]. A theoretical description of the experimental asymptotic
time delay is challenging because of the entanglement of Coulomb field effects
with those of the tunneling delay.  In a regime far from over-the-barrier
ionization (OTBI), the asymptotic time delay is vanishing [36, 37] However, in
regimes approaching the OTBI the asymptotic delay is not negligible; it was in
fact found to be negative [42, 43, 44] and shown to arise due to interference
of direct and the under-the-barrier recolliding trajectories [51].

In the tunnelling region, the tunneling time delay is deduced by following the
peak of the electron wavefunction, the so-called Wigner trajectory. A few de-
Broglie wavelengths away from the tunnel exit, the electron dynamics are
quasiclassical and the classical trajectory is accurately described by the
eikonal of the wave function in the Wentzel-Kramers-Brillouin (WKB)
approximation. However, near the tunnel exit, the WKB approximation fails and
a full quantum mechanical description is given by the Wigner trajectory. Near
the tunnel exit this description diverges strongly from the classical one but,
nevertheless approaches it asymptotically, as one moves toward a detector. The
Wigner time delay has been calculated numerically with the virtual detector
method in Refs. [36, 34, 35, 33], showing positive time delay and nonvanishing
longitudinal velocity at the tunnel exit.

In this paper we investigate the Wigner time delay in tunnelling ionization,
which a virtual detector in a Gedanken experiment would observe. We concern
ourselves chiefly with understanding the principles of the tunnelling delay,
and to this end we consider a simple, time-dependent model of a one-
dimensional (1D) atom, with an electron bound by a short-range potential,
which is ionized by a half-cycle laser pulse.

At first sight, such abstraction may seem overly simplistic; nonetheless, it
takes into account all the necessary features of tunnelling ionization.
Firstly, ionization occurs mainly in the direction of the electric field so a
1D treatment is appropriate. Likewise, no real laser pulse is a half-cycle
sinusoid but by considering one such laser we ensure no continuum electron-to-
atomic core recollisions take place, which convolute the physical picture.
Such an interpretation is rendered clearer by having analytic expressions,
unattainable with a pure Coulomb atomic potential but feasible with a short
range potential.

In this manner, the time-dependent wavefunction is described within the strong
field approximation (SFA). The components of the wavefunction which are
reflected by the tunnelling barrier are identified by analyzing the saddle
points of the time-integral of the wavefunction. We show that the interference
of reflected and transmitted components of the wavefunction generates the
Wigner time delay, while the same delay vanishes when the reflected components
are neglected. Additionally, the scalings with respect to the laser field
strength of the Wigner time delay and group velocity at the tunnel exit are
derived and interpreted.

We work well below the OTBI regime, in what we term the deep tunneling regime.
While in this regime the asymptotic tunneling time delay is vanishing we show
that the Wigner time at the tunnel exit (calculated in the first-order SFA) is
not only nonzero, but sizeable, as already noted in [37]. This is in contrast
to the asymptotic time delay, where the first-order SFA result is vanishing
and the second-order SFA is required to obtain a nonvanishing asymptotic time
delay (attributable to sub-barrier recollisions and interference of paths).

The structure of the paper is as follows: in Sec. II we introduce the basic
ionization model, calculate the SFA time-dependent wavefunction, and construct
the Wigner trajectory from the latter. To elucidate our method of analysing
the contribution of reflections to the wavefunction, in Sec. III we consider a
simpler, analytic, model of ionization: an electron in a constant electric
field. The analyticity of this model allows us to relate the reflected
components of the wavefunction to the contribution of the specific saddle
points of the integral representation of the wavefunction. We apply this
concept to the SFA wavefunction in Sec. IV, whereby we show that neglecting
reflections results in a zero Wigner time delay. Lastly, in Sec. V, we
summarize  and interpret our results and discuss the implications of these on
the interpretation of attosecond streaking experiments. For the reader
unacquainted with strong field physics, the Wigner time delay for scattering
by a square potential barrier was previously analyzed in [69, 7], where the
significance of reflections to the time delay was alluded to. In Appendix A
this problem is revisited and the role of reflections is explicitly presented.
Atomic units (a.u.) are used exclusively throughout this work.

##  II Time Delay in Strong Field Ionization

The Wigner time, developed for scattering problems [10, 11, 13], considers the
time taken for the peak of the wavepacket to travel a given distance. For
tunneling through a time-independent barrier, the Wigner time can be derived
via the energy derivative of the phase of the tunnelling wavefunction $\psi$:

$\tau_{W}(x)=-i\,\frac{\partial}{\partial\,\mathcal{E}}\ln\frac{\psi(x)}{|\psi(x)|}.$
(1)

The peak of the ionized wave packet leads to the asymptotic formation of a
well-defined peak in the attoclock PMD, which is subsequently measured and
interpreted.

As the peak of the wavepacket follows the trajectory $\tau_{W}(x)$, the Wigner
group delay velocity of the wave packet can be defined

$v_{W}(x)=\left(\frac{\partial\tau_{W}(x)}{\partial x}\right)^{-1}.$ (2)

When a monochromatic wave is incident on a finite potential barrier, the
propagation inside the barrier is a superposition of an exponentially
suppressed wave (transmission) with a growing exponential wave, namely the
reflection by the surface of the barrier (see e.g. Appendix A). The goal of
this paper is to establish the causal effect of wavefunction refections inside
the barrier on the time delay in strong field ionization.

In order to do this, we calculate the time-dependent ionized electron
wavefunction at the position where the electron appears in the continuum,
thereby allowing us to deduce the peak of the wavepacket with respect to time.
The Wigner time delay can be calculated as the energy derivative of the phase
of the wave function via Eq. 1 in a static field but in a time-dependent laser
field this prescription fails as energy is not conserved. Correspondingly, we
derive the Wigner time delay at the tunnel exit explicitly as the delay of the
peak of the wavepacket, rather than from the energy derivative of the phase of
the wavefunction.

###  II.1 The Model

Consider an electron initially bound in a 1D short-range potential

$V(x)=-\kappa\>\delta(x)$ (3)

with a binding energy $I_{p}=\kappa^{2}/2$, and with a corresponding ground
state wavefunction
$\psi_{0}(x,t)=\sqrt{\kappa}\exp\left(-\kappa|x|+iI_{p}t\right)$.

To describe the generation of the peak of the ionized wavepacket at the tunnel
exit, not the usual PMD at a detector, a half-cycle laser field is modelled,
avoiding recollisions in the continuum. The laser electric field,

$\displaystyle E(t)$ $\displaystyle=$ $\displaystyle-E_{0}\cos^{2}(\omega t),$
(4)

is switched on at $\omega t_{0}=-\pi/2$.  In a half-cycle laser field the
ionization happens mostly near the peak of the field. Here, nonadiabatic
effects are governed by the second-derivative of the field at the peak
$E^{\prime\prime}(0)/E(0)=2\omega^{2}\equiv\omega_{eff}^{2}$, where we define
the effective frequency $\omega_{eff}=\sqrt{2}\omega$.

We consider the tunnel ionization regime and keep constant the Keldysh
parameter [70]

$\gamma=\sqrt{\frac{I_{p}}{2U_{p}}}\ll 1,$ (5)

where $U_{p}=E_{0}^{2}/(4\,\omega_{eff}^{2})$ is the ponderomotive potential.
Thus, when varying the field strength $E_{0}$, we vary the frequency
$\omega=\gamma E_{0}/(\sqrt{2}\kappa)$ accordingly.

Moreover, we consider the so-called deep tunneling regime, wherein $E_{0}\ll
E_{th}$, and $E_{th}$ is the threshold field of OTBI. This threshold can be
estimated as the field strength where the coordinate-saddle point of the SFA-
matrix element, i.e. the starting point $x_{s}\approx\sqrt{\kappa/E_{0}}$ of
the quantum orbit, becomes comparable the tunnel exit $x_{e}$, which for the
short range potential corresponds to the condition $E_{th}\approx\kappa^{3}/4$
[71].

The superposition of the laser field with the atomic potential creates a
potential barrier, through which an electron can tunnel, with a characteristic
classical tunnel exit $x_{e}=I_{p}/E_{0}=10$ a.u. 111 A more precise
definition for the tunnel exit can be given by the SFA, namely
$x^{SFA}_{e}=\int_{t_{s}}^{\rm
Re\\{t_{s}\\}}(A(t^{\prime})-A(0))\;dt^{\prime}\approx(1-\gamma^{2}/4)\,I_{p}/E_{0}=0.995\,I_{p}/E_{0}$,
where $t_{s}$ is the temporal saddle point of the SFA integral. The minuscule
discrepancy between the classical tunnel exit, $x_{e}$, and its SFA correction
for the parameters in this work ($\gamma=\sqrt{2}/10$) leads us to use the
simpler of the two definitions.. While this is a rudimentary estimate, it is
in agreement with a tunnel exit derived quantum mechanically in our parameter
regime as shown in Section IV.2.

(a)  (b)

Figure 1:  (a) Ionization amplitude $|\psi_{i}(x,t)|^{2}$ as defined in Eq.
(17). At every cross-section in $x$, the temporal peak of the wavefunction was
determined (plotted in blue) which can be interpreted as the trajectory of the
peak. (b) Comparison of trajectories in the region near the tunnel exit,
$x_{e}$. The displayed curves are: the peak trajectory (blue, given by the
maximum of $|\psi_{i}(x,t)|^{2}$), the classical trajectory (orange, given by
the Newton equation starting at the tunnel exit), and the Wigner trajectory
(green, given by Eq. (25)). Under the barrier ($x<10$, shaded blue), there is
an increasing delay of the peak w.r.t the laser peak. Outside the barrier, the
peak trajectory rapidly converges with the classical and Wigner electron
trajectory.  Near the atomic core ($x<2$), the bound state,
$|\psi_{0}(x)|\gg|\psi_{i}(x)|$, dominates the full wavefunction
$\psi(x)=\psi_{0}(x)+\psi_{i}(x)$ so the meaning of a trajectory for
ionization is lost. Figure 2: Probability distribution
$|\psi_{i}(x_{e},t)|^{2}$ vs time at the classical tunnel exit
$x_{e}=I_{p}/E_{0}$. This distribution is peaked at a greater time,
$t_{m}\approx 7.6$ a.u. $\approx 183$ as, marked by the dot, than the peak of
the laser pulse.

###  II.2 Time Evolution of the Wavefunction

Our investigation is based on the SFA wavefunction describing the electron
dynamics in tunneling ionization. We define the Wigner trajectory and
tunneling time delay employing the virtual detector approach [34, 35, 67, 68]
based on the SFA wavefunction. The ionization dynamics are described by the
Schrödinger equation

$i\frac{\partial}{\partial t}\Psi(x,t)=(H_{0}+H_{i})\Psi(x,t),$ (6)

with the atomic Hamiltonian

$H_{0}=-\frac{1}{2}\frac{\partial^{2}}{\partial x^{2}}+V(x),$ (7)

and the interaction Hamiltonian with the laser field

$H_{i}=xE(t)\;$ (8)

The standard SFA is employed for the solution of the Schrödinger equation. In
the interaction picture with Hamiltonian $H=H_{0}+H_{i}$, the time evolution
operator $U(t,t_{0})$ satisfies the Dyson integral equations [73]

$U(t,t_{0})=U_{0}(t,t_{0})-i\int^{t}_{t_{0}}\>\mathop{}\\!\mathrm{d}t^{\prime}\>U(t,t^{\prime})H_{i}(t^{\prime})U_{0}(t^{\prime},t_{0}),$
(9)

where $U_{0}$ is the evolution operator corresponding to the Hamiltonian
$H_{0}$. A perturbation series can be constructed by replacing the full
evolution operator $U(t,t_{0})$ in the Dyson integral with the evolution
operator corresponding to the Hamiltonian $H_{i}$, namely

$U_{f}(t,t^{\prime})=\int\mathop{}\\!\mathrm{d}p\ket{\Psi_{p}(t)}\bra{\Psi_{p}(t^{\prime})},$
(10)

where $\ket{\Psi_{p}}$ are the Volkov states [74]. Each order in the
perturbation theory corresponds to a sub-barrier recollision in our half-cycle
laser field. As each sub-barrier recollision implies a longer tunnelling path,
each term in this series is suppressed by the Keldysh exponent
$\exp(\nicefrac{{-2\kappa^{3}}}{{3E_{0}}})$ and so, in contrast to the near
OTBI regime, in the deep tunnelling regime a first order expansion in the SFA
suffices (i.e. neglecting sub-barrier recollisions) [51].

Consequently, the electron state during the interaction takes the form
$\ket{\psi(t)}=\ket{\psi_{0}(t)}+\ket{\psi_{i}(t)}$ where $\ket{\psi_{0}(t)}$
is the eigenstate of the atomic Hamiltonian $H_{0}$ and

$\displaystyle\ket{\psi_{i}(t)}$ $\displaystyle=$
$\displaystyle-i\int^{\infty}_{-\infty}dp\int^{t}_{t_{0}}dt^{\prime}\ket{\Psi_{p}(t)}\braket{\Psi_{p}(t^{\prime})}{H_{i}(t^{\prime})}{\psi_{0}(t^{\prime})},\qquad$
(11)

describes the electron dynamics induced by the laser field, which generates
the Wigner trajectory. It is the dominant part of the wavefunction at
distances far from the core, and therefore, we concentrate on the analysis of
its behavior.

(a)  (b)

Figure 3:  Log-log plots of the scaling w.r.t. field strength, $E_{0}$, of the
(a) Wigner time delay and (b) group velocity at the classical tunnel exit,
$x_{e}=I_{p}/E_{0}$, for the time evolved SFA wavefunction (red dots) and the
adiabatic constant field wavefunction (blue lines). For the time evolved
wavefunction, the maxima in time, $t_{m}$, of the probability distribution
$|\psi(x_{e},t)|^{2}$ and its derivative were calculated; for the adiabatic
case, results follow from Eqs. (26) and (27). The agreement in the trends is
good, deteriorating as one approaches the OTBI threashold $E_{th}\approx 0.25$
a.u.

Denoting the electronic kinetic momentum as

${\cal P}(t)\equiv p+A(t),$ (12)

with the laser vector potential
$A(t)=-\int_{-t_{0}}^{t}E(t^{\prime})\,dt^{\prime}$, the 1D Volkov state in
the length gauge is

$\Ket{\Psi_{p}(t)}=\Ket{{\cal P}(t)}e^{-i\,S(t)}$ (13)

with a plane wave component
$\Braket{x}{p}=(2\pi)^{-\frac{1}{2}}\exp({i\,p\,x})$ and the Volkov-phase

$S(t)=\frac{1}{2}\int^{t}\mathop{}\\!\mathrm{d}\tau\,{\cal P}(\tau)^{2}\;.$
(14)

We can expand the overlap in (11) using a resolution of identity in a dummy
variable $x^{\prime}$

$\braket{\Psi_{p}(t^{\prime})}{H_{i}(t^{\prime})}{\psi_{0}(t^{\prime})}=\frac{e^{i\left(I_{p}t^{\prime}-S(t^{\prime})\right)}}{\sqrt{2\pi}}E(t^{\prime})\int^{\infty}_{-\infty}dx^{\prime}\,e^{i\,\mathcal{P}(t^{\prime})x^{\prime}-\kappa|x^{\prime}|}x^{\prime}$
(15)

While this integral is soluble, we do not evaluate it but rather first
integrate (11) w.r.t. to $p$ and then $x^{\prime}$. With the notation $\Delta
f_{n}(t,t^{\prime})=f_{n}(t)-f_{n}(t^{\prime})$, and defining the integrals
$f_{n}(t)=\int^{t}dt^{\prime}A(t^{\prime})^{n}$, we may perform the initial
integral over $p$ which is a simple Gaussian integral. The resulting
expression

$\psi_{i}(x,t)=\int^{t}_{t_{0}}\mathop{}\\!\mathrm{d}t^{\prime}\int_{-\infty}^{\infty}\mathop{}\\!\mathrm{d}x^{\prime}\frac{x^{\prime}\sqrt{\kappa}\,E(t^{\prime})}{\sqrt{2\pi\,i(t-t^{\prime})}}\exp\left[i\left(I_{p}t^{\prime}+i\kappa|x^{\prime}|+x(A(t)-A(t^{\prime}))+\frac{i\left((x-x^{\prime})-\Delta
f_{1}(t,t^{\prime})\right)^{2}}{2(t-t^{\prime})}-\frac{\Delta
f_{2}(t,t^{\prime})}{2})\right)\right]$ (16)

may then be finally integrated in $x^{\prime}$ to yield

$\displaystyle\psi_{i}(x,t)$
$\displaystyle=\int^{t}_{t_{0}}dt^{\prime}\,\sqrt{\kappa}\,E(t^{\prime})\left(\frac{z_{-}^{2}(1+\mathop{}\\!\mathrm{Erf}(z_{-}))e^{-z_{+}^{2}}}{z-\kappa}+\frac{z_{+}^{2}(1-\mathop{}\\!\mathrm{Erf}(z_{+}))e^{-z_{-}^{2}}}{z+\kappa}\right)\exp\left(i\zeta\right)\equiv\int^{t}_{t_{0}}dt^{\prime}\exp({i\,\Phi(x,t,t^{\prime})})$
(17)

where we have defined $z_{\pm}=\sqrt{\frac{i}{2}(t-t^{\prime})}(z\pm\kappa)$
for

$z=-i\,\left(A(t^{\prime})+\frac{x-\Delta
f_{1}(t,t^{\prime})}{t-t^{\prime}}\right)$ (18)

and where

$\displaystyle\zeta$ $\displaystyle=I_{p}t^{\prime}+x\,A(t)+$ (19)
$\displaystyle(t-t^{\prime})\left(\frac{(x-\Delta
f_{1}(t,t^{\prime}))^{2}}{2(t-t^{\prime})^{2}}+\kappa^{2}+z^{2}\right)-\frac{\Delta
f_{2}(t,t^{\prime})}{2}$ (20)

Our aim is to calculate the wavefunction and Wigner time delay in the
interaction region around the tunnel exit by studying the phase $\Phi$. Thus,
in deviation to standard SFA studies, the time integral is calculated up to
the finite observation time $t$, meaning the integration must be performed
numerically. The usual saddle-point integration method is valid only for
sufficiently large values of $t$, placing one well outside the tunnelling
region.

###  II.3 Numerical Integration

For the numerical integration of the SFA wavefunction Eq. (17) an electric
field strength $E_{0}=0.05$ a.u. and Keldysh parameter
$\gamma=\sqrt{2}/10\approx 0.14$ were chosen to ensure deep tunnelling is
considered. With these parameters, our half-cycle laser pulse spans
$[-314,314]$ a.u. We present results for the case of hydrogen, for which
$I_{p}=\kappa^{2}/2=1/2$ a. u., implying an OTBI threshold field
$E_{th}\approx 0.25$ a.u. The time integration was carried out with the
standard numerical integration routine of Mathematica 12 to a precision of 30
digits.

The space-time probability distribution, $|\psi_{i}(x,t)|^{2}$, is presented
in Fig. 1. From this distribution the Wigner trajectory is derived as follows:
the probability time-distribution at each space point $x$ is invoked (see Fig.
2 for the case of the tunnel exit coordinate $x=x_{e}$) and the maximum
$t_{m}(x)$ of this distribution is derived. The Wigner trajectory is
represented by the function $t_{m}(x)$, which runs along all maximum points of
the space-time distribution.  Near the core, $|x|\lesssim 3$ a.u., the bound
state dominates the wavefunction $|\psi_{0}|\gg|\psi_{i}|$, so the concept of
the Wigner trajectory loses its meaning.

We underline that the wavefunction $\psi_{i}$ vanishes after the laser field
is switched off. Thus, $\psi_{i}$ does not include the possible net
excitations of the atomic states due to interaction with the laser field. This
is a feature of the SFA, where the exact time-evolution operator in the final
state is replaced by the Volkov propagator, representing the continuum
electron in the laser field.

We note the maximum of the wavefunction displays a time delay with respect to
the peak of the field. The tunneling time delay under the barrier ($x<10$)
increases when moving towards the tunnel exit. Outside the barrier, this
trajectory rapidly approaches the classical electron trajectory (beginning at
the classical tunnel exit with zero momentum). The slight deviation is due to
quantum mechanical corrections to the quasiclassical wave function not far
from the tunnel exit.

The main idea advocated in this paper is that the Wigner time delay during
tunneling is closely related to reflections arising during tunneling dynamics.
It is straightforward in the simple case of tunneling through a box potential
to show that reflections are responsible for the tunneling time delay, as is
done in Appendix A. However, unlike in the separable box potential case, a
given wavefunction is not easily deconstructible as a simple superposition of
reflected and transmitted components corresponding to simple decaying /growing
exponentials. Instead, the transmitted and reflected components are completely
encapsulated in the wavefunction making it rather more problematic to
disentangle. This is ultimately achieved in Sec. V for the SFA wavefunction
presented in this paper but in order to highlight the main aspects of the
method we discuss beforehand a simpler example, that of tunneling in a
constant electric field.

##  III Time delay in a constant field

The adiabatic model of ionization, namely atomic ionization in a constant
field, will be helpful for our purposes as it is analytically tractable.
Moreover, it will provide us with a reference with which to compare our
earlier time dependent model.

###  III.1 Ionisation in an adiabatic field

Consider a bound electron of energy $-I_{p}$ in 1D $\delta$-potential ionized
by a constant electric field $E_{0}$. The continuum eigenstate of the electron
in this field is given by the solution to the time-independent Schrödinger
equation:

$-\frac{1}{2}\frac{\mathop{}\\!\mathrm{d}^{2}\psi}{\mathop{}\\!\mathrm{d}x^{2}}+\left(I_{p}-E_{0}\>x\right)\psi(x)=0,$
(21)

which has as a general analytical solution as a superposition of the Airy
functions of the first and second kind

$\psi(x)=c_{1}\mathop{}\\!\mathrm{Ai}(\tilde{x})+c_{2}\mathop{}\\!\mathrm{Bi}(\tilde{x}),$
(22)

where

$\tilde{x}=\left(\frac{2}{E_{0}^{2}}\right)^{\frac{1}{3}}\left(I_{p}-E_{0}\>x\right).$
(23)

The requirement that the wavefunction be a travelling wave as
$x\rightarrow\infty$ imposes the condition $c_{1}=-ic_{2}$ so that the
wavefunction takes the form

$\psi(x)=T\left[\mathop{}\\!\mathrm{Bi}(\tilde{x})+i\,\mathop{}\\!\mathrm{Ai}(\tilde{x})\right],$
(24)

where $T$ can be determined by matching this wavefunction to the bound-state
solution of the atom; this prefactor plays a role in the ionization amplitudes
but is irrelevant to the phase of the wavefunction. We calculate the Wigner
time delay for this wavefunction via the energy derivative of the phase

$\displaystyle\tau_{W}(x)=i\frac{\partial}{\partial{I_{p}}}\ln\frac{\psi}{|\psi|}=\frac{2^{\frac{1}{3}}}{\pi
E_{0}^{\frac{2}{3}}}\;\frac{1}{\mathop{}\\!\mathrm{Ai}(\tilde{x})^{2}+\text{Bi}(\tilde{x})^{2}}.$
(25)

The constant field model allows one to estimate the scaling of the Wigner time
delay,  given by Eq. (1), at the tunnel exit $x_{e}=I_{p}/E_{0}$:

$\tau_{W}(x_{e})=\frac{3^{4/3}\,\Gamma\left(\frac{2}{3}\right)^{2}}{2^{5/3}\pi}\frac{1}{E_{0}^{2/3}}.$
(26)

Using Eq. (2), we derive the scaling of the Wigner group velocity of the
electron at the tunnel exit,

$v_{W}(x_{e})=\frac{2\left(\frac{2}{3}\right)^{2/3}\sqrt{\pi}\,\Gamma\left(\frac{7}{6}\right)}{\Gamma\left(\frac{2}{3}\right)^{2}}\,E_{0}^{1/3},$
(27)

and find it consistent with the estimate of the scaling of the electron
momentum $p_{e}\sim E_{0}\,\tau_{W}$ in Ref. [51].

We compare the Wigner time delay and the group delay velocity in the constant
field case with the exact time dependent SFA calculations in Fig. 3.  The time
delay in the SFA scales with the electric field in a similar manner to the
adiabatic case, $\tau_{W}(x_{e})\propto 1/E_{0}^{2/3}$, but shifted in time by
a constant. This shift stems from the fact that the Wigner trajectory is
derived via the  energy derivative of the phase of the wave function (which
corresponds to the following the peak of the wave packet, neglecting its
spreading), while the time-integration responsible for the formation of the
ionization wave packet in SFA is calculated exactly numerically.

The scaling of the Wigner group velocity using the SFA is also in agreement
with the adiabatic estimate, $v_{W}(x_{e})\propto E_{0}^{1/3}$. However, the
relative error between these two grows with the field strength, becoming
appreciable at higher field strengths when approaching the OTBI threshold.
This is consistent with the simple estimate for the threshold for OTBI,
$E_{th}\approx 0.25$ a.u., where a tunnelling description no longer applies.

Note that the Wigner trajectory in the constant field case is determined from
the solution of the time-independent Schrödinger equation (energy eigenstate),
while in the SFA case from the time-dependent wavefunction. In the first case,
the Wigner trajectory is defined as the derivative of the wavefunction phase
with respect to the energy, which corresponds to determining the coordinate of
the peak of the electron wavepacket at a fixed time moment, i.e. determining
the motion of the wavepacket peak. In contrast, in the SFA case we explicitly
determine the maximum of the wavefunction in time for a fixed coordinate.

Figure 4:  The solution to the electron in a constant field problem is given
by a superposition of Airy functions of the first and second kind,
$\mathop{}\\!\mathrm{Ai}(\tilde{x})$ and $\mathop{}\\!\mathrm{Bi}(\tilde{x})$,
plotted in blue and orange respectively. These functions have very accurate
asympotic expansions, shown dashed, which can be derived by considering the
saddle points of the Airy integrals (28)-(29). Under the barrier, $x<10$
(shaded blue), these expansions show that the wavefunction components
$\mathop{}\\!\mathrm{Ai}(\tilde{x})$ and $\mathop{}\\!\mathrm{Bi}(\tilde{x})$
respectively correspond to the reflected and transmitted components of the
wavefunction.

(a)  (b)  (c)

Figure 5:  The Airy integral $\int_{\gamma}\,ds\exp(\tilde{x}\,s-s^{3}/3)$ is
defined on the complex $s$ plane; it converges when the endpoints of its
infinite contour, $\gamma$, lie in the shaded areas. The canonical contours
defining the Airy function of the first ($\gamma_{1}$) and second
($\gamma_{2}-\gamma_{3}$) kind are shown in (a). Contributions to the Airy
integral arise principally from portions of the contour near the saddle-points
of the integrand function $\exp(\tilde{x}\,s-s^{3}/3)$. The configuration of
the saddles $s_{\pm}(x)=\pm\sqrt{\tilde{x}}$ (red dots) in the complex plane
is shown for positions (b) inside the barrier ($\tilde{x}>0$), and (c) outside
the barrier ($\tilde{x}<0$). Through each saddle point pass two perpendicular
level curves, the path of steepest descents (solid blue) and the path of
steepest ascents (dashed gray). Deforming the defining contours in (a) through
the steepest descents contours in (b) and (c) provides asymptotic expansions
for the Airy function, as in Eqs. (31) and (32).

###  III.2 Reflections in constant field

To investigate the effect of under-the-barrier reflected components, we need
to isolate their contribution to the full wavefunction. The analytical wave
function of Eq. (24) consists of a superposition of Airy functions
$\mathop{}\\!\mathrm{Ai}(\tilde{x})$ and $\mathop{}\\!\mathrm{Bi}(\tilde{x})$
which can be given interpretation as the under-the-barrier reflected and
transmitted components of the tunneling wavefunction, respectively.

This interpretation can be generally understood by considering the Airy
functions, as shown in Fig. 4. The $\mathop{}\\!\mathrm{Bi}$-component of the
wavefunction decays exponentially as it approaches the tunnel exit, $x=x_{e}$,
from the atomic core at $x=0$, and hence can be seen to correspond to the
transmission component; likewise, the exponentially growing
$\mathop{}\\!\mathrm{Ai}$-component corresponds to wavefunction reflection.

This interpretation can be formally established by considering the integral
representation of the Airy functions

$\displaystyle\mathop{}\\!\mathrm{Ai}(\tilde{x})$ $\displaystyle=\frac{1}{2\pi
i}\int_{\gamma_{1}}ds\;\exp(\tilde{x}\,s-\frac{s^{3}}{3}),$ (28)
$\displaystyle\mathop{}\\!\mathrm{Bi}(\tilde{x})$
$\displaystyle=\frac{1}{2\pi}\int_{\gamma_{2}-\gamma_{3}}ds\;\exp(\tilde{x}\,s-\frac{s^{3}}{3}),$
(29)

where the complex plane integration paths $\gamma_{i}$ are indicated in Fig. 5
(a). These integrals converge when their endpoints lie in the slices of the
complex plane defined by $-\frac{\pi}{6}<\theta<\frac{\pi}{6}$,
$\frac{\pi}{2}<\theta<\frac{5\pi}{6}$, and
$\frac{7\pi}{6}<\theta<\frac{3\pi}{2}$, where in polar $\\{r,\theta\\}$
coordinates $s=r\exp(i\,\theta)$. In these regions, shaded blue in Fig. 5, the
integrand vanishes rapidly as $r\rightarrow\infty$.

The majority of the contributions to the Airy integrals thus come from around
the saddle points,

$\displaystyle s_{\pm}=\pm\sqrt{\tilde{x}},$ (30)

of the argument of integrand. The Airy contours, $\gamma_{i}$, can be deformed
into paths that to go through these saddle-points and, using the standard
technique of saddle-point integration method [75], asymptotic expressions for
the Airy integrals can be determined.

The saddle-points, and the respective paths of steepest descents  and ascents,
are illustrated in Figs. 5 (b) and (c); since these are dependent on
$\tilde{x}$ the application of the saddle point method is different for the
two cases of inside ($\tilde{x}>0$) and outside ($\tilde{x}<0$) the potential
barrier. As shown in Fig. 5 (b), for $\tilde{x}>0$ we may deform the contour
$\gamma_{1}$ smoothly into the path of steepest descents for the saddle point
$s_{-}$ and in doing so obtain the asymptotic approximation

$\displaystyle\mathop{}\\!\mathrm{Ai}(\tilde{x})$
$\displaystyle=\frac{\exp\left(-\frac{2}{3}\tilde{x}^{\frac{3}{2}}\right)}{2\,\sqrt{\pi}\,\tilde{x}^{\frac{1}{4}}}+\mathop{}\\!\mathcal{O}(\tilde{x}^{-\frac{3}{2}}).$
(31)

Likewise, we may deform the contours $\gamma_{2}$ and $-\gamma_{3}$ to both go
through the steepest descent path of the saddle point $s_{+}$ and hence deduce
the asymptotic form of the $\mathop{}\\!\mathrm{Bi}(x)$-function under the
potential barrier

$\displaystyle\mathop{}\\!\mathrm{Bi}(\tilde{x})$
$\displaystyle=\frac{\exp\left(+\frac{2}{3}\tilde{x}^{\frac{3}{2}}\right)}{\sqrt{\pi}\,\tilde{x}^{\frac{1}{4}}}+\mathop{}\\!\mathcal{O}(\tilde{x}^{-\frac{3}{2}}).$
(32)

From a tunnelling particle’s perspective, one may identify the
$\mathop{}\\!\mathrm{Ai}(x)$ function and its saddle-point $s_{-}$ with the
reflected part of the wavefunction, and the saddle-point $s_{+}$, or the
$\mathop{}\\!\mathrm{Bi}(x)$ function with the transmitted part of the
wavefunction: the transmitted wave decays as it moves to the tunnel exit,
while the reflected wave propagates from the exit decaying exponentially
toward the atomic core, as shown in Fig. 4.

In the following section, this relation between sub-barrier reflections and
the contribution of one saddle point to the time-integrand of the wavefunction
amplitude is applied to the problem of ionization in a time-dependent field.
This then allows us to investigate the role of sub-barrier reflection in the
formation of a tunneling time delay.

##  IV Wigner Time and Reflections in the SFA

Figure 6:  Complex $t^{\prime}$ plane of the argument $\Phi(x,t,t^{\prime})$
of the wavefunction integral
$\psi_{i}=\int^{t}_{t_{0}}dt^{\prime}\exp(i\,\Phi)$, for parameters $x=9$ and
$\omega t=\pi/40$. For these configurations there are two saddle-points,
denoted by a circle and triangle, the former which, $t_{+}$, corresponds to
reflections. The saddle-point method can be used to solve the wavefunction
integral, when the path is taken over both saddle-points. We use a partial
path given by (33) to remove the contribution of reflections by integrating
over only one of the saddle-points. Other possible configurations of the
saddle points are shown in Fig 11.

As shown in Sec. III, we may reasonably ascribe the saddle-point of the
integrand of the wavefunction to under the barrier reflection or transmission.
To reveal the contribution of the reflection to the tunneling time delay, we
investigate the complex continuation of the integrand function of the SFA
wavefunction, Eq. 17. The insight developed previously, namely the relation of
a specific saddle-point of the integrand function to reflection-like behaviour
in the wavefunction, is used to extract the contribution of reflections to the
wavefunction by modifying the path of integration the complex plane.

Figure 7:  Log-log plot of the relative contributions to the wavefunction
amplitude from the portion of the integration path up to the saddle point
(blue) and from the saddle point to the given real time $t$ (orange) for
various positions $x$ under the classical barrier (i.e. far from the atomic
core, but smaller than the tunnel exit). The behaviours are approximately
decaying and growing exponentials, respectively, analogous to transmitted and
reflected parts of the wavefunction.

(a)  (b)

Figure 8: (a) Temporal probability distributions at the position
$x=8\textnormal{ a. u.}$ (under the barrier, near tunnel exit), for the
ionised wavefunction, $\psi_{i}$, and the pseudo-wavefunction neglecting
reflections, $\psi_{nr}$. The physical wavefunction has accrued a Wigner time
delay with respect to the laser field, whereas the maximum of the pseudo-
wavefunction is synchronous with the laser field peak. (b) Trajectories in the
$(x,t)$ plane via: the maximum of the wavefunction $\psi_{i}$ (given by
$t_{m}(x)$), the maximum of the wavefunction neglecting reflections
$\psi_{nr}$, and the energy derivative of the constant field wavefunction
$\tau_{W}(x)$. The classical electron trajectory, $x_{cl}(t)$, starting at the
tunnel exit at the peak of the laser field is also shown. Under the barrier,
the probability distribution neglecting reflections $|\psi_{nr}|^{2}$ shows
zero time delay. In regions where Eq (34) does not apply, mostly near the
classical tunnel exit, we may not identify reflections nor plot a subsequent
trajectory.

In general, the complex $t^{\prime}$-plane picture of the phase $\Phi$ in Eq.
(17) has many similarities to that of adiabatic ionisaion. However, it depends
continuously not only on the coordinate $x$ but now also on observation time
$t$, making its analysis somewhat more involved. In adiabatic ionisation,
there was only one degree of freedom, $\tilde{x}$, and the saddle points were
either purely real or imaginary.

As shown in Fig. 6, in the time dependent case there are also two saddle-
points of relevance in the SFA integral denoted $t_{\pm}$. Their arrangement
in the complex $t^{\prime}$-plane is dependent on $x$ and $t$ but, as in the
adiabatic case, the real coordinate $x$ determines whether they are vertically
or horizontally aligned (in or outside the barrier); the observation time $t$
merely shifts the axis of symmetry which is always observed around the line
$t^{\prime}=t$. A more detailed discussion of the configuration of saddle
points in $(x,t)$ parameter space may be found in Appendix B.

###  IV.1 Extracting Reflections

In the adiabatic case, a partial integration over only contour containing the
saddle point $s_{-}$ would exclude the exponentially growing contributions to
the wavefunction, i.e. the reflections, associated with the saddle point
$s_{+}$.

We use the same principle in the time dependent case; in Fig. 6 we identify
the saddle point corresponding to reflections, denoted $t_{+}$ (represented by
a circle), and the secondary saddle point $t_{-}$ (represented by a triangle).
When we consider the wavefunction under the potential barrier, we can split
the integration contour in Eq. (17)  into two parts: from the start of the
pulse, $t_{0}=-\pi/(2\omega)$, to the upper saddle point, $t_{+}$, and from
$t_{+}$ to the real time $t$.

The contributions of these two contours to the wavefunction are shown in Fig.
7. The former integral has the form of a growing exponential and so can
ostensibly be identified as the reflected part of the wavefunction while the
latter is a decaying exponential identifiable as the transmitted part. To
neglect contributions from the reflections to the full wavefunction we
identify the following wavefunction:

$\displaystyle\psi_{nr}(x,t)=\begin{dcases}\quad\int_{t_{0}}^{t_{+}}dt^{\prime}\exp(i\,\Phi(x,t,t^{\prime}))&x>x_{e}\\\\[14.0pt]
\quad\int^{t}_{t_{+}}dt^{\prime}\exp(i\,\Phi(x,t,t^{\prime}))&x<x_{e}\\\
\end{dcases}$ (33)

Our method of distinguishing the reflection contribution is based on the
distinguishability of the saddle-points contributions to the wavefunction
integral. This is not possible when the two saddle points are so close that
the cubic term in the expansion of the phase $\Phi(x,t,t^{\prime})$ becomes
non-negligible [76], i.e. when

$\left\lvert\frac{\frac{\partial^{3}}{\partial t^{\prime
3}}\Phi(x,t,t^{\prime})}{\left(\frac{\partial^{2}}{\partial t^{\prime
2}}\Phi(x,t,t^{\prime})\right)^{\frac{3}{2}}}\right\rvert_{t^{\prime}=t_{\pm}}\gtrsim
1.$ (34)

for either saddle point $t_{\pm}$. This occurs in the region very near the
barrier boundary and at the peak of the laser pulse, as discussed further in
Sec. IV.2. In this case, the reflection cannot be separated in the
wavefunction in a meaningful manner.

We find that when reflections are extracted from the wavefunction using Eq.
(33) the Wigner time delay under the barrier vanishes, as evident in both
panels of Fig. 8. That is, when reflections are neglected, there is no time
delay between the peak of the electron probability distribution and the peak
of the laser field. Moreover, outside the barrier, the trajectory of the peak
rapidly becomes classical.

Plotted also in Fig. 8(b) is the trajectory for an electron in a constant
field, given by Eq. (25). For parameters in the deep tunnelling regime, the
SFA trajectory (given by the peak of the wavefunction), and the classical
trajectory, the constant field trajectory display very similar behaviour as
the pulse elapses (the trajectories do not completely coincide far away due to
the time dependence of the field and the need to match a tunnel exit,
respectively).

At a detector infinitely far away such as one in an attoclock experiment, a
signature of time delay would be absent for the deep tunneling conditions
detailed in this work. Thus, it is of importance to distinguish between a time
delay of the peak of the wavefunction near the tunnel exit and an asymptotic
time delay at a detector. Since in attoclock experiments the asymptotic
momentum distribution is measured, i.e., the asymptotic time delay, one
expects to find a zero time delay in the deep tunnelling regime. However, near
the tunnel exit, quantum mechanical considerations must be taken into account
and a quantum mechanical treatment such as the one exposed here exhibits an
explicit non-zero Wigner delay.

###  IV.2 Tunnel Exit from the SFA

We may also use the saddle point analysis of Eq. (17) to extract relevant
physical information about the electron dynamics. In particular, and in
analogy to the constant field scenario, the topology of the saddle points
through the complex $t^{\prime}$ plane reveal the functional behaviour of the
electronic wavepacket.

The paths of these two saddle point, for two representative choices of time
$t$ while varying $x$, are shown in Fig. 9, closely resembling those
corresponding to Figs. 5 (b) and (c). Much as in the constant field case,
where the saddles are purely real or imaginary, as given by Eq. (30) and Fig.
5, the pair of saddle points in the SFA is aligned around the line given by
$\textnormal{Re}[t^{\prime}]=t$; this alignment is either vertical or
horizontal depending on the coordinate $x$. For a fixed observation time $t$,
there corresponds a coordinate $x$ of closest approach between the saddle
points. For the peak of the pulse, $t=0$ there exists a coordinate $x_{t}$
where the saddle points merge completely, as they do in the case $\tilde{x}=0$
for the constant field electron.

Thus, two lines, $t=0$ and $x=x_{t}$, define separatrices for the behaviour of
the saddle points and their contributions to the integral: the behaviour of
the wave function can be seen to change depending on whether $x>x_{t}$. This
is a phenomenon exactly analogous to the merging of the saddle points for
constant field case of Sec. III: there, the merging of the saddle points
occurs at the classical exit, $\tilde{x}=0$, and separates the regions of
evanescence and oscillation in the wave function.

We may use this parallel to identify the merging point as a measure of the
tunnel exit. In our study, for a field strength of $E_{0}=0.05$ and
$\gamma=0.14$, we find that this exit takes the value $x_{t}\approx 10.2$
a.u.; this is in good agreement with the classically expected tunnel exit
$x_{e}=I_{p}/E_{0}=10$ a.u. More precise agreement is achieved with the
probability averaged tunnel exit

$x_{K}=\dfrac{\int
dt^{\prime}\frac{I_{p}}{|E(t^{\prime})|}\exp\left(-\frac{2\kappa^{3}}{3|E(t^{\prime})|}\right)}{\int
dt^{\prime}\exp\left(-\frac{2\kappa^{3}}{3|E(t^{\prime})|}\right)}\approx
10.35\;\textnormal{a.u.},$ (35)

where the integration runs over the whole laser pulse.

Figure 9:  Positions of the upper and lower saddle-points of the integral (17)
in the complex $t^{\prime}$ plane with varying $x$ and for fixed
$\omega\,t=\pm 0.1\pi/2$ (blue and red, respectively). The saddles draw smooth
curves with varying $x$, where the arrows indicate growing values of $x$. For
values of $x\lesssim 7$, the lower saddle-point disappears below the imaginary
axis. Each pair of curves is centred around the line $\textnormal{Re}(\omega
t^{\prime})=\omega t$, shown in dashed. For the case $\omega t=0$, not shown,
the two lines meet one point corresponding to $x=x_{t}$.

##  V Conclusions

In this work, we have analyzed the tunneling time delay in strong field
ionization for a simple model of an atom with a short-range potential. The
wave function was calculated to first-order in the SFA for any intermediate
time, enabling a quantum treatment of dynamics in the region where
quasiclassical descriptions break down, viz. around the classical tunnel exit.
For a given coordinate, the peak of the wave packet shows a time delay with
respect to the peak of the laser field and this time delay is positive at the
classical tunnel exit.

We argue that reflections of the electron wavepacket under the tunneling
barrier are fundamentally responsible for this non-zero time delay around the
tunnel exit. This is a general phenomenon, present in any regime of strong
field ionization, as well as in any tunneling process. In particular, a simple
showcase is presented in Appendix A for the case of tunneling through a box
potential.

To identify and separate contributions of reflected components to the
wavefunction, we first considered the analytically soluble case of an electron
in a constant electric field. In addition to providing a benchmark to our SFA
wavefunction, the constant field wavefunction can be deconstructed by
analysing its definition as an integral in the complex plane; this integral is
dominated by two saddle points, allowing us to unambiguously identify the
transmitted and reflected wavefunction components as contributions to the
integral from the regions around each saddle point.

We apply this observation to the SFA wavefunction by defining an integration
contour that purposefully neglects the contribution of the reflection saddle
point, and obtain a pseudo-wavefunction distribution that has zero time delay
with respect to the peak of the laser field everywhere under the barrier.
Moreover, the tunnel exit is unequivocally determined from the SFA theory as
the spatial co-ordinate at which the saddle points merge at the peak of the
laser field.

Finally, we emphasize the distinction between the time delay present at the
tunnel exit (theoretically measurable using a virtual detector) and an
(experimentally observed) asymptotic delay. We have shown within the first-
order SFA that the signature of the time delay in the electron wavefunction
vanishes since the ionized electron propagates to the detector as the
classical (i.e. with zero time delay) and the quantum mechanical electron
share equivalent asymptotic trajectories. Thus, the asymptotic time delay
associated with the direct ionization path is always vanishing. Nevertheless,
a more accurate estimate of the asymptotic time delay via the second-order SFA
in Ref. [51] has shown that interference of the direct and the sub-barrier
ionization paths generates non-vanishing negative asymptotic time delay in the
near-threshold of the OTBI regime, which fades out in the deep tunneling
regime.

##  VI Acknowledgements

The authors are grateful to the referees for their useful remarks, which
generated fruitful discussions. This article comprises parts of the PhD thesis
work of Daniel Bakucz Canário, submitted to Heidelberg University, Germany.

##  Appendix A Tunneling through a box potential barrier

In this appendix we consider the role of quantum reflections in the tunneling
time delay for an electron wavepacket tunnelling through a one dimensional box
potential. This is perhaps the simplest example of tunnelling time delay since
the wavefunction is readily separable (and of analytic solution) in the
regions inside and outside the barrier, allowing the contributions of
reflections in to the time delay to be clearly identified.

It should be mentioned that this model differs from true strong field
ionization in that it considers the scattering of a wavepacket incident on a
potential barrier; in actual ionization the electron originates from within a
potential barrier. Be that as it may, this study provides a simple, intuitive
picture of tunnelling time delay, suitable even for the uninitiated.

###  A.1 The Box Potential

Figure 10:  Pictorial representation of the square barrier potential for an
incident monochromatic wave. The wavefunction is a piecewise solution of the
Schrödinger equation for the three regions shown, given by Equations
(37)-(38).

Consider a wave packet

$\Psi(x,t)=\int\mathop{}\\!\mathrm{d}p\>f(p-p_{0})\>\psi(p)\>e^{-i\>E(p)\;t}$
(36)

with energy $E(p)={p}^{2}/2$, incident on a potential barrier $V(x)=V_{0}$ for
$0\leq x\leq a$ and $0$ elsewhere, where $f(p-p_{0})$ is some distribution
peaked at $p_{0}$ (e.g. a Gaussian), as shown Fig. 10. Each $p$-component
wavefunction obeys the time-independent Schrödinger equation with the
piecewise solution

$\displaystyle\psi_{I}(x)$ $\displaystyle=$ $\displaystyle
e^{ip\,x}+Re^{-ip\,x},$ (37) $\displaystyle\psi_{II}(x)$ $\displaystyle=$
$\displaystyle C_{1}e^{qx}+C_{2}e^{-qx},$ (38) $\displaystyle\psi_{III}(x)$
$\displaystyle=$ $\displaystyle Te^{ipx},$ (39)

with momenta $p=\sqrt{2\,E}$, $q=\sqrt{2(V_{0}-E)}$. The amplitude of the
incoming wave has been set to unity, without loss of generality, and the co-
efficients $C_{1}$ and $C_{2}$ are the typical reflection and transmission
coefficients under the barrier, respectively.

Matching the above solutions and their derivatives at the boundaries yields
the coefficients

$\displaystyle C_{1}$
$\displaystyle=\dfrac{(-2i\chi)(1+i\chi)e^{-\xi}}{(1-i\chi)^{2}\>e^{\xi}-(1+i\chi)^{2}\>e^{-\xi}},$
(40) $\displaystyle C_{2}$
$\displaystyle=\dfrac{(-2i\chi)(1-i\chi)e^{\xi}}{(1-i\chi)^{2}\>e^{\xi}-(1+i\chi)^{2}\>e^{-\xi}},$
(41) $\displaystyle R$
$\displaystyle=\frac{(1+\chi^{2})\;\left(e^{-\xi}-e^{\xi}\right)}{(1-i\chi)^{2}\>e^{\xi}-(1+i\chi)^{2}\>e^{-\xi}},$
(42) and, $\displaystyle T$
$\displaystyle=\frac{(-4i\chi)e^{-ipa}}{(1-i\chi)^{2}\>e^{\xi}-(1+i\chi)^{2}\>e^{-\xi}}.$
(43)

We have introduced the dimensionless parameters $\chi=p/q$ and $\xi=q\,a$
which, loosely speaking, determine the relative length and height of the
barrier respectively.

###  A.2 Time Delay

The wavepacket after transmission is of the form

$\Psi_{III}=\int\>\mathop{}\\!\mathrm{d}p\>|T(p)|\>\exp{[i\left(\varphi(p)+px-E(p)t\right)]},$
(44)

where $T=|T|e^{i\varphi}$. The maximum of this amplitude occurs when the phase
in Eq. (44) vanishes, that is when (after some re-arrangement):

$x=p_{0}\,t-\left[\frac{\partial\varphi}{\partial p}\right]_{p=p_{0}}$ (45)

In the absence of a potential barrier, the peak travels at the classical
velocity (in atomic units) $p_{0}$. Equation (45) shows that the barrier
causes a delay of the peak in reaching a given position $x$, a delay which is
given the by the energy derivative of the transmission phase $\varphi$. Thus,

$\tau=\frac{1}{p_{0}}\left[\frac{\partial\varphi}{\partial
p}\right]_{p=p_{0}}+\frac{a}{p_{0}},$ (46)

gives the time delay of the peak after crossing the barrier, i.e. the
tunneling time. For the box potential, one finds

$\tau=\frac{a}{p_{0}}\frac{\frac{1}{2\xi}\left(\chi+\frac{1}{\chi}\right)^{2}\tanh(\xi)+\frac{(1-\chi^{2})}{2}\mathop{}\\!\textnormal{sech}^{2}(\xi)}{1+\frac{1}{4}\left(\chi-\frac{1}{\chi}\right)^{2}\tanh^{2}(\xi)}.$
(47)

###  A.3 Quantum Reflections and Time Delay

What is the origin of this time delay? For longer barriers, $\xi\gg 1$, the
tunnelling time $\tau$ becomes

$\lim_{\xi\gg 1}\tau\approx\frac{2\chi}{p_{0}^{2}},$ (48)

vanishing when $\chi\ll 1$. Thus, the time delay is vanishing for $\xi\gg 1$
and $\chi\ll 1$. With this knowledge, we can analyze the exact wave function
inside the barrier. The reflection coefficient $C_{1}$ in the given limiting
conditions, $\xi\gg 1$ and $\chi\ll 1$,

$C_{1}\approx-2i\chi e^{-2\xi},$ (49)

vanishes asymptotically. This suggests an intuitive link between the
reflection of wave components and the retardation of the wave packet.

In the box potential case, in contrast to the tunneling problem, there exists,
however, a second reflection: the reflection from incidence on the first
surface of the barrier, described by the coefficient $R$. Classically, $R=1$,
and there is no time delay at reflection. However, in the quantum case
$R-1\neq 0$, inducing a time delay of the electron wave packet from entry into
the barrier. In the limit $\xi\gg 1$, and $\chi\ll 1$, we have
$R\approx-(1+2i\chi)$, i.e.,

$|1-R|\approx\chi\ll 1$ (50)

Analyzing the exact wave function, we can conclude that the time delay
vanishes when the reflection from the barrier surface is classical, and when
the reflection inside the barrier is negligible.

One can deduce the time delay caused by reflection at the entry by applying
the Wigner delay formula, Eq. 1, at $x=0$. For $\xi\gg 1$,
$R\approx-(1+i\chi)/(1-i\chi)$ and

$\displaystyle\tau_{entry}$
$\displaystyle=\frac{-i}{p_{0}}\left.\,\frac{\partial}{\partial
p}\ln\left(\frac{\psi_{I}(x)}{|\psi_{I}(x)|}\right)\right|_{x=0}\approx\quad\frac{\chi}{p_{0}^{2}}=\quad\frac{1}{2}\;\lim_{\xi\gg
1}\,\tau.\;$ (51)

Thus, one can conclude the time taken for the peak of the electron wavepacket
to travel through the box potential is caused in equal parts by the
reflections of the wavepacket on the barrier entry ($x=0$) and exit ($x=a$)
surfaces.

##  Appendix B Configurations of saddle points

Figure 11: Configurations of the saddle points of the argument
$\Phi(x,t,t^{\prime})$ of the wavefunction integral
$\psi_{i}=\int^{t}_{t_{0}}dt^{\prime}\exp(i\,\Phi)$ in complex $t^{\prime}$
plane, for parameter ranges $x=7,10,14$ and $\omega t=-\pi/20,0,+\pi,20$. The
dashed line corresponds to $t^{\prime}=0$ and each plot is centred around the
(scaled) observation time $\omega t$. The scale and colour coding are
identical to those of Fig. 6. As $x$ increases the two saddle points approach
vertically and, after a closest approach, separate horizontally. For the peak
of the pulse, $\omega t=0$, this closest approach is zero and the two saddle
points merge at the point $x_{t}$. Otherwise, varying $t$ skews the relative
orientation of the saddle points around the line
$\textnormal{Re}[t^{\prime}]=t$.

The configurations of the saddle points of the argument $\Phi(x,t,t^{\prime})$
of the wavefunction integral
$\psi_{i}=\int^{t}_{t_{0}}dt^{\prime}\exp(i\,\Phi)$ in complex $t^{\prime}$
plane, for a large $(x,t)$ parameter range are presented as a table in Fig.
11. The laser pulse evolves from left to right in this figure and mostly
shifts the reference line around which the saddle points are centred, namely
$\textnormal{Re}[t^{\prime}]=t$. It should be noted, by the definition in Eq.
17, a singularity is always to be observed at the point $t^{\prime}=t$.

As one moves down the table, configurations of the complex $t^{\prime}$ plane
are shown for spatial coordinates inside, neighbouring, and outside the
tunnelling barrier ($x=7,10,$ and $14$ a.u. respectively). There are marked
differences between each observation coordinate but the behaviour is
reminiscent of the complex plane configuration for the Airy integral,
displayed in Fig. 5. Indeed, the two pictures can easily be reconciled by a
$\nicefrac{{\pi}}{{2}}$ rad. clockwise rotation.

Inside the barrier, the saddle points are vertically aligned along the line
$\textnormal{Re}[t^{\prime}]=t$. As one increases the observation coordinate
and approaches the tunnel exit, $x\approx 10$, the two saddle points approach
each other vertically; as one exits the region neighbouring the tunnel exit,
$x\gg 10$, the saddles then separate from each other on the horizontal.

As one approaches the peak of the laser field, $t=0$, the distance of closest
approach around the tunnel exit shrinks. At the exact peak, this distance is
zero; that is, the saddle points merge. This phenomenon is directly comparable
to the merging of saddle points for the Airy integral at the classical tunnel
exit. Thus, we are able identify a new tunnel exit, $x_{t}$, from the complex
plane landscape, a topic discussed in Sec. IV.2.

The identification the exact value for the co-ordinate $x_{t}$ to a desired
precision is in principle achievable by a binary search or global minimization
of the distance function for the saddle points. However, such precision was
deemed unnecessary for the purposes of this work.

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