Can we measure the Wigner time delay in a photoionization experiment B. Feti c1W. Becker2and D. B. Milo sevi c1 3 2 1University of Sarajevo Faculty of Science Zmaja od Bosne 35 71000 Sarajevo Bosnia and Herzegovina

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Can we measure the Wigner time delay in a photoionization experiment?
B. Feti´c,1, ∗W. Becker,2and D. B. Miloˇsevi´c1, 3, 2
1University of Sarajevo, Faculty of Science, Zmaja od Bosne 35, 71000 Sarajevo, Bosnia and Herzegovina
2Max-Born-Institut, Max-Born-Str. 2a, 12489 Berlin, Germany
3Academy of Sciences and Arts of Bosnia and Herzegovina,
Bistrik 7, 71000 Sarajevo, Bosnia and Herzegovina
(Dated: October 12, 2022)
No, we cannot! The concept of Wigner time delay was introduced in scattering theory to quantify
the delay or advance of an incoming particle in its interaction with the scattering potential. It was
assumed that this concept can be transferred to ionization considering it as a half scattering process.
In the present work we show, by analyzing the corresponding wave packets, that this assumption is
incorrect since the wave function of the liberated particle has to satisfy the incoming-wave boundary
condition. We show that the electron released in photoionization carries no imprint of the scattering
phase and thus cannot be used to determine the Wigner time delay. We illustrate our conclusions by
comparing the numerical results obtained using two different methods of extracting the photoelectron
spectra in an attoclock experiment.
The idea that a particle wave can penetrate through
a potential barrier higher than its energy, i.e., through a
classically forbidden region, has been one of the most
intriguing features of quantum mechanics. This phe-
nomenon known as the tunnel effect has been a subject
of continuous research and debate since the first days of
quantum mechanics. For a historical perspective on how
the investigation of the tunnel effect shaped the early
days of quantum mechanics, see [1]. The phenomenon of
tunneling through a potential barrier has sparked a long-
standing debate in the scientific community with the very
simple question - how long does it take a particle to tun-
nel through the barrier? Although quantum tunneling is
formally well understood and exploited in countless ap-
plications, e.g., in semiconductors and superconductors
as well as scanning tunneling microscopy, there is no con-
sensus on the definition of a tunneling time and numer-
ous answers to this simple question are still debated and
disputed [2–6] even though almost a century has passed
since the first attempt to calculate the tunneling time [7].
The difficulties in understanding the tunneling time are
mainly due to two reasons. The first is related to the total
mechanical energy of the particle, which is lower than its
potential energy, implying that its kinetic energy is neg-
ative during the tunneling. The second reason lies in the
fact that time in quantum mechanics is not associated
with a Hermitian operator, but occurs as a parameter.
One might add that tunneling is a gauge-dependent con-
cept; hence a tunneling time is not a physical quantity.
In recent years the debate has further intensified with
the advent of ultrafast lasers and attosecond metrol-
ogy [8], which allow for measuring tunneling delays dur-
ing photoionization induced by a strong laser field. Tun-
neling can be understood as the first crucial step in
strong-field ionization. Under the influence of an in-
tense field, the electron can be liberated from the atomic
ground state into the continuum via tunneling through
the barrier formed by the atomic potential lowered by the
laser field. Initial measurements [9–11] suggested that
this tunneling is instantaneous, but subsequent results
appeared to imply that the tunneling process takes a fi-
nite time [12–14]. More about the current status and the
controversies resulting from this ongoing debate can be
found in [15–20]. Often, a tunneling time is inferred from
the phase shifts of the partial-wave scattering phases. In
this Letter, we will show that the wave packet created in
an ionization experiment does not carry any information
about the scattering phase shifts. Hence, in such an ex-
periment no time delay can be inferred that is related to
scattering phases.
We introduce the concept of the Wigner time delay
for a particle scattered off a spherically symmetric short-
range potential V(r). The motion of the particle is gov-
erned by the Hamiltonian H0=−∆/2 + V(r). We as-
sume that the initial direction of the incoming particle is
along the zaxis so that the initial state is associated with
the plane wave eikz/(2π)3/2. After elastic scattering the
final momentum of the particle is k= (k, θk, ϕk) and the
wave function is the plane wave
φk(r) = eik·r/(2π)3/2=
∞
X
`=0
i`g`(θ)j`(kr),(1)
where g`(θ) = (2`+ 1)P`(cos θ)/(2π)3/2,θis the angle
between the unit vectors ˆ
kand ˆ
r,P`(cos θ) is a Leg-
endre polynomial and j`(kr) a spherical Bessel func-
tion with the asymptotic behavior j`(kr)r→∞
−−−→ sin(kr −
`π/2)/(kr). From scattering theory we know that there
are two linearly independent eigenstates of the station-
ary Schr¨odinger equation, H0ψ(±)
k(r) = Ekψ(±)
k(r), Ek=
k2/2>0, which obey different boundary condition at
large distances rfrom the origin [21]:
ψ(±)
k(r)r→∞
−−−→ (2π)−3/2heik·r+f(±)
k(θ)e±ikr /ri,(2)
where outgoing (i.e., eikr /r) and incoming (i.e., e−ikr /r)
spherical waves have, respectively, the scattering ampli-
arXiv:2210.05219v1 [quant-ph] 11 Oct 2022
2
tude f(+)
k(θ) and f(−)
k(θ). The method of partial waves
can be used to present ψ(±)
k(r) in the form
ψ(±)
k(r) =
∞
X
`=0
i`g`(θ)e±iδ`(k)u`(k, r)
kr ,(3)
where δ`(k) is the scattering phase shift of the `th
partial wave and the normalization hψ(±)
k|ψ(±)
k0i=
δ(k−k0) is used. The radial functions u`(k, r)
are solutions of the radial Schr¨odinger equation
d2/dr2−`(`+ 1)/r2−2V(r) + k2u`(k, r) = 0, satisfy-
ing the relation u`(k, r)r→∞
−−−→ sin (kr −`π/2 + δ`). The
scattering phase shift is a real angle that vanishes for
all `if the potential V(r) is equal to zero. It mea-
sures the amount by which at large distances from the
origin the phase of the radial wave function for angu-
lar momentum `is shifted in comparison with the freely
moving radial wave. Using (1), (3), and the asymptotic
forms of the functions j`(kr) and u`(k, r) for r→ ∞, it
can be shown that the scattering amplitude is f(±)
k(θ) =
k−1P∞
`=0(2`+ 1)(±1)`e±iδ`sin δ`P`(cos θ).
Next, we analyze the time evolution of the wave pack-
ets built from the eigenstates ψ(±)
k(r) [22]:
Ψ(±)
k0(r, t) = Zd3kAk0(k)e−iω(k)tψ(±)
k(r),(4)
with ω(k) = k2/2. We assume that the momentum k
is narrowly spread around some finite momentum k0so
that the wave-packet amplitude Ak0(k) peaks at k=
k0and decreases rapidly with increasing |k−k0|. A
convenient choice for this amplitude is
Ak0(k) = b
k0kexp −(k−k0)2
2σ2δ(Ωk−Ωk0),(5)
where σis a real constant that specifies the width of
the wave packet, Rd3kAk0(k) = 1, and b−1=σ√2π.
Note that all contributing waves propagate in the same
direction ˆ
k0. From (3)–(5), we get
Ψ(±)
k0(r, t) =
∞
X
`=0
g`(θ0)R(±)
k0`(r, t),(6)
R(±)
k0`(r, t) = bi`Z∞
0
dke−(k−k0)2
2σ2−iω(k)t±iδ`(k)u`(k, r)
k0r.
(7)
Since the amplitude is narrowly peaked around k0, the
scattering phase shifts δ`(k) and ω(k) can be approxi-
mated by their first-order Taylor expansions:
δ`(k)≈δ`0+δ0
`0(k−k0), ω(k)≈ω0+v(k−k0),(8)
where δ`0≡δ`(k0), δ0
`0≡(dδ`/dk)k=k0,ω0≡ω(k0), and
v≡(dω/dk)k=k0>0.
Using the asymptotic form of the function u`(k, r), we
obtain the time-dependent wave packet (6) at large dis-
tances rfrom the target, with
R(±)
k0`(r, t)r→∞
−−−→ X
s=±1
s`+1
2ik0R(±,s)
k0`(r, t).(9)
After the substitution k0=k−k0,k0→k, using (7) we
get R(±,s)
k0`(r, t) = b
reiδ`0(s±1)+i(sk0r−ω0t)Rdk exp{− k2
2σ2+
i[sr −vt + (s±1)δ0
`0]k}, where the integral over k∈
(−∞,∞) can be solved using R∞
−∞ dx exp(−ax2−2cx) =
pπ/a exp(c2/a), a > 0. The result is
R(±,±)
k0`(r, t) = ei(±k0r−ω0t)±2iδ`0
re−σ2
2(r∓vt+2δ0
`0)2,
R(±,∓)
k0`(r, t) = ei(∓k0r−ω0t)
re−σ2
2(r±vt)2.(10)
The wave packet for the plane wave (1) is
Φk0(r, t) = Zd3kAk0(k)e−iω(k)tφk(r)
r→∞
−−−→
∞
X
`=0
g`(θ0)X
s=±1
s`+1
2ik0R(−s,s)
k0`(r, t),(11)
while for the scattered wave in (2) it is
F(±)
k0(r, t) = Zd3k
(2π)3/2Ak0(k)e−iω(k)tf(±)
k(θ)e±ikr
r
=
∞
X
`=0
g`(θ0)F(±)
k0`(r, t),(12)
F(s)
k0`(r, t) = s`+1
2ik0hR(s,s)
k0`(r, t)− R(−s,s)
k0`(r, t)i.(13)
Using Eqs. (6)–(13) it can be shown that
Ψ(±)
k0(r, t)r→∞
−−−→ Φk0(r, t) + F(±)
k0(r, t).(14)
Now, the physical interpretation of the wave func-
tions ψ(±)
k(r) can be deduced from the time evolution
of the corresponding wave packets. The plane-wave
packet Φk0(r, t) is always present, while the scattered
wave packet F(±)
k0(r, t) does or does not contribute, de-
pending on whether we consider the wave packet be-
fore (t→ −∞) or after (t→+∞) the electron is in-
cident on the potential V(r). Let us first consider the
wave packet Ψ(+)
k0(r, t). For large positive times t→ ∞,
the term F(+)
k0`(r, t)∝ R(+,+)
k0`(r, t)− R(−,+)
k0`(r, t) is dom-
inant [23]. It represents an outgoing almost spherical
wave ∝eik0r/r, which is equal to the difference between
the wave localized around r=vt −2δ0
`0and the free
wave localized at r=vt, and moves away from the ori-
gin with the group velocity v. For large negative times
t→ −∞, both R(+,+)
k0`(r, t) and R(−,+)
k0`(r, t) vanish and
Ψ(+)
k0(r, t) reduces to the plane wave packet Φk0(r, t), i.e.,
3
more precisely, to its part R(+,−)
k0`(r, t), which represents
an incoming spherical wave ∝e−ik0r/r and is localized
around r=−vt. Hence, the wave packet Ψ(+)
k0(r, t) corre-
sponds to a scattering scenario: an incoming plane wave
for negative times approaches the scattering center at the
origin. In the interaction, it generates an outgoing spher-
ical wave. This latter wave contains the scattering phases
δ`(k) and its peak lags behind by the radial distance 2δ0
`0
with respect to a freely propagating wave.
The wave packet Ψ(−)
k0(r, t) displays a very different be-
havior. We have F(−)
k0`(r, t)∝ R(−,−)
k0`(r, t)− R(+,−)
k0`(r, t).
For t→ ∞, according to (10), both terms vanish. There-
fore, for t→ ∞ the wave packet Ψ(−)
k0(r, t) reduces to
the plane-wave packet Φk0(r, t). Hence, it is suitable for
describing a photoionization experiment in which the lin-
ear momentum k=k0of the liberated photoelectron is
measured at large distances from the atomic target at
times long after the photoionization event occurred. It
is crucial for our argument that for t→ ∞ Ψ(−)
k0(r, t)
reduces to a plane-wave packet, which is independent of
the scattering phases δl. Indeed, for photoionization, we
have a bound-continuum transition and there is no “be-
fore event” like in scattering.
One might argue that for ionization rather than scat-
tering different combinations of the two linearly indepen-
dent wave functions ψ(+)
k(r) and ψ(−)
k(r) have to be used.
However, in [24] we showed that in extracting the elec-
tron spectrum from the solution of the time-dependent
Schr¨odinger equation (TDSE) the former has to be pro-
jected on the incoming-wave scattering solution ψ(−)
k(r).
Any admixture of ψ(+)
k(r) may lead to unphysical arti-
facts in the spectrum.
For scattering, the derivative ∆tW= 2δ0
0`/v =
2dδ`/dEk(for Ek=Ek0) was first proposed by Eisen-
bud [25] to quantify the delay or advance of an incom-
ing particle in its interaction with the scattering poten-
tial [26]. This was further elaborated by Wigner [27]
and Smith [28] and is often referred to as the Eisenbud-
Wigner-Smith time delay or just the Wigner time delay
(both terms are used interchangeably). Originally, it was
introduced for the scattering of an s-wave (`= 0) off a
hard sphere. For more details about the time delays in-
duced by the scattering potential, see [5].
Ionization has been envisioned as a half-scattering pro-
cess. Hence, it has been argued that one half of the
Wigner time delay ∆tWis the pertinent delay [29, 30].
However, as we just noticed, the final state of the elec-
tron released in a photoionization process has no imprint
whatsoever of the scattering phase and, in consequence,
does not lend itself to an extraction of the Wigner tun-
neling time from scattering phases. In the Supplement,
we consider a long-range potential, which includes the
Coulomb potential in addition to the short-range poten-
tial. In this case, the corresponding long-range wave
packet Ψ(−)
Ck0(r, t) for t→ ∞ reduces to the Coulomb
wave packet in place of a pure plane-wave packet.
The term attosecond angular streaking refers to a
method of extracting temporal information from ioniza-
tion experiments with few-cycle laser pulses with near-
circular polarization [9, 10, 17, 19]. The basic idea be-
hind the attoclock is that the tunneling process is most
likely to occur when the field E(t0) assumes its maximal
strength. The rotating electric field and the atomic po-
tential create a rotating potential barrier, which electrons
can tunnel through to reach the continuum. Depend-
ing on the ionization time t0, the liberated electrons are
forced into different directions in the polarization plane
(like the hand of a clock). By utilizing a circularly po-
larized pulse no rescattering off the atomic potential is
possible, meaning that the electrons are forced directly
towards the detector. A few-cycle pulse ensures that the
ionization probability assumes its maximum only once,
at the peak of the electric field. If the electron appears
in the continuum at the time t0, it will be detected in
the direction perpendicular to that of E(t0) with the mo-
mentum k=−A(t0), where A(t) = −RtE(t0)dt0is the
vector potential. This statement holds under the condi-
tions that the initial electron velocity is zero, the laser
field only depends on time, and the binding potential is of
short, ideally zero, range [31]. Otherwise, an offset angle
θdresults between the direction of A(t0) and the electron
momentum at the detector, which can have various ori-
gins. After all of the above (and some other) mechanisms
have been discounted, an additional offset angle might be
left. This would be attributed to a nonzero time that the
electron spends under the classically forbidden barrier,
i.e., a tunneling time.
In the presence of the long-range Coulomb potential,
the time delay τd=θd/ω (with ωthe frequency of the
laser field) is often expressed as the sum of two contri-
butions [29, 30]: τd=τW+τCLC, where τW= ∆tW/2 is
a one half of the Wigner time delay, since, as mentioned
before, photoionization is considered a “half-scattering”
process, and τCLC is the Coulomb-laser-coupling delay re-
sulting from the interaction of the outgoing photoelectron
with the laser field plus the atomic potential of the resid-
ual positive ion. Both terms originate from the energy
derivative of the phase difference of the continuum states
in comparison to the free wave. This phase difference in-
cludes the scattering phase shift of the `th partial wave,
which combines the scattering shift due to the short-
range potential and the long-range Coulomb potential.
However, in the Supplement we show that for photoion-
ization only the contribution of the Coulomb logarithm
plays a role.
In order to provide numerical support for our previ-
ous conclusion that ionization experiments do not give
access to scattering phases (and the pertinent time de-
lays) we use solutions of the TDSE as described in
the Supplement. The photoelectron momentum distri-
摘要：

CanwemeasuretheWignertimedelayinaphotoionizationexperiment?B.Fetic,1,W.Becker,2andD.B.Milosevic1,3,21UniversityofSarajevo,FacultyofScience,ZmajaodBosne35,71000Sarajevo,BosniaandHerzegovina2Max-Born-Institut,Max-Born-Str.2a,12489Berlin,Germany3AcademyofSciencesandArtsofBosniaandHerzegovina,Bistri...

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