Coulomb time delays in high harmonic generation [original]

New J. Phys. 19 ( 2017 ) 023012 https: // doi.org / 10.1088 / 1367-2630 / aa55ea
PAPER
Coulomb time delays in high harmonic generation
Lisa Torlina
1
and Olga Smirnova
1 , 2
1
Max Born Institute for Nonlinear Optics and Short Pulse Spectroscopy, Max-Born-Strasse 2 A, D-12489 Berlin, Germany
2
Technische Universität Berlin, Ernst-Ruska-Gebäude, Hardenbergstr. 36A, D-10623, Berlin, Germany
E-mail: [email protected]
Keywords: strong ﬁ eld ionisation, high harmonic generation, Coulomb effects, analytical R -matrix theory, ionisation and recombination
times
Abstract
Measuring the time it takes to remove an electron from an atom or molecule during photoionization
has been the focus of a number of recent experiments using newly developed attosecond
spectroscopies. The interpretation of such measurements, however, depends critically on the
measurement protocol and the speci ﬁ c observables available in each experiment. One such protocol
relies on high harmonic generation. In this paper, we derive rigorous and general expressions for
ionisation and recombination times in high harmonic generation experiments. We show that these
times are different from, but related to, ionisation times measured in photoelectron spectroscopy: that
is, those obtained using the attosecond streak camera, RABBITT and attoclock methods. We then
proceed to use the analytical R -matrix theory to calculate these times and compare them with
experimental values.
1. Introduction
The problem of time-resolving the removal of an electron during photoionization and studying the time delays
associated with this process is an intriguing one [ 1 ] . Besides the fundamental implications for our understanding
of atom-light interaction, such measurements have the capacity to serve as a sensitive probe of multielectron
dynamics [ 2 , 3 ] and have an important role to play in calibrating attosecond recollision-based pump-probe
experiments [ 4 ] .
Indeed, recent experimental developments — including the attosecond streak camera [ 5 ] , the attoclock [ 6 , 7 ] ,
attosecond transient absorption [ 8 ] , RABBIT [ 9 – 11 ] and high harmonic spectroscopy [ 4 , 12 , 13 ] — have made it
possible to measure ionisation times down to the level of tens of attoseconds, both in the one-photon [ 5 , 9 , 10 ]
and multi-photon regimes [ 4 , 6 – 8 , 12 – 14 ] . In each case, however, a thorough theoretical understanding [ 15 – 21 ]
of the underlying physics has been crucial to correctly interpret the experimental results. To do so, it is necessary
to formulate a connection between the classical concept of time, the quantum wavefunction and the
experimental observables. The times obtained are inherently very sensitive to the way in which this is done.
In the one-photon case, the theoretical basis for doing so is now well-established ( e.g. see [ 1 , 22 ] ) . Ionisation
is thought of as a half-scattering process and time delays are de ﬁ ned in terms of the Eisenbud – Wigner – Smith
( EWS ) time [ 23 , 24 ] : the derivative of the scattering phase-shift of the photoelectron with respect to its energy,
f
D= - () t E
d
d .1
s
k
WS
In the multiphoton regime, however, the situation is notably less straightforward. If we think of ionisation as
a tunnelling process, we are presented with a number of different possible de ﬁ nitions for the tunnelling time,
and it is not clear from the outset which is the ‘ correct ’ time to use ( e.g. see [ 25 – 27 ] ) . What ’ s more, the use of any
of these de ﬁ nitions in the context of strong ﬁ eld ionisation necessarily restricts us to the tunnelling limit. It is not
clear a-priori how to take non-adiabatic effects into account or how to investigate the effect on time delays as we
move towards the few-photon limit [ 28 ] .
OPEN ACCESS
RECEI VED
10 July 2016
REVISED
10 December 2016
ACCEPTED FOR PUBLICATION
28 December 2016
PUBLISHED
2 February 2017
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Alternatively, the concept of ionisation time arises in another way within certain analytical approaches to
strong ﬁ eld ionisation. These include the widely used and broadly successful strong ﬁ eld approximation ( SFA ) —
which we use here as an umbrella term for the related work of Keldysh [ 29 ] , Perelomov et al [ 30 , 31 ] , Faisal [ 32 ]
and Reiss [ 33 ] — as well as the more recently developed analytical R -matrix ( ARM ) method [ 34 – 39 ] . The latter is
a fully quantum theory that is able to accurately describe the long-range Coulomb interaction between the
outgoing electron and the core — the absence of which is the main limitation of the SFA. In each of these
approaches, it is necessary to integrate over a time variable that describes the instant at which the bound electron
ﬁ rst interacts with the ﬁ eld. In analytical approaches, this integral is typically evaluated using the saddle point
method, and the real part of the complex saddle point solution, = i
[
] t

t Re
i , is interpreted as the most probable
time of ionisation — that is, the time at which the electron appears in the continuum. The concept of saddle point
time as ionisation time also features in the Coulomb-corrected SFA approach [ 40 , 41 ] ( see also recent review [ 42 ]
and recent update of the theory [ 43 ] ) , which takes the SFA ionisation amplitude as its starting Q1
point.
Recently, the above idea was applied to analyse [ 3 ] the results of the attoclock experiment [ 6 , 7 , 14 ] , the basic
premise of which relies on using short pulses of circularly or nearly circularly polarised light to induce ionisation
and de ﬂ ect electrons in different directions, depending on their time of ionisation. Using ARM to describe this
experimental setup and applying saddle point analysis, it was possible to derive an expression for ionisation time
as a function of the angle and momentum q
(

) p , at which the photoelectron is observed [ 3 ] :
qq =+ D () () ( ) tt p t p ,, . 2
C
i i
0
The ﬁ rst term,
qw q =+ D ()
t

tp ,
i
0 env
, is the SFA saddle point result, accurate in the limit of short-range
potentials ( where
w

is the angular frequency of the laser ﬁ eld and
q
D

() tp ,
env
is a small correction due to the
shape of the pulse envelope ) . The second term
D

t
C

is an additional delay that comes about due to the long-range
electron-core interaction. It takes the form
f
D= -
¶
¶ () t I ,3
C C
p
where
f
C

is the phase accumulated by the outgoing electron due to its interaction with the ionic core and I
p
is the
ionisation potential of the bound state from which the electron escaped [ 3 ] . Expression ( 3 ) has also been derived
within the ARM method in [ 37 ] . Applying this relationship to analyse the results of ab initio numerical
experiments, it was shown that accounting for the delay
D

t
C

was vital to correctly interpret experimental results
and reconstruct the ionisation time [ 3 ] .
In fact, equation ( 3 ) , which we can think of as the delay accumulated in a long-range potential compared to a
short-range potential, coincides with an expression for ionisation delay derived in an entirely different way. In
[ 28 ] , the idea of the Larmor clock — originally proposed in the context of tunnelling times — was applied to
ionisation. It was shown to reduce to the EWS time in the single-photon limit and reproduce equation ( 3 ) in the
strong ﬁ eld regime. Thus, the different de ﬁ nitions for times are not unrelated: there is a link between the saddle
point-based ionisation time, the Larmor tunnelling time and the EWS single-photon delay.
The attoclock, however, is not the only means by which ionisation times can be measured in the strong ﬁ eld
regime. Two-colour high harmonic spectroscopy experiments offer another elegant and powerful approach to
this problem, using a weak probe pulse to perturb the ionisation dynamics in a controlled way [ 4 ] . In light of this,
a number of questions naturally arise. Are the times measured in HHG the same as those measured by the
attoclock? If not, how are they related? How does the electron-core interaction imprint itself on the ionisation
and recombination times in this case?
Here, we address these questions by extending the saddle point analysis discussed above to HHG. After
brie ﬂ y reviewing the basic approaches for describing time in HHG in section 2 , we present a general analysis that
incorporates electron-core interaction in section 3 , and discuss how such a calculation can be implemented in
practice using ARM in section 4 . In section 5 , we present results of this calculation and compare our ﬁ ndings to
times reconstructed from two-colour high harmonic spectroscopy experiments. Finally, we discuss the results
and analyse the magnitude of Coulomb corrections to times in section 6 . Section 7 concludes the work.
2. Ionisation and recombination times in HHG: a brief overview
2.1. The classical model
The simplest theoretical description of ionisation time in the context of HHG comes from the classical model,
which describes the process in terms of three steps: tunnel ionisation, classical propagation in the continuum
and recombination. In the standard classical model [ 44 ] ( see also pertinent [ 45 – 48 ] ) , electron-core interaction is
neglected during the classical propagation step, and it is assumed that the electron starts its continuum motion
with a velocity of zero. If we also assume that recombination occurs when the electron returns to its starting
point and relate its kinetic energy at this instant to the energy of the emitted photon, we obtain the following
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New J. Phys. 19 ( 2017 ) 023012 L Torlina and O Smirnova

three equations:
= () ( ) t v 0, 4
i
ò = () ( ) tt v d0 , 5
t
t
i
r
=-
w
() ( ) vt E I
1
2 ,6
p r 2
where =+ () (
)

tt
v

pA is the electron velocity,
A

is the vector potential that describes the laser ﬁ eld


=-
¶
¶ t
A ,
and
w

=
w
E

N
is the energy of the emitted photon.
For any given photon energy
w
E

, the above equations can be solved for the ionisation time t
i
, the
recombination time t
r
and the canonical momentum
p

. As such, we can associate an ionisation and
recombination time to each harmonic number: this de ﬁ nes the mapping between experimental observable and
time in this case. However, although excellent as a ﬁ rst approximation, this model is clearly rather crude. It
arti ﬁ cially matches a quantum mechanical description of the ionisation step with classical propagation, assumes
very simple initial conditions for the electron ’ s continuum motion and ignores the in ﬂ uence of the positively
charged ionic core. It is not surprising, therefore, that a comparison of the above predictions with times obtained
in high harmonic spectroscopy experiments revealed a notable discrepancy [ 4 ] .
2.2. The SFA
As mentioned in the introduction, the SFA offers a more sophisticated quantum approach to the problem of
time in strong ﬁ eld ionisation, which is based on saddle point analysis. In essence, the key approximation of the
SFA is to neglect the interaction between the electron and its parent ion after the instant ionisation or,
equivalently, to assume that the core potential is short range. Doing so, the induced dipole can be expressed as
[ 49 ] :
òò ò
w =- ¢ ¢ -- ¢ - ¢
w
() ( ) ( )
() ( )
Nt t P t t Dp p id d d , , e e e , 7
Et It t S t t p
i ii , ,
p V
where S
V
is the Volkov phase
ò tt =
¢
() ( ) Sv
1
2 d, 8
t
t
V 2
and P is a prefactor that varies relatively slowly. The times
¢ t

and t over which we integrate can be associated with
ionisation and recombination respectively.
The presence of a large phase, which leads to rapid oscillations of the integrand, makes it possible to evaluate
the above integral using the saddle point method. It tells us that the integral will be accumulated predominantly
in the vicinity of points where the derivative of the phase vanishes, which are de ﬁ ned by the saddle point
equations [ 49 , 50 ] :
¶
¶¢ = ()
S
t I ,9
p
V
¶
¶ = ()
S
p 0, 1 0
V
¶
¶ =-
w ()
S
t EI .1 1
p
V
We shall denote the solutions to these equations by ir
t

t p ,,
s
00 0
.
In fact, it is easy to check that equations ( 10 ) and ( 11 ) coincide exactly with equations ( 5 ) and ( 6 ) from the
classical model, while equation ( 9 ) is a modi ﬁ ed version of equation ( 4 ) : ¢= - () vt I 2 p
2 . The latter, in fact, gives
rise to a key difference between the two descriptions. Whereas the solutions in the classical model are fully real,
their counterparts in SFA are complex in general ( see e.g. discussion in [ 51 , 52 ] ) . Nevertheless, as mentioned in
the introduction, the real parts of the saddle point solutions come with an interpretation. We can associate
= i
[
] t

t Re
i
00
, = [
]

pp Re s
0 0 and = r
[
] t

t Re
r
00
with the time of ionisation, canonical momentum and time of
recombination respectively. For short-range potentials, this interpretation is particularly transparent ( e.g. see
discussion in [ 3 ] ) . As such, equations ( 9 ) – ( 11 ) again de ﬁ ne a mapping between time and harmonic number,
albeit a somewhat different one.
Using this model, it was shown that agreement with times reconstructed from high harmonic spectroscopy
experiments is notably improved [ 4 ] . However, what both the classical model and the SFA have in common is
the neglect of the long-range electron-core interaction throughout the electron ’ s motion in the continuum. In
the context of the attoclock, we have seen that this is not suf ﬁ cient: ignoring the in ﬂ uence of the core potential on
saddle point solutions leads to qualitatively and quantitatively incorrect results [ 3 ] . Motivated by this, let us now
consider how the above solutions are modi ﬁ ed if we allow for an electron-core interaction term.
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New J. Phys. 19 ( 2017 ) 023012 L Torlina and O Smirnova

3. In ﬂ uence of electron-core interaction on times in HHG: a general analysis
It was shown in [ 34 ] that, within ARM, the SFA expression for the ionisation amplitude — the analogous quantity
to the induced dipole given by equation ( 7 ) — is modi ﬁ ed by the addition of a Coulomb term
-
e
W i
C
. This term
accounts for the effect of the long-range interaction between the outgoing electron and its parent ion. Let us now
consider what a term of this nature would imply for times in HHG.
In particular, suppose that the integral in equation ( 7 ) now includes some additional factor
-¢ ()
e
Ft t p i, ,
, which
takes electron-core interaction into account. Although ARM can provide us with an explicit expression for F ,t o
which we shall return in section 4 , we shall keep the analysis general for the time being. We assume only that F is
small compared to S
V
and allow it to be complex in general. The real part of F then speci ﬁ es the phase associated
with the electron-core interaction, which we shall denote by
f

= [
]

F Re .
The saddle point equations ( 9 ) – ( 11 ) are then modi ﬁ ed as follows
¶
¶¢ + ¶
¶¢ = ()
S
t
F
t I ,1 2
p
V
¶
¶ + ¶
¶ = ()
SF
pp
0, 1 3
V
¶
¶ + ¶
¶ =-
w ()
S
t
F
t EI ,1 4
p
V
and since F is small compared to S
V
, we can search for solutions of the form
+D +D +D
i i r r
()

tt p p tt ,,
s s
00 0
,
where
ir
(

) tp t ,,
s
000
satisfy equations ( 9 ) – ( 11 ) . Expanding about the SFA saddle points, keeping only ﬁ rst order
terms in F , and using the chain rule to rewrite the derivatives, we arrive at the following result ( see appendix A ) :
D= -
¶
¶ - ¶
¶
i
w
() t F
I
F
E ,1 5
p
D= -
¶
¶
r
w
() t F
E .1 6
In general, since F is complex, these corrections will also be complex. However, as before, we can assign an
interpretation to their real parts:
D

=D
i
[] tt Re
i
and
D

=D
r
[] tt Re
r
encode the delays due to the electron-core
interaction imprinted upon ionisation and recombination times respectively:
ff
D= -
¶
¶ - ¶
¶
w
() t IE
,1 7
p
i
f
D= -
¶
¶
w
() t E .1 8
r
There are a few things worth noting about the above results. First, notice that the expression for the
correction to the recombination time, f
D

=- ¶ ¶ w
tE
r , is reminiscent of the EWS time given by equation ( 1 ) .
Indeed, we can understand this if we recall that recombination is simply single-photon ionisation run in reverse.
Varying with respect to the photon energy is directly equivalent to varying with respect to the kinetic energy of
the recombining electron. Second, note that the correction to ionisation time in HHG,
ff
D

=- ¶ ¶ -¶ ¶ w
tI E
p i , has an extra term compared to its counterpart in strong ﬁ eld ionisation given by
equation ( 3 ) . Subtracting the above expressions for
D

t
r
and
D

t
i
yields
f
D- D = D - =
¶
¶
() ( ) tt t t I .1 9
p
ri r i
In other words, the derivative of the phase with respect to the ionisation potential in HHG tells us how much
more ( or less ) time the electron will spend in the continuum before it recombines, as a consequence of electron-
core interaction.
4. Coulomb corrections in HHG using ARM
Keeping the function F unspeci ﬁ ed, this is as far as we can go. If we want to determine the corresponding time
delays in practice, we must evaluate F explicitly. Luckily, this is precisely what ARM allows us to do.
In particular, it was shown in [ 34 ] that if we apply ARM to strong ﬁ eld ionisation,
-
e F i
is replaced by -
e W i C
SF
I

,
where
4
New J. Phys. 19 ( 2017 ) 023012 L Torlina and O Smirnova

ò
=¢ ¢
k
() ( ( ) ) ( ) WT t U t pr p ,d , . 2 0
C t
T
s
SFI
Here,
W

C
SF
I

is the electron action associated with the Coulomb-laser coupling [ 53 , 54 ] ,
p

is the electron
momentum measured at the detector, T is the time of observation, (
) U

r is the core potential and
r s
is the
Coulomb-free electron trajectory,
ò
¢= +  
¢
i
() ( ( ) ) ( ) tt t rp p A ,d . 2 1
s t
t
The lower limit of the integral in equation ( 20 ) ,
k =-
ki
t

t i
2
, is determined by the boundary-matching
procedure for the outgoing electron [ 34 , 37 ] .
Indeed, this boundary matching was crucial. It made it possible to smoothly merge the asymptotic tail of the
bound electron wavefunction with the quasiclassical wavefunction of the escaping electron driven by the laser
ﬁ eld, and allowed us to avoid using
W

C
SF
I

too close to the ionic core, beyond its range of applicability. In strong
ﬁ eld ionisation, it was only necessary to carry out this matching once, when the electron departed from its parent
atom or molecule. In HHG, on the other hand, we know that we must also account for the recombination step,
when the electron returns to the core. This makes it necessary to perform the matching procedure once more, to
connect the phase due to Coulomb-laser coupling accumulated between ionisation and recombination to the
ﬁ eld-free continuum solution for the returning electron. Effectively, doing so will tell us the correct endpoint
t
end
to use in place of the observation time T in the upper integration limit of the HHG counterpart to
equation ( 20 ) .
Fortunately, an equivalent boundary-matching problem has already been solved. In [ 17 ] , single photon
ionisation in the presence of a probing infra-red ﬁ eld was analysed in the context of the attosecond streak
camera. There, a matching argument was used to show that the effective starting point for an electron trajectory
with initial velocity v
0
is given by
= () () () rv va v
1 ,2 2
00
00
where
=
gx -
() ( )
()
av 2e e , 23
v
0 22
E 0
g

E
is Euler ’ s constant and
å
x =-
=
¥
⎡
⎣
⎢ ⎛
⎝
⎜ ⎞
⎠
⎟ ⎤
⎦
⎥
() ( ) v n vn vn
1 1a r c t a n
1 .2 4
n
0
1
0
0
Noting that recombination in HHG is simply the reverse of this process ( that is, the emission, rather than
absorption, of a photon in the presence of an infra-red ﬁ eld ) , we can apply this result directly to determine the
boundary-matched endpoint t
end
. In particular, we now think of r
0
as the end point of our electron trajectory
and set = ()
r

rt
s 0e n d
. The corresponding velocity v
0
is the velocity at recombination,
== -
w
() ( ) vv E I 2. 2 5
p 0r
For any given photon energy
w
E

, we then have
ò
== +
w
iw
() ( ) ( ( ) ( ) ) ( )
()
vt p E A t t rr d , 2 6
s tE
t
s
0r e n d
end
which can be used to solve for t
end
, using equations ( 22 ) – ( 25 ) to evaluate
()
r

v
0r
.
In doing so, it should be noted that r
s
( t ) is complex in general for real times, whereas r
0
is always real.
Consequently, in order to satisfy the above equation, we must allow t
end
to be complex as well. This tells us that,
in contrast to ionisation, our integral for W
C
will no longer end on the real axis: both start and end points are
now complex. The generalisation, however, is straightforward. When describing Coulomb effects in strong ﬁ eld
ionisation, the integration contour was chosen in two parts: ﬁ rst, down from t
κ
to =
i
[] tt Re i on the real axis,
and then along the real axis up to time T [ 34 , 38 ] . These two legs were interpreted in terms of tunnel ionisation
and the electron ’ s motion in the continuum respectively, following PPT [ 31 ] . For HHG, we simply add a third
leg: down from [
]

t Re end on the real axis to t
end
( see ﬁ gure 1 ) .
We note that, in general, the interpretation of the real part of the complex saddle point as the ionisation time
is not connected to a certain choice of the integration contour. Analytic properties of the integrand in W
C
,o f
course, make it possible to deform the contour considerably without in ﬂ uencing the result of the integration.
However, in contrast to the contour, the saddle point time itself is unique and well-de ﬁ ned: it has a real and an
imaginary part. In fact, the real part of the saddle point can be directly probed experimentally using the attoclock
setup [ 3 ] , as discussed in section 1 . In the long wavelength limit, the attoclock observable ( the so-called offset
angle ) is simply equal to the real part of the saddle point time multiplied by the angular frequency [ 28 ] . What ’ s
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New J. Phys. 19 ( 2017 ) 023012 L Torlina and O Smirnova

more, the expression for the ionisation time we obtain from the real part of the saddle point coincides with that
derived using an alternative method in [ 28 ] . The latter method uses neither the concept of tunnelling nor
trajectories, nor does it rely on saddle point analysis, yet the result for the ionisation time remains the same.
Having determined the correct integration endpoint t
end
by boundary matching, it is straightforward to
write down the HHG counterpart to equation ( 20 ) . There are only two major changes we need to make: ( 1 ) the
momentum at the detector
p

should be replaced by the saddle point solution w
() E p
s , and ( 2 ) the observation
time T should be replaced by the time t
end
. That is,
ò
=¢ ¢
ww
k
() ( ( ) ) ( ) WE t U E t r d, , 2 7
C t
t
s
HHG
end
where
ò
¢= +  
ww
¢
iw
() ( ( ) ( ) ) ( )
()
Et E t t rp A ,d . 2 8
s tE
t
s
We can readily determine the value of w
() E p
s from equations ( 9 ) – ( 11 ) .
5. Results: Coulomb time delays in HHG
Having determined t
end
and chosen a contour, we have all the ingredients we need in order to evaluate the
correction W
C
as given by equation ( 27 ) . In itself, this tells us the ﬁ rst order effects on HHG spectra due to the
long-range electron-core interaction. However, as we saw in section 3 , we need one further step to learn about
times: we must differentiate the Coulomb phase
f

= [
]

W Re C with respect to I
p
and
w
E

to ﬁ nd the corrections to
the saddle point solutions, as given by equations ( 17 ) and ( 18 ) . In practice, this can be done numerically by
evaluating W
C
for two or more closely spaced values of I
p
,
w
E

.
We shall now compare these results with times reconstructed from high harmonic spectroscopy
measurements, originally published in [ 4 ] . In essence, in addition to the fundamental ﬁ eld responsible for HHG,
the experiment uses a weak probe ﬁ eld to de ﬂ ect electrons laterally during their motion in the continuum. By
varying the delay of the probe ﬁ eld with respect to the fundamental and observing the associated variation in
harmonic signal, it is possible to determine the time at which ionisation and recombination occurred. The
details of the associated reconstruction procedure are described in [ 4 ] ( see also SI of [ 4 ] ) and analysed in detail in
[ 19 ] . For the bene ﬁ t of the reader, we also brie ﬂ y outline the main idea of the procedure in appendix B .
Figure 2 shows the results for the helium atom. The ionisation and recombination times obtained using
ARM ( blue lines ) are compared to SFA ( black lines ) , the classical model ( grey lines ) and times reconstructed
from high harmonic spectroscopy experiments ( pink and green dots, see [ 4 ] ) . Note that, as discussed in
appendix B , the experiment could only determine the difference between ionisation and recombination times,
and their dependence on harmonic number: the results are valid up to an overall additive constant. To plot
absolute times, we calibrate the experiment by shifting all ionisation and recombination times by a ﬁ xed amount
until we achieve the best possible ﬁ t with the recombination times predicted by our models ( as ﬁ gure 2 shows,
there is relatively little variation in recombination times between different models, so this is unambiguous ) .
Figure 1. Contour for the W
C
integral in HHG.
6
New J. Phys. 19 ( 2017 ) 023012 L Torlina and O Smirnova

Compared to the SFA, we ﬁ nd that the corrected ionisation times in ARM are shifted to earlier values. This
effect has only a weak dependence on harmonic number: the shift varies between ∼ 33 and ∼ 37 attoseconds,
decreasing slightly with N . The recombination times are notably less affected overall, though they display a
stronger dependence on the harmonic number. The shift in this case is between ∼ 5 and ∼ 19 attoseconds, again
decreasing with N . Putting these two facts together, we see that the total amount of time the electron spends in
the continuum ( given by
f
¶

¶ I
p ) increases by ∼ 18 – 28 attoseconds. The relative size of these corrections and
their dependence on harmonic number is analysed further in section 6 .
Comparing our results with times reconstructed from high harmonic spectroscopy measurements, we ﬁ nd
that ARM offers a notable improvement over the SFA, where electron-core interaction was neglected. Although
the corrections to ionisation and recombination times are only of the order of tens of attoseconds, they are
nevertheless clearly within the resolution of current state-of-the-art HHG experiments.
We note that our method also makes it possible to analyse the Coulomb corrections to imaginary times. The
results of doing so are presented in [ 13 ] , where imaginary ionisation times where reconstructed from
experimental measurements.
6. Discussion: how can we understand the size of the corrections?
If we consider the results presented in the previous section, it is natural to ask why the shift in ionisation times
due to the electron-core interaction is notably larger than the shift in recombination times. Can we estimate the
relative size of these two corrections and understand their dependence on harmonic number? What are the small
parameters associated with each of them?
Referring back to equations ( 17 ) and ( 18 ) from section 3 , it is clear that to answer these questions we must
look more closely at the two partial derivatives
f
Dº -
¶
¶ () t I ,2 9
p
1
f
Dº -
¶
¶
w
() t E ,3 0
2
of which
D

t
i
and
D

t
r
are composed. Let us analyse each of these in turn.
First, consider f
D

=¶ ¶ tI
p 1 . We recall from equation ( 3 ) in section 1 that this coincides with
D

t
C

, the ﬁ rst
order correction to the saddle point time in strong ﬁ eld ionisation relative to the SFA result. To obtain a simple
analytical expression for the small parameter associated with this term, let us now take the tunnelling limit. In
this limit, the imaginary part of the SFA saddle point
i
t

0
is given by  t = () It 2 p i where


() t
i
is the magnitude
of the laser ﬁ eld at the moment of ionisation: this is the well-known Keldysh tunnelling time. The real part of the
saddle point can be chosen to be zero. At the same time, it is possible to show that, in the tunnelling limit, the
correction
D

t
C

due to the electron-core interaction takes the value
D

»
i
tZ I
p
SF 3 2 , where Z is the charge of the
core ( see derivation in [ 36 , 37 ] ) . Its imaginary component vanishes to ﬁ rst order in this limit. Taking the ratio of
the norms of these two complex quantities, we obtain
*

z = D ==
i
∣∣
∣∣
() ()
t
t
Zt
I
n
S
2 2Im
2
3 ,3 1
p
1
1
0
i
2 SFA
52
where * =
n

ZI
p
1
2

is the effective principal quantum number of a given quantum state and Im S
SFA
is the
imaginary part of the electron action associated with the electron ’ s dynamics in the laser ﬁ eld only. The quantum
Figure 2. Ionisation and recombination times predicted using the classical model ( grey lines ) , SFA ( black lines ) and ARM ( blue lines )
compared with times reconstructed from high harmonic spectroscopy experiments ( pink and green dots, see [ 4 ] ) for helium atoms at
l

= 800 nm and
»´ I 3.8 10
14
Wc m
– 2
.
7
New J. Phys. 19 ( 2017 ) 023012 L Torlina and O Smirnova

number n
*
characterises the action of the electron in the bound state, while S
SFA
characterises the action of the
electron driven by the laser ﬁ eld. As long as the action due to the laser-driven dynamics exceeds the action in the
ground state — which is usually the case in strong ﬁ eld ionisation — the ratio z 1 will remain small. Indeed, this
condition is essentially equivalent to the condition for applicability of most Coulomb-corrected SFA theories.
Let us now turn our attention to the second correction term f
D

=- ¶ ¶ w
tE
2 . This term is the analogue of
the well-known Wigner – Smith ionisation time given by equation ( 1 ) , albeit the phase f here is not the ﬁ eld-free
scattering phase, but incorporates the effects of the laser ﬁ eld and describes Coulomb-laser coupling. To obtain a
small parameter associated with this term, let us consider a ratio of velocities:
z =D qv
2 r
, where
D

q
is the shift
of electron momentum due to Coulomb-laser coupling, and v
r
is the velocity of the electron at the moment of
recombination, as given by equation ( 25 ) . Using the expression for
D

q
from [ 54 ] ,w e ﬁ nd

z = - =
w
()
∣( ) ∣ )
() ()
Zt
vE I
Zt
v
2 ,3 2
p
2
r
r 32 32
r
r
4
where


() t
r is the magnitude of the laser ﬁ eld at the recombination time.
If we now compare expressions ( 31 ) and ( 32 ) for our two small parameters, we notice that they have a similar
structure. Both are of the form

z µ ()
Z
v .3 3
4
For z 1 , v is the electron velocity of the bound state
= vI 2
bp
and


is the magnitude of the laser ﬁ eld at the
moment of ionisation. For z
2
, we have the velocity of the returning electron v
r
and the laser ﬁ eld at the moment
of recombination.
What can this tell us about the relative size of
D

t
i
and
D

t
r
? In HHG, we know that the velocity of the
returning electron is larger than that associated with the bound state. At the same time, the instantaneous value
of the laser ﬁ eld is larger at ionisation than at recombination, or they are comparable, for majority of electron
trajectories ( except those corresponding to low harmonic numbers ) . Based on the small parameters above, this
suggests that
D

t
2

should be smaller than
D

t
1
, which, in turn, helps explain why corrections to ionisation times,
given by
D

+D tt
12
, are notably larger than corrections to recombination times, determined by
D

t
2

alone.
The expressions for z 1 and z
2
may also shed light on the dependence of
D

t
i
and
D

t
r
on harmonic number.
Higher harmonics are associated with higher recombination velocities v
r
and lower values of the instantaneous
ﬁ eld


() t
r , which, via z
2
, would explain why corrections to recombination times decrease with harmonic
number. Since, conversely, the instantaneous ﬁ eld at ionisation


() t
i
displays a slow increase with harmonic
number, while v
b
is ﬁ xed, we would expect that
D

t
1
grows with N . Putting these two facts, we can understand the
observed values of
D

t
i
. Its behaviour is governed by an interplay of both terms and — since
D

t
2

decreases faster
than
D

t
1
increases — this would explain the slow decrease of
D

t
i
with N observed in our results.
Finally, it should be noted that both
D

t
1
and
D

t
2

are inherently small, which can make it dif ﬁ cult to resolve
D

t
i
and
D

t
r
in practice. In particular, these corrections have not been seen in the numerical simulations of [ 55 ] .
7. Conclusions and outlook
In this paper, we have shown how ionisation and recombination times in HHG are modi ﬁ ed when the long-
range interaction between the active electron and the ionic core is taken into account. The resulting corrections
are closely related to the delays in strong ﬁ eld and single photon ionisation respectively, though they are not
identical. In particular, the expression for the ionisation delay in HHG, ff
D

=- ¶ ¶ -¶ ¶ w
tI E
p
i
HHG ,
contains an additional term compared to its counterpart in strong ﬁ eld ionisation, f
D

=- ¶ ¶ tI
p
i
SFI ( derived
in [ 3 , 28 , 37 ] ) . In essence, this ‘ additional ’ ionisation delay stems from the different measurement protocols used
in strong ﬁ eld ionisation and HHG experiments respectively. While the former detect photoelectrons, the latter
detect the XUV photons that are generated when electrons return to the core. Consequently, in the case of HHG,
the ‘ delay-line ’ on the way to the detector is associated not only with ionisation ( which includes propagation in
the continuum ) , but also with the recombination step, which brings with it a delay of its own.
Comparing the predictions of the ARM theory — in which the electron-core interaction is accounted for —
with times measured in high harmonic spectroscopy experiments, we ﬁ nd that the agreement is excellent. The ﬁ t
is visibly better than for the SFA, where such effects are omitted. Thus, although relatively small, we can conclude
that the electron-core interaction leaves a measurable and distinct signature on times in HHG. As such, it should
be taken into consideration when calibrating attosecond recollision-based pump-probe experiments and
interpreting experimental data.
8
New J. Phys. 19 ( 2017 ) 023012 L Torlina and O Smirnova

Acknowledgments
The authors gratefully acknowledge the support of Deutsche Forschungsgemeinschaft, project SM 292 / 2-3. We
thank H So ﬁ er and N Dudovich for providing the experimental data from [ 4 ] , shown in ﬁ gure 2 . The authors
gratefully acknowledge the support of MEDEA-AMD-641789-17 project. We thank V Serbinenko, F Morales
and M Ivanov for discussions.
Appendix A. Derivation of general expression for saddle point corrections
In this section we present the derivation of equations ( 15 ) , ( 16 ) . To do so, it will be convenient to introduce the
following vectors
=¢ () ( ) tp t s ,, , A 1
=
ir
() ( ) tpt s ,, , A 2
s
0 000
D =D D D
ir
() ( ) tp t s ,, , A 3
s
¡ =
w
() ( ) IX E ,, A 4
p
and the vector-valued function
 
-
ww
 () ( ) ( ) IX E IX E I
f :
,, ,, . A 5
pp p
33
The gradient ∇ will be de ﬁ ned as a row vector, and 
F

s 0 will occassionally be used as shorthand for
 (
)

F s
s 0 ,
where F is the function introduced in section 3 .
Using this notation, the saddle point equations ( 12 ) – ( 14 ) can be expressed succinctly as
DD ¡ + +  + = () () ( ) ( ) SF ss ss f ,A 6
ss V 00
while their analogues in the SFA become
¡ = () ( ) ( ) S sf ,A 7
s V 0
where we have added a constant

X

0
to the lhs of the second saddle point equation for convenience in both
cases.
If we now expand the solution to equation ( A6 ) about
s

0
, we obtain ( to ﬁ rst order in
D

s
and 
F

s )
D¡ +  +  = ( ) [ [ ]( )] ( ) ( ) ( ) SJ S F ss s s f ,A 8
T
ss s V 0 sV
00
where
J

s is the Jacobian with derivatives taken with respect to
s

. Using equation ( A7 ) and the fact that the
Jacobian of a gradient is the Hessian, we have
D =-  [ [ ]( )] ( ) ( ) HS F ss s .A 9
T s sV 00
Since the Hessian is symmetric and invertible ( in this case ) , we can rewrite this as
D =-  -
[[ ] ] ( ) FH S s .A 1 0
ss V 1
00
In principle, if we could evaluate 
F

s 0 , we would be done. However, we do not have direct control over the
value of the complex saddle point
s

when we do the calculation numerically, which makes this a dif ﬁ cult
quantity to work with. Instead, what we can do is vary the parameters
¡

and see how F changes as a result — this
makes it possible to evaluate  ¡ F numerically. With this in mind, we would like to rewrite equation ( A10 ) in
terms of  ¡ F instead of  F
s .
To do so, we note that equation ( A7 ) establishes a functional relationship between
s

0
and
¡

. In principle, we
could solve this equation to ﬁ nd
¡ ()
s

0
. Taking the gradient of F with respect to
¡

and applying the chain rule
gives
¡¡ = 
¡¡
(( ) ) ( ) [() ] ( ) FF J ss s ,A 1 1
s
00 0
so
= 
¡¡ -
[[ ] ] ( ) FF J s .A 1 2
s 01
0
We now need only evaluate the Jacobian
¡
[
] J

s
0
. To do so, let us differentiate both sides of the SFA saddle
point equation equation ( A7 ) with respect to
¡

:
¡¡ =
¡¡
[ ( ( ))] [ ( )] ( ) JS J sf .A 1 3
s V 0
9
New J. Phys. 19 ( 2017 ) 023012 L Torlina and O Smirnova

Applying the chain rule again, we can rewrite this as
¡¡ =
¡¡
[ ( )] [ ( )] [ ( )] ( ) JS J J ss f ,A 1 4
ss V 00
and so ( again making use of the fact that the Jacobian of a gradient is the Hessian ) ,
=
¡¡
-
[] [ ] [ ] ( ) JH S J sf .A 1 5
s
0 V 1
0
Taking the inverse and substituting this into equation ( A12 ) gives
= 
¡¡
-
[[ ] ] [] ( ) FF J H S f .A 1 6
ss
1 V
00
Finally, this allows us to rewrite equation ( A10 ) as
D =- 
¡¡
--
[ [ ]] [ ] [ [ ]] ( ) F J HS HS sf ,A 1 7
ss
1 VV
1
00
which simpli ﬁ es to
D =- 
¡¡
-
[] ( ) FJ sf .A 1 8
1
In our case,
¡
J f

is very simple:
=
-
¡
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟ () J f
10 0
01 0
10 1
,A 1 9
and
=
¡ -
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
[] ( ) J f
10 0
01 0
10 1
.A 2 0
1
This allows us to write down a solution for
D

s
in terms of  ¡ F :
D= -
¶
¶ - ¶
¶
w
() t F
I
F
E ,A 2 1
s
p
D= -
¶
¶ () p F
X ,A 2 2
s
D= -
¶
¶
w
() t F
E .A 2 3
r
Appendix B. Reconstruction procedure for times in high harmonic spectroscopy
experiments
Suppose HHG is driven by a strong laser ﬁ eld with fundamental frequency ω , described by the vector potential
w =
w

() ( )
A

te A t sin
x 0
. To probe the associated ionisation times, let us also apply an additional perturbative
control ﬁ eld with frequency
w

2
, given by
wa =+
w

() (
) A

te A t sin 2
y 20 , which is phase locked to the
fundamental ﬁ eld and polarised in an orthogonal direction. This ﬁ eld will effectively ‘ kick ’ the electron in a
lateral direction when it leaves its bound atomic state and exits from the tunnelling barrier at the ionisation time
t
i
. The magnitude and sign of this kick is controlled by α , the relative phase between the two ﬁ elds, referred to as
the ‘ two-colour delay ’ .
As we vary α ,w e ﬁ nd that the HHG yield is modulated. In particular, the harmonic signal attains its
maximum value when the lateral kick is equal to zero or, in other words, when the electron displacement
between ionisation and recombination is minimised. To good approximation, this occurs when the vector
potential of the control ﬁ eld at the moment of ionisation is close to zero: that is,
wa =+ »
w

() ( )
A

te A t sin 2 0
y 2i 0 i m a x
if
a
max
is the two-colour delay at which the harmonic signal is
maximised. This, in turn, gives us a relationship between the observed
a
max
and the ionisation time t
i
:
a
w

»
∣

∣ t
im a x
. In the full reconstruction, corrections accounting for the kick during tunnelling are also
included.
Note that the second harmonic ﬁ eld breaks the symmetry in electron dynamics between the two consecutive
laser half cycles, which leads to the generation of even harmonics. This asymmetry is greatest at those values of α
where the lateral velocity of the electron upon recombination is largest. In practice, recombination times are
reconstructed following the maximum HHG signal for even harmonics, as suggested and performed in [ 4 ] and
augumented in [ 55 ] by including complex ( rather than real ) SFA recollision times.
Note also that the experiment can only reconstruct ( i ) the delay between ionisation and recombination time,
( ii ) the dependence of the ionisation and recombination times on harmonic energy. In order to determine the
absolute values of ionisation and recombination times in HHG, it is necessary to calibrate
a
max
. We do this by
10
New J. Phys. 19 ( 2017 ) 023012 L Torlina and O Smirnova

shifting all times by a ﬁ xed amount until we obtain the best ﬁ t with the recombination times predicted by our
models. This is a consistent way to proceed since recombination times do not differ considerably between
models.
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