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Keldysh theory of strong field ionization: history, applications, difficulties and perspectives
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2014 J. Phys. B: At. Mol. Opt. Phys. 47 204001
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Journal of Physics B: Atomic, Molecular and Optical Physics
J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204001 (35pp)
doi:10.1088/0953-4075/47/20/204001
Review Article
Keldysh theory of strong ﬁeld ionization:
history, applications, difﬁculties and
perspectives
S V Popruzhenko
National Research Nuclear University MEPhI, Kashirskoe shosse 31, 115409, Moscow, Russian Federation
E-mail: [email protected]
Received 4 April 2014, revised 16 June 2014
Accepted for publication 30 June 2014
Published 8 October 2014
Abstract
The history and current status of the Keldysh theory of strong ﬁeld ionization are reviewed. The
focus is on the fundamentals of the theory, its most important applications and those aspects
which still raise difﬁculties and remain under discussion. The Keldysh theory is compared with
other nonperturbative analytic methods of strong ﬁeld atomic physics and its important
generalizations are discussed. Among the difﬁculties, the gauge invariance problem, the
tunneling time concept, the conditions of applicability and the application of the theory to
ionization of systems more complex than atoms, including molecules and dielectrics, are
considered. Possible prospects for the future development of the theory are also discussed.
Keywords: intense laser ﬁelds, nonlinear ionization, Keldysh theory, optical tunneling, strong
ﬁeld approximation, coulomb effects, simple man model
(Some ﬁgures may appear in colour only in the online journal)
1. Introduction
theory and several productive generalizations of the original
Keldysh approach. Second, despite the very broad and longterm use of the theory, several important conceptual questions
remain open and are being actively debated in the literature.
What is the difference between the Keldysh theory, the KFR
theory and the strong ﬁeld approximation (SFA)? How is the
Keldysh model connected to the simple-man model (SMM) of
ionization? Why is the theory, in general, gauge-noninvariant
and how can a noninvariant theory be used for the calculation
of observables? Whatis the accuracy of the theory and which
parameters deﬁne its applicability conditions? What is the
physical interpretation of the Keldysh tunneling time; can this
quantity be measured in experiments? Is the Keldysh theory
actually applicable to the description of the ionization of
dielectrics and other spatially extended systems? These
questions are answered in different ways in the literature and
some of them do not yet have any answer. Here we try to sort
such conceptual questions into two groups: those with a
known correct answer and those that still remain unclear. In
This review paper is dedicated to the 50th anniversary of
Keldyshʼs seminal work [1], where a nonperturbative
approach to the description of the nonlinear ionization of
atoms and dielectrics by intense electromagnetic ﬁelds was
pioneered. Now known as the Keldysh theory (often also
referred to as the Keldysh model, the Keldysh ionization
ansatz or the Keldysh theory of optical tunneling) or the
Keldysh–Faisal–Reiss (KFR) [2, 3] theory, it is being routinely used for the description of multiquantum processes
induced by intense laser radiation. Apart from giving a historic overview of the theory, this paper has two aims. First,
the development of the method proposed by Keldysh has
generated a bulk of literature which is difﬁcult to navigate,
particularly for those researchers beginning their work in the
ﬁeld. Therefore, a sort of manual to guide researchers through
the theory and its applications would be useful. Having this
purpose in mind, we describe here the main results of the
0953-4075/14/204001+35$33.00
1
© 2014 IOP Publishing Ltd Printed in the UK
J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204001
Review Article
the ﬁrst case we will formulate the solution, in the second we
will present the most essential controversial viewpoints.
In order to keep the length of this paper reasonably
limited, we leave out those aspects of the theory which have
already received detailed and clear consideration in the literature. This includes relativistic and Lorentz ionization, spin
effects (see reviews [4, 5] and references therein) and ionization by ultrashort laser pulses, reviewed in [4, 6]. Ionization
of molecules is considered only brieﬂy in connection with the
gauge invariance problem. We will also not discuss recollision phenomena, i.e. high-energy above-threshold ionization,
generation of high-order harmonics and nonsequential ionization. For overview of this currently thriving area of strong
ﬁeld physics, we direct the reader to the reviews [7–10].
For routine use of the Keldysh theory, one needs reliably
checked analytic expressions for photoelectron momentum distributions, angular distributions, spectra and rates calculated for
standard cases (including monochromatic ﬁelds, elliptical polarization, ionization from states with an arbitrary angular
momentum, etc). For the most frequently used cases we have
tried to collect such expressions in this review. More formulas
can be found in the early papers by Nikishov and Ritus [11, 12],
Perelomov, Popov and Terentʼev [13–15], in the pivotal paper by
Reiss [3] (where, in particular, properties of the generalized
Bessel functions are discussed in detail), in the paper by Grybakin and Kuchiev [16] and the review by Popov [4]. For the
reader interested in the theory of ionization beyond the Keldysh
approximation as well as in physics of other strong ﬁeld phenomena, including both experiment and theory, we recommend
the recent review [17] and the books [18–22].
This review is organized as follows. Section 2.1 is devoted
to the formulation of the fundamentals of the Keldysh theory,
including the nonlinear photoionization matrix element and
general expressions for photoelectron momentum distributions.
In section 2.2 we explain the terminology and introduce the
dimensionless parameters which are important for further discussion. In section 3 a brief historic overview is given. Sections 4
and 5 review the current status of the theory. Subsection 4.1
contains a derivation of the matrix element. This derivation
allows one to formulate the conditions of applicability
(section 4.2) and to clarify the relationship between the Keldysh,
KFR and SFA approaches (section 4.3). In section 5 we formulate the theory in terms of complex trajectories, achieving this
via application of the saddle-point method (SPM; section 5.1),
and introduce the imaginary time method (ITM; section 5.2).
Section 6 is devoted to extensions of the Keldysh ionization
model including the Coulomb–Volkov approximation (CVA;
section 6.1), the Coulomb-corrected SFA (section 6.2) and the
analytic R-matrix (ARM) method (section 6.3). In section 7 we
discuss other efﬁcient nonperturbative methods which can be
reduced to the Keldysh theory or viewed as complementary.
Section 8 is devoted to difﬁculties, including the problem of
gauge invariance (section 8.1), the concept of the tunneling time
and relation to classical models of ionization (section 8.2) and
application of the Keldysh model to ionization of spatially
extended systems (section 8.3). The ﬁnal section contains a brief
conclusion and outlook.
We use the CGS (centimeter–gram–second) system in
the following section, and atomic units  = e = m = 1 with e
and m being the absolute value of the electron charge and the
electron mass, respectively, throughout the rest of the paper.
2. The basic equations of the Keldysh theory
2.1. Ionization amplitude and spectra
According to the Keldysh ansatz, the transition probability
amplitude between an atomic bound state and a continuum
state speciﬁed by the value of the photoelectron momentum p
measured at the detector is given by
MK (p) = −
+∞
i

∫−∞
Ψp Vint (t ) Ψ0 dt .
(1)
This expression is not present in [1], but its form can be
deduced unambiguously from equation (8) of that paper. Here
and below the subscript K denotes values deﬁned within the
Keldysh model; Ψ0 = ψ0 (r)e iIp t  is the bound state wave
function unperturbed by the laser ﬁeld and having the ionization
potential Ip, Ψp is the Volkov function [23–25] corresponding to
the electron canonical momentum equal to p and Vint is the
electron ﬁeld interaction operator. In the nonrelativistic regime,
the interaction operator and the Volkov function can be used in
the dipole approximation by neglecting the spatial dependence
and the magnetic component of the wave. An explicit form of
these functions is determined by the gauge chosen for
description of the electromagnetic wave. The most commonly
used are the length gauge when
Vint (r , t ) = e E (t ) r ,
Ψp (r , t ) =
1
exp
(2π ) 3 2
−
m
2
t
(2)
{
∫−∞ v2p (
i ⎡
⎣ m vp (t ) r

⎤⎫
t ′ dt ′⎥⎬
⎦⎭
)
(3)
and the velocity gauge
Vint (r , t ) = −
Ψp (r , t ) =
ie 
e2 2
A (t )  +
A (t ),
m
2m
⎧i ⎡
1
m
exp ⎨ ⎢ pr −
32
⎣
⎩
2
(2π )
t
(4)
⎤⎫
∫−∞ v2p (t′)dt′⎥⎦ ⎬⎭.(5)
˙ (t ) is the electric ﬁeld strength of the laser
Here E (t ) = −A
wave and A(t ) is the respective vector potential, both are spatially homogeneous in the dipole approximation and
vp (t ) = (p + eA (t ) ) m
(6)
is the time-dependent electron velocity. In the ﬁeld of a plane
electromagnetic wave, p is conserved (in the nonrelativistic
regime) and equal to the average (drift) electron momentum in
the laser ﬁeld and to the asymptotic momentum the electron has
after the ﬁeld is turned off.
If the amplitude of ionization M (p) is known (not
necessarily in the Keldysh approximation) the differential
probability to ﬁnd the photoelectron in the elementary volume
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J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204001
Review Article
d3p near the momentum p is given by
dW (p) = M (p) 2d3p .
absorption of an integer number n of photons
(7)
(
W=
∫
2 3
M (p) d p .
n
× R (p)dεp dOp, εp =
MK (p, τ )
τ
τ →∞
MK (p, τ ) = −
i

∫0
R (p) =
2
n ⩾ Nmin = ⎡⎣ I p + UP
(
d3p ,
τ
Ψp Vint (t ) Ψ0 dt .
⎛ iT ⎡ p 2
⎤⎞
= Ψp Vint Ψ0 (t ) exp ⎜⎜ ⎢
+ I p + UP ⎥⎟⎟
⎦⎠
⎝  ⎣2m
2
=
(10)
pn2
T
≡ εp + UP.
(11)
E0
(cos ωt , − ξ sin ωt , 0).
ω
 Kn (p) = −
(12)
For the ponderomotive energy, (11) gives
UP =
e2E 02
( 1 + ξ ),
2
4 mω2
(13)
∑
k =−∞
 Kn = 2π i
+∞
2iπkx
e
=
∑
δ (n − x )
(17)
(18)
= nω − I p − UP.
(19)
i

∫0
T
dt Ψp Vint (t ) Ψ0
.
(20)
p = pn
For a monochromatic ﬁeld with arbitrary polarization, the
amplitude (20) and the respective momentum distributions
can be expressed via Bessel functions. Assuming the vector
potential is in the form (12) and using the velocity gauge
representation (4), (5), one obtains (after taking the time
integral in (20) by parts) for a linearly polarized ﬁeld, ξ = 0,
where − 1 ⩽ ξ ⩽ 1 is the ﬁeld ellipticity; ξ = 0 corresponds to
linear and ξ = ±1 to right (left) circular polarization. Using
(10) the time integral in (9) can be presented as a sum over the
laser periods. Applying the relation
+∞
ω⎤⎦ + 1,
Equations(15)–(19) are model-independent and apply in any
theory describing ionization in quasimonochromatic ﬁelds.
Within the Keldysh model, the partial amplitude in (16) is
given by
The ﬁrst term in the r.h.s. of (11) is the drift energy and the
second is the ponderomotive (quiver) energy. In a monochromatic ﬁeld elliptically polarized in the (x,y) plane
A(t ) =
)
with pn deﬁned from the energy conservation law
2m
T
(16)
dwn (n)
= R ( p = pn n) ≡ R n (n)
dOp
2
p
e
+
A2
2m
2m
2
where Nmin is the minimal number of photons required for
ionization, i.e. the ionization threshold. Correspondingly,
peaks with n > Nmin are called above-threshold peaks and the
regime of ionization when these peaks are present in the
spectrum is known as above-threshold ionization (ATI).
Angular distribution in a given peak is obtained by integration
(15) over energy
(9)
where UP is the average quiver electron energy in the ﬁeld.
Using (6), we obtain for kinetic energy averaged over the
laser period:
1
=
(p + eA(t ))2
2m
mω2p
 (p)
2π
where dOp is the solid angle along the momentum direction
and  (p) is a contribution into the amplitude of ionization
from a single laser period1. Photoelectron peaks in (15) exist
for
Ψp Vint Ψ0 (t + T )
T
(15)
with
In a periodic ﬁeld the integrand in (9) possesses the property:
mv2
2
p2
2m
(8)
These two expressions are meaningful for laser pulses of
ﬁnite duration, particularly for few-cycle pulses. For sufﬁciently long pulses containing a large number of optical
periods—so that its electromagnetic ﬁeld is close to a periodical function of time: E (t + T ) = E (t )—it is physically
more appropriate to use probabilities per time unit (rates)
instead of time-integrated values (7) and (8). The differential
rate of ionization is deﬁned as a limit
dw K = lim
)
dw (p) = ∑δ εp + I p + UP − nω
This deﬁnes the momentum distribution of photoelectrons.
The total probability of ionization
(14)
I p + p2 2 m
ω
⎛ e E 0 p UP ⎞
⎟.
ψ0 (p) Jn ⎜
,
⎝ mω2 2ω ⎠
(21)
n =−∞
1
Note that the discreteness of the photoelectron spectrum (15) is a
consequence of the periodicity of the classical external laser ﬁeld and does
not require for its explanation a concept of photons, in contrast to what is
commonly stated.
one can extract a delta-function responsible for energy conservation, so that the momentum distribution takes the form
of a set of inﬁnitely narrow peaks, each corresponding to the
3
J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204001
Review Article
Here the Fourier–Bessel expansion is used
introduced in [1] and is known as the Keldysh parameter
+∞
∑
eiz sin φ =
γ=
Jk (z)eikφ
2mI p ω eE 0.
(29)
k =−∞
It can be interpreted as a ratio of the characteristic atomic
momentum ℏκ = 2mI p to the ﬁeld induced momentum
pF = eE0 ω or, alternatively, as a ratio of the time the
electron takes to cover the distance b 0 = I p eE0 moving with
the atomic velocity vat = 2I p m . The value b0 is the width
of a static barrier created by the ﬁeld E0, so that
τ = b 0 vat = γ ω is the time of ﬂight under the barrier or the
Keldysh tunneling time. The tunneling time concept in the
Keldysh theory is discussed in section 8.2. The limit γ ≪ 1 is
known in the literature as the tunneling domain. This characterization means that ionization proceeds similarly to the
case of a static electric ﬁeld when the electron tunnels through
a time-independent potential barrier. The tunneling limit is
also often mentioned as a quasistatic or adiabatic regime of
ionization, underlining the fact that in this limit time can be
treated as a parameter. As we will see below, in the tunneling
limit, accurate expressions for the rate of ionization differ
from those for the static ﬁeld by small corrections ∼γ2. The
very existence of a correct static limit is a standard benchmark
for a theory of strong ﬁeld ionization. The opposite limit
γ ≫ 1 is known as the multiphoton ionization regime. Over
the years the terminology has been evolving, so that identical
or similar terms are sometimes used for the description of
different ionization regimes. Namely, the tunneling limit,
γ ≪ 1, is also often mentioned as a low-frequency limit which
is not the same thing, according to (28). On the other hand, as
soon as condition (28) is satisﬁed, many photons are needed
to ionize the atom. It is, however, not appropriate to call this
situation ‘multiphoton ionization’ because this term is
reserved for the more speciﬁc case of γ ≫ 1. Instead, when
(28) is satisﬁed, we will specify this as the multiquantum
regime, in contrast to the few-quantum, K0 ∼ 1, or the single
quantum, K0 < 1, regimes.
Note that the value of the multiquantum parameter itself
is insufﬁcient for full characterization of an ionization regime,
as the ionization problem is two-parametric [3]. From the
parameters of the ﬁeld and atom, Ip, E0 and ω, more than two
dimensionless combinations can be constructed. Apart from
K0 and γ there are another two frequently used in the theory.
One is the ratio of the laser electric ﬁeld amplitude to the
characteristic electric ﬁeld Ech of the respective atomic level
and the generalized Bessel function is introduced [3, 26–28]
+∞
Jn (a , b) =
∑
Jn + 2k (a) Jk (b).
(22)
k =−∞
For circularly polarized ﬁelds, ξ = ±1, the expression for the
amplitude is simpler
 Kn = 2π i
I p + p2 2 m
ω
⎛ eE 0 p ⎞
⎟.
ψ0 (p) Jn ⎜
⎝ mω2 ⎠
(23)
Equations (21) and (23) contain the Fourier transform of the
bound state wave function
ψ0 (p) =
1
(2π )3 2
∫ d3r e−ipr  ψ0 (r).
(24)
For arbitrary ellipticity, the amplitude still has the form (21)
with a more complex function instead of Jn (a , b ) (see, e.g.
equation (45) in [29]).
In the length gauge (2), the Fourier transform of the
bound state wave function contains the time-dependent
velocity vp (t ) in the argument, instead of p . As a result, the
generalized Bessel expansion does not apply, excluding the
special case when the electron is bound by the zero-range
potential (ZRP) usually deﬁned in 3D space as
U (r ) =
∂
2 π 2
δ (r ) r , κ =
κm
∂r
2I p m

(25)
and the bound state wave function has the form2
κ e−κr
.
2π r
ψ0 ( r ) =
(26)
Then
ψ0 (q) =
κ
π
3/2
1
κ + q2  2
(27)
2
and (I p + v 2p 2 m ) ψ0 (vp ) = (I p + p 2 2 m ) ψ0 (p), so that the
amplitude calculated in the length gauge coincides exactly
with (21). Thus for a system bound by zero-range forces, the
Keldysh theory is gauge-invariant. There is further discussion
of the gauge dependence in section 8.1.
2.2. Useful definitions and dimensionless parameters
F = E 0 E ch,
The Keldysh theory is usually applied to the description of
nonlinear ionization in low-frequency electromagnetic ﬁelds.
The term ‘low-frequency ﬁeld’ means that the ionization
potential Ip of an atom is essentially larger than the photon
energy ω , i.e. the multiquantum parameter is large,
K0 = I p ω ≫ 1.
Another
2
independent
dimensionless
where the characteristic ﬁeld is deﬁned as
E ch =
(28)
parameter
(30)
32
⎛ κ ⎞3
m2 ⎛ 2I p ⎞
⎜
⎟ = ⎜ ⎟ Ea ,
e ⎝ m ⎠
⎝ κ0 ⎠
(31)
where Ea = m2e 5  4 ≈ 5.14 · 109 V cm−1 is the atomic
with
electric
ﬁeld
unity
and
κ0 = 1 a B
a B =  2 me2 = 5.29 · 10−9 cm is the Bohr radius. Another
parameter is the ratio of the ponderomotive energy (13) to the
was
In ZRP there is a unique bound state with l=0 [30].
4
J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204001
Review Article
photon energy (with the factor (1 + ξ 2 ) 4 omitted)
zF =
e2E 02
was obtained in the form equivalent to (20) and evaluated by
the SPM for small photoelectron momenta corresponding to
the maximum of the ionization probability. An expression for
the total ionization rate was obtained with exponential accuracy. Using the notations of section 2.2 one can present this
result in the form
(32)
mω3
known as the strong ﬁeld parameter [3]. Only two parameters
are independent, so that another two can be correspondingly
expressed as, for example
F = 1 2K0 γ , z F = 8F
K03
2
2
= 2K0 γ .
w K ∼ exp { −2K0 fK (γ ) },
(33)
⎛
1 ⎞
fK (γ ) = ⎜ 1 + 2 ⎟ arcsinhγ −
⎝
2γ ⎠
3. Brief history
1 + γ2
2γ
,
(34)
where the function fK is known as the Keldysh function and
the Keldysh parameter γ is deﬁned by (29). A pre-exponential
factor was also found in [1], but it does not reproduce the
known static-ﬁeld asymptotic [40] because the Coulomb
interaction between the photoelectron and the atomic core is
disregarded. This circumstance is clearly formulated in [1]. At
the same time, detachment of negative ions [41] and ionization of atoms [42] by a static electric ﬁeld was considered
using a different method and simple expressions for ionization rates were obtained. In the limit γ → 0 the result of
Keldysh agrees with the formulas of [41, 42] with exponential
accuracy.
Soon after the publication of Keldyshʼs paper,his result
was generalized in several important cases. Namely,photoelectron spectra and total ionization rates were calculated in a
closed analytic form for linear [13] and arbitrary elliptical
polarization of laser light [14],the effect of quantum interference in photoelectron spectra was considered [13] and the
method was extended into the relativistic domain [11]. In
[12, 15] the problem of describing the Coulomb interaction
was addressed and in [15] a correct expression for the rate of
ionization was found in the tunneling limit, γ ≪ 1. For derivation of the respective Coulomb correction, the ITM was
developed [15, 43]. It allows the interpretation of the Keldysh
theory in terms of classical electron trajectories propagating in
complex time and space. This interpretation appeared to be
extremely useful for the development of new analytic methods in the theory of strong ﬁeld ionization,in particular, to
describe the Coulomb interaction and treatment of relativistic
ionization including spin effects (see the reviews [4, 44, 45]).
Although the ITM is already expounded in reviews [4, 44], in
order to make this paper self-contained for the reader, we
describe the ITM procedure in section 5.2. Thus, by the end
of 1960s the Keldysh ionization ansatz was substantially
explored.
However, it soon became clear that the application of the
results of [1, 11–15] to descriptions of experimental data is
essentially restricted. The Keldysh theory allows the calculation of ionization rates and photoelectron momentum distributions consisting of ATI peaks separated by the photon
energy. At that time momentum distributions were not
available to be measured because at intensities of 1010 – 1011
W cm −2 even the second ATI maximum was too low to be
detected. Thus the only measurable values predicted by the
theory were angular distributions and total ionization rates.
However, angular distributions were affected by
Soon after the invention of lasers in 1960 the level of intensity
sufﬁcient for observation of highly nonlinear optical effects in
atoms, molecules and solids was reached. The ﬁrst observations of laser-induced breakdown of gases was already
reported in 1963 [31, 32], although it was not clear if space
charge effects played a role or not. The ﬁrst unambiguous
observations of nonlinear ionization of atomic xenon via a
seven-photon absorption [33] and of the molecule H2 via a
nine-photon absorption of radiation emitted by a ruby laser
[34] were reported in 1965 by Delone and coworkers.
The theoretical work of the pre-laser epoch was restricted
by the application of the lowest-order perturbation theory to
the description of processes involving two photons (see e.g.
[35, 36]). In two papers [26, 37] published in 1962, Reiss
suggested applying Volkov functions for the description of
the electron continuum dressed by an intense electromagnetic
ﬁeld. This idea opened the way for the development of
nonperturbative theories of strong ﬁeld phenomena. In 1964
the basis of relativistic Volkov functions was used by
Nikishov and Ritus for the description of elementary quantum
processes in the ﬁeld of a plane electromagnetic wave [38].
For scattering problems (e.g. annihilation of electron–positron
pairs and the inverse process of pair creation, nonlinear
Thompson scattering, etc), the use of a complete set of
Volkov waves instead of bare plane waves allowed the
development a theory where the classical ﬁeld of a plane
electromagnetic wave is accounted for exactly, while interaction with the spontaneous radiation ﬁeld is treated perturbatively. In contrast, in the case of ionization, when a bound
atomic state is involved, there is no rigorous method to
describe the ﬁeld of a strong electromagnetic wave. The
easiest case is a system subjected to circularly polarized light.
In the reference frame rotating with the ﬁeld frequency, the
Hamiltonian becomes time-independent [39]. For a system
bound by the zero-range force this allows the calculation of
the width of the quasistationary state developed from the
initial bound state, i.e. the rate of ionization. For a general
case of an atom and an arbitrary polarized ﬁeld this method
does not apply, so an essentially nonperturbative ansatz is
needed. The ansatz expressed by (1) was proposed in the
work by Keldysh, where he calculated the probability of
ionization of the ground state of hydrogen in a linearly
polarized monochromatic ﬁeld with the interaction operator in
the length gauge form (2). The partial ionization amplitude
5
J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204001
Review Article
that the photoelectron energy εf observed at a detector does
not depend on the laser intensity
ponderomotive scattering (see below in this Section) and the
rate requires, for its correct calculation, the Coulomb interaction to be accounted for. The latter was performed in the
tunneling limit [15], while the available data corresponded to
a deeply multiphoton regime of ionization. For example, in
early experiments [33, 34], the peak intensity was about
1011 W cm−2, so that γ ≃ 30 . These inconsistent theoretical
and experimental limitations were the reason why the Keldysh theory was not extensively used in 1970s and was
essentially even forgotten. In this period an important contribution to the theory was made by the work of Faisal [2],
where the nonlinear ionization amplitude (1) was derived
under more general assumptions than those of [1] (see the
discussion in section 4.2). In 1980 Reiss [3] developed an Smatrix formalism based on the approximation of the exact
continuum state by the Volkov function. Using the velocity
gauge(4), he obtained an expression for the amplitude of
ionization virtually identical to that of Faisal (see equations
(21) and (22) of [2] and equations (26) and (42) of [3]). In [3],
the ﬁrst systematic study of ionization spectra in different
regimes and an extended investigation of the properties of the
generalized Bessel functions were given. Simultaneously, the
theoretical discussion of problems inherent to the theory,
including its gauge-noninvariance, also reduced interest in the
Keldysh model in this period (see, e.g., a paradox discussed in
[46] and its resolution in [47]).
Only in 1979 was the ﬁrst observation of a photoelectron
spectrum consisting of more than one peak—i.e. an ATI
spectrum—reported [48]3. In contrast to the energy conservation law given by (15), where the positions of the ATI
peaks are intensity-dependent due to the presence of the
ponderomotive energy, the positions of the photoelectron
peaks observed in [48] and the subsequent experiments of that
time did not depend on the laser intensity. The reason for this
was the relatively long duration of laser pulses. Equation (15)
assumes that the laser ﬁeld is a plane electromagnetic wave,
so the electric ﬁeld amplitude does not depend on the coordinate perpendicular to the propagation direction of the pulse.
In a real experiment, a laser pulse is focused and the ponderomotive energy depends on the lateral coordinate dropping
down from its maximum at the focus center to zero at its edge.
If the pulse lasts long enough, so that the electron can travel
out of the laser focus before the ﬁeld is off, it experiences
ponderomotive acceleration and its canonic momentum p is
no longer constant [50, 51]. In the limit
τ ≫ R p,
ε f = nω − I p .
At parameters typical for the ionization experiments of the
1970s and early 1980s, τ ≃ 10−8 – 10−10 s, v ≃ 108 cm s−1
and R ≃ 10−3 cm, one obtains that inequality (35) is amply
satisﬁed, so that the intensity-independent positions of ATI
peaks are not surprising. The equations of section 2 assume
the limit opposite to (35), i.e. short laser pulses. This discrepancy, although not physically fundamental, additionally
postponed the time when the Keldysh theory was accepted
overall for descriptions of strong ﬁeld ionization. At the same
time, the application of long laser pulses in strong ﬁeld
experiments has stimulated the development of the theory of
ponderomotive forces (see [52, 53] for reviews including
ponderomotive forces in the relativistic regime).
Since the mid−1980s, after the invention of the chirped
pulse ampliﬁcation method [54], femtosecond laser pulses
became routinely used in experiments. Pulses with a duration
of less than 1ps are short in the sense of (35). Simultaneously
to the shortening of pulse duration, peak laser intensity has
increased enormously. The strong ﬁeld regime z F ≫ 1 and
even the regime of tunneling γ ≪ 1 were ﬁrst achieved with
CO2 lasers [55] and later with infrared lasers with a wavelength of approximately a micron. Since then, the Keldysh
theory and closely related approaches have been frequently
used for the interpretation of experimental data. The theoretical and experimental achievements of this period are
reviewed, e.g., in [56, 57].
Fast experimental progress offered higher and higher
intensities, so that investigations of nonlinear ionization
moved from the multiphoton to the intermediate regime
γ ≃ 1. Here the Keldysh model provided a commonly
accepted picture of the phenomenon: a typical ionization
spectrum (the spectra shown in the ﬁgures of sections 7.1 and
8.1 serve as good examples) extends up to several ponderomotive energies containing ∼ z F ≫ 1 ATI peaks; highly
charged ionic states are produced via the sequential ionization
mechanism, when electrons are being detached from an atom
one by one. The slopes of ionization rates plotted versus
intensity are determined mostly by the intensity dependence
of the Keldysh function (34),although the tunneling Coulomb
correction to the prefactor is also important for the quantitative description of data.
Finally, with the further development of laser technique
and photoelectron diagnostics, new features of photoionization spectra have been observed which are beyond the standard Keldysh ionization model. They can be sorted in three
groups. First, there are the effects of atomic structure completely disregarded in the model. Freeman resonances with
excited states can serve as an example [58]. Such resonances
generate a ﬁne intensity-independent structure for ATI peaks.
Theoretical approaches describing the resonant structure of
ATI spectra beyond the Keldysh model are reviewed, e.g., in
[56]. Second, signiﬁcant effects are induced by the Coulomb
interaction between the photoelectron and the parent ion.
Coulomb effects are determined mostly by the ion residual
(35)
where τ is the pulse duration and R is the laser focal spot
radius, the ﬁnal photoelectron momentum p f satisﬁes energy
conservation
p f2
p2
+ UP =
,
2
2
which means, in combination with energy conservation (15),
3
It is interesting to note that one of the phenomena directly connected to
ATI, namely nonsequential double ionization, was observed for the ﬁrst time
earlier, in 1975 [49].
6
J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204001
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charge and the laser pulse parameters but otherwise are species-independent. Asymmetric photoelectron angular distributions in elliptically polarized ﬁelds [59] present a good
example of a qualitative Coulomb effect. Third, a variety of
effects are generated due to laser-driven recollisions of the
photoelectron with its parent ion [60, 61]. Understanding of
the rescattering mechanism, including its role in the generation of the high-energy photoelectron plateau, high-order
harmonics, nonsequential double ionization and its potential
for probing atomic and molecular structure, has made an
enormous impact in strong ﬁeld physics over the past twenty
years.
Theoretical work of this late period has shown that the
Keldysh model can be regularly extended to include rescattering effects [62–64]. This extension is currently known as
the improved SFA or, alternatively, as the SFA with rescattering. Nonperturbative inclusion of Coulomb effects is less
straightforward but also possible. Here, considerable progress
(reviewed in section 6) has been achieved over the past
decade. The extension of the Keldysh theory toward inclusion
of the Coulomb interaction is mostly based on the technique
of Coulomb corrections to the phase of the amplitude (1).
Evaluation of these corrections employs the ITM. In an early
paper by Perelomov and Popov [15], the ITM was applied to
the calculation of the total ionization rate of atoms in the
tunneling limit, γ ≪ 1. It was shown there that the Coulomb
ﬁeld enhances the rate of ionization, typically by several
orders in magnitude—the effect has been reliably proven in
experiments [65]. Later this result was generalized to arbitrary
values of the Keldysh parameter [66]. Apart from the overall
enhancement of the total ionization rate, the technique of
Coulomb corrections allows the description of several effects
accessible for experimental observation, including Coulomb
asymmetry in elliptically polarized ﬁelds [59, 67, 68], cusps
and double-hump structures [69–71], low-energy structures
[72–76] and side lobes [77] in the momentum spectra of
photoelectrons. Inclusion of atomic structure effects into the
theory seems to be most problematic. The only atomic
information encoded in the Keldysh ionization amplitude (1)
is the value of the ionization potential and the symmetry of
the initial bound state, therefore the role of excited bound
states including possible multiphoton resonances is fully
ignored. The application of few-cycle laser pulses with a
broad spectrum, which has become routine over the past
10–15 years, makes the inﬂuence of the atomic structure in
strong ﬁeld ionization less important, because in a spectrally
broad ﬁeld transient resonances are essentially suppressed.
Thus the Keldysh theory is more appropriate for the
description of ionization in short pulses than in quasimonochromatic ﬁelds.
The Keldysh theory without Coulomb corrections is
quantitatively accurate for systems bound by short-range
(SR) forces. Therefore observation of the strong ﬁeld
photodetachment of negative ions offers a strict test of the
model [78]. The measurement of above-threshold detachment (ATD) of negative ions requires mid-infrared femtosecond lasers (in order to achieve the strong ﬁeld regime,
z F ≫ 1, γ ≃ 1 at low intensities) with high repetition rates
(for accumulation of good statistics with low-density targets). Such measurements became possible over the past
10 –15 years when high-resolution ATD spectra of H−, Br −
and F− were recorded [79–83]. Comparisons of the data
with calculations by Grybakin and Kuchiev made using the
Keldysh theory [16], demonstrated a very good quantitative
agreement for the low-energy part of the spectrum, while
for the high-energy part an equally good agreement was
achieved when rescattering was taken into account.
Another efﬁcient test of the Keldysh model justifying its
quantitative applicability for the description of negative
ions photodetachment employs comparisons with predictions of the method of quasistationary quasienergy states
(QQES), which provides a highly accurate solution of the
ATD problem (see the discussion in section 7.1).
Currently, the Keldysh theory is a cornerstone for the
theoretical apparatus of strong ﬁeld physics. Modern experiments performed with short intense laser pulses provide
conditions under which the theory is applicable and can serve
as a reliable zeroth-order approximation for the investigation
of high ﬁeld phenomena.
4. The Keldysh theory and closely related
approaches
In this section we discuss how the Keldysh ionization ansatz
relates to another standard theoretical tool of strong ﬁeld
physics, the SFA. In order to formulate the applicability
conditions of the theory, we start from a derivation of the
Keldysh ionization amplitude.
4.1. Derivation of the ionization amplitude
In [1] the probability distribution (given by equations (8)–(15)
therein) is postulated—the reason why the theory is often
called ‘the Keldysh ionization ansatz’. It was shown later [13]
how this result can be rigorously derived. Below we sketch
this derivation.
Consider an electron bound by a potential U(r) and
subject to a time-dependent electric ﬁeld E(t ).4 The timedependent Schrödinger equation (TDSE) then reads
i
⎤
⎡
1
∂
ψ (r , t ) = ⎢ − Δ + U (r ) + E (t ) r ⎥ ψ (r , t )
⎦
⎣
2
∂t
(36)
where the length gauge representation (2) for the interaction
operator is used. It has to be solved with the initial condition
ψ (r , t → − ∞) = ψ0 (r)e iI p t ,
(37)
where ψ0 (r) is the initial atomic bound state. After the laser
ﬁeld turns off, the wave function can be presented in the form
ψ (r , t → + ∞) = ψb (r , t ) + ψout (r , t ),
(38)
where the ﬁrst term in the r.h.s describes the bound part of the
4
As we consider the problem nonrelativistically, the dipole approximation
applies, so the magnetic ﬁeld of the wave can be neglected and the electric
ﬁeld can be considered space-independent.
7
J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204001
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wave function and the second the detached electron which
goes away from the atom. At r → ∞, the bound part ψb
vanishes while ψout takes the form of an outgoing spherical
wave. It can be expanded in plane waves; the respective
coefﬁcients determine the amplitudes of ionization.
Equation (36) with initial condition (37) can be presented in
the integral form :
+∞
ψ (r , t ) =
∫−∞
dt1
S 0 (p , t ) =
η ( t − t1)
−
(2π )
3
1
2
t
∫t
M (p) =
1
(40)
=
{
+
1
2
(41)
t
p
t
1
⎤⎫
v 2p t ′ dt ′⎥ ⎬×
⎦⎭
()
×U ( r1) ψ0 ( r1)e iI p t1 .
(42)
At large distances and for t → ∞, when the laser ﬁeld is off it
takes the form
ψ (r , t ) =
1
(2π )3 2
∫ d3peipr−ip t 2M (p).
2
(43)
Comparing (42) and (43), one obtains for the amplitude
M (p) =
−i
(2π ) 3 2
×
+∞
∫−∞
dt1e−iS0( p, t1)
∫ d3r1e−iv ( t ) r U ( r1) ψ0 ( r1),
p 1
1
⎫
⎤
p 1
1
0
1
∞
∫−∞ dt1 ∫ d3r1e−iv ( t ) r −iS ( p,t ) E ( t1) r1ψ0 ( r1) ,
p 1
1
0
1
1. Relatively weak laser ﬁelds, F ≪ 1. This limitation
means that the application of the Keldysh theory to the
description of ionization in the barrier suppression or
superintense, F ⩾ 1, regimes is not justiﬁed. It is shown
by comparisons with results of numerical calculations
and asymptotic expansions for the widths of atomic
levels in a static electric ﬁeld, that the rate of ionization
calculated from the Keldysh model (as well as from the
KFR and SFA), even with Coulomb corrections
included, already deviates considerably from exact
results for F ≃ 0.1, and this deviation grows with
increasing reduced ﬁeld F. In this domain the SFA
overestimates the rate and predicts the effect of
stabilization which is not conﬁrmed by other methods
based on summation of series of the perturbation theory
and a 1 N expansion (see the discussion in [84] including
ﬁgure 1 therein). On the other hand, the restriction F ≪ 1
is actually not severe. Ionization usually saturates during
a few optical periods when the reduced ﬁeld approaches
F ≃ 0.1 or even earlier (this can easily be estimated using
the expressions for the rate of ionization given in
section 5.1 and 6.2.3). Therefore, it is rather difﬁcult to
achieve the regime F ≃ 1 in experiments, as most of the
atoms will be ionized at lower intensities at the front edge
of the laser pulse.
2. SR potential. This condition is of principal importance,
because otherwise the spatial integral in (39) forms at
large distances from the atom where the wave function
differs substantially from the unperturbed bound state
function.
∫ d3pe iv (t) r ∫−∞ dt1 ∫ d3r1 exp {− i ⎡⎣ vp ( t1) r1
∫t
⎧⎡
∫ d3r1 ⎨⎩ ⎢⎣ 12 Δ1 − i ∂∂t1 ⎥⎦ e−iv ( t ) r −iS ( p,t ) ⎬⎭ ψ0 ( r1)
The derivation above allows the formulation the applicability
conditions of the Keldysh ionization model. These are:
t
−i
(2π ) 3
−i
(2π )3 2
∞
∫−∞ dt1
4.2. Applicability conditions
∫ d3p exp i ⎡⎣ vp (t ) r − vp ( t1) r1
()
(45)
p
i.e. the Keldysh amplitude (1) in the length gauge.
∫ d3pΨp* ( r1, t1) Ψp (r, t )=
with η (t ) = ∫−∞ δ (x )dx being the Heaviside step function.
Assume now that (i) the electric ﬁeld amplitude is small
compared to the atomic ﬁeld, i.e. the reduced ﬁeld (30) is
small, F ≪ 1, and (ii) U(r) is an SR potential of the radius rc,
so that U = 0 for r > rc [13]. The second assumption guarantees that only a small vicinity of the atom contributes to the
spatial integral in (39). Then, under the ﬁrst assumption, the
atomic wave function remains only slightly disturbed by the
laser ﬁeld at r ⩽ rc , and we can replace the exact wave
function ψ (r, t ) by its unperturbed limit (37). Then
ψ (r , t ) =
−i
(2π ) 3 2
×
⎤⎫
v2p t ′ dt ′⎥⎬
⎦⎭
⎤
2
( ) + I ⎥⎦ dt′.
Applying the self-conjugated operators Δ and i∂ ∂t1 onto the
left part of the integrand in (44) one obtains
with the condition GV (r, t; r1, t1 ) = 0 at t < t1. It expresses
via the Volkov functions (2)
= −i
⎡1
⎢ vp t ′
⎣2
⎛1
∂ ⎞
U ( r1) ψ0 ( r1)e iI p t1 = ⎜ Δ1 + i ⎟ ψ0 ( r1)e iI p t1 .
∂t1 ⎠
⎝2
where GV is the retarded Greenʼs function in a homogeneous
electric ﬁeld, satisfying the equation
GV ( r , t ; r1, t1) = −iη ( t − t1)
+∞
Amplitude (44) coincides with (1). Indeed, the Schrödinger
equation for the unperturbed bound state reads
∫ d3r1GV ( r, t; r1, t1) U ( r1) ψ ( r1, t1),(39)
⎤
⎡ ∂
1
⎢⎣ i + Δ − E (t ) r⎥⎦ GV ( r , t ; r1, t1)
2
∂t
= δ ( t − t1) δ ( r − r1),
∫t
(44)
There are no special limitations on the laser frequency, in
contrast to a widely accepted viewpoint that the Keldysh
with the phase
8
J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204001
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theory only works in the multiquantum, K0 ≫ 1, or even only
in the tunneling, γ ≪ 1, regimes. For example, for an SR
potential supporting unique s-state amplitudes, (1) and (44)
reproduce the ﬁrst-order perturbation theory limit exactly. In a
weak ﬁeld the exponential factor can be replaced by the ﬁeldfree one exp { − iS0 } → exp {i(I p + p 2 2) t}. Then integrating in (44) twice by parts with ψ0 from (26) and (27) we
obtain for the amplitude
2 κ
M (p) =
2
∞
∫−∞ pE (t )ei(I +p
2 2
p
2
2) t
dt ,
in [2])
U (r − a (t )) → U (r ) .
This simpliﬁcation is valid either for an SR potential or when
the quiver amplitude is small compared to the characteristic size
of the atomic initial bound state. The latter is, however, satisﬁed
only for sufﬁciently weak ﬁelds: even at intensities of about
1013W cm−2 and wavelengths of 1μm the quiver amplitude
a = E0 ω2 ≃ 5 a.u. Thus a more general expression (47)
reduces in practice to (21).
In the paper by Reiss [3], the probability amplitude was
deﬁned as
(46)
π (κ + p )
which coincides with the result of the ﬁrst-order perturbation
theory employing exact continuum states in the ZRP
1
ψp (r) =
(2π )3 2
⎡
1 eipr ⎤
⎥,
⎢ eipr −
r κ + ip ⎦
⎣
κ=
+∞
(S − 1)if = −i
(48)
where Vint is the interaction operator in the velocity gauge (4)
and Ψ f( − ) is the exact ﬁnal state wave function which contains both the effect of the laser ﬁeld and of the binding
potential and has an asymptotic behavior ∼ exp {ipr} at
r → ∞. The central approximation of Reissʼs paper is that
this exact function can be replaced by the Volkov function
(5). It reduces the amplitude (48) to (1) with accuracy to a
prefactor, again because of the different gauge used. This is
also clearly stated in [3].
Thus, the differences between the results of Keldysh [1],
Faisal [2] and Reiss [3] are: (i) that the Keldysh version uses
the length gauge form of the interaction operator, while Faisal
and Reiss employ the velocity gauge form and (ii) that Faisalʼs version contains a principal possibility to go beyond the
Volkov wave approximation for the ﬁnal state (equations (14)
and (17) in [2]), although this possibility was not explored in
the actual calculations. Using the velocity gauge allows a
direct relativistic generalization of the theory, while the length
gauge approach in the form (2) is usually restricted by the
dipole approximation5. In the nonrelativistic domain the two
gauges give formally different results, but this difference is
not of physical signiﬁcance: for the ZRP both gauges lead to
the same expression, while for the Coulomb potential the preexponential factor is again incorrect in both cases (see
section 8.1). One of the arguments in favour of the SFA form
of the ionization amplitude is that its derivation does not
require the two assumptions made to derive the Keldysh
ionization amplitude (see the previous subsection). It can,
however, be seen that the same approximations are indirectly
required. Indeed, replacement of the exact wavefunction by
the Volkov solution is unambiguously justiﬁed only for SR
binding potentials. A common argument, that with increasing
laser ﬁeld amplitude the role of an atomic potential will
sooner or later become negligible, does not apply to the
Coulomb potential because of its long-range nature: even in a
very strong oscillating ﬁeld with a zero time-averaged value,
the Coulomb force generates a net accumulating effect. This
mechanism can be understood well on the level of classical
mechanics. As is shown in section 6 using the language of
trajectories, the Coulomb correction to the complex phase of
4.3. Keldysh theory, KFR and SFA
In the literature, the analytic theory of strong ﬁeld ionization
is associated with several closely related approaches. Along
with the term ‘Keldysh theory’, KFR theory and the SFA are
frequently used. Here we discuss brieﬂy the relationship
between these three methods. To this end we compare ionization amplitude (1) with the respective amplitudes derived
by Faisal [2] and Reiss [3]. In the work by Faisal the
expression (equation (10) therein) for the amplitude of ionization differs by the gauge choice (velocity instead of length).
In addition, the amplitude of [2] is determined for an arbitrary
ﬁnal state, which is not necessarily a plane wave. So, instead
of (21), the n-photon amplitude of ionization in a linearly
polarized ﬁeld has the form [2]
nω − UP
ω
× ψi (s),
Ψ (f − ) Vint (t ) Ψi dt
2I p .
Thus the original Keldysh approximation is restricted by
SR potentials and applicable in a wide range of frequencies
under the condition F ≪ 1. Extensions of the theory making
it applicable to the quantitative description of the ionization
dynamics of atoms introduce additional restrictions on the
ﬁeld parameters which will be discussed in sections 6 and 9.
 if , n =
∫−∞
⎛E s U ⎞
∫ d3 sψ *f (s) Jn ⎜⎝ ω02 , 2ωP ⎟⎠
(47)
where ψi, f (s) are the Fourier transforms (24) of the initial and
ﬁnal atomic wave functions. Taking a plane wave for the ﬁnal
state ψf, so that
ψ f (s) = δ (p − s)
we obtain (21) (with accuracy to the factor 2π i ω which is
due to the different deﬁnitions of the transition amplitude here
and in [2]) or, equivalently, equation (21) of [2].
The amplitude (47) is obtained under the assumption the
quiver amplitude of the electron in the laser ﬁeld,
a (t ) = ∫ dt′A (t′), can be omitted in the Kramers–Henneberger
transformation (see [85, 86] and references therein for a review
of the Kramers–Henneberger method in application to the
ionization problem) of the binding potential (equations (5)–(8)
5
It is also possible to adopt the length gauge for relativistic calculations [87].
9
J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204001
Review Article
the ionization amplitude, although it becomes smaller with
increasing laser ﬁeld amplitude, remains numerically large at
any reasonable intensity.
Apart from the KFR and SFA, the Perelomov–Popov–Terenʼev (PPT) and Ammosov–Delone–
Karinov (ADK) models are also often mentioned in the literature. In [13, 14] expressions for the total ionization rate
and momentum distributions of photoelectrons were derived
from the Keldysh model for the case of an initial state in an
SR potential with the arbitrary azimuthal and magnetic
quantum numbers l and m. In the paper by Perelomov and
Popov [15] the expression for the total ionization rate was
generalized to the case of atoms, i.e. with the Coulomb
interaction accounted for, but only in the tunneling limit,
γ ⩽ 1. This result extends to low-frequency ﬁelds the staticﬁeld rate derived by Smirnov and Chibisov [42]. These two
formulas of PPT (one for negative ions, another for atoms) are
now frequently used for the calculation of ionization rates in
the ﬁeld of intense laser radiation and are often mentioned as
PPT rates. An identical expression for the total rate of ionization of atomic levels as derived by Perelomov and Popov
was later obtained in [88] and is known as the ADK rate. For
a more detailed description of the relation between the PPT
and the ADK rates, we direct the reader to the review [4].
5.1. Saddle-point approximation
If the phase S0 in (45) is numerically large, the integrand in
(1), (44) oscillates fast in time and the SPM [91] can be
applied. After spatial integration, the amplitude (1) takes the
form
+∞
MK (p) =
∫−∞
dtP (p, t )e−iS0(p, t ) ,
(49)
where the phase S0 (p, t ) is given by (45) and P is a gaugedependent prefactor
P (p , t ) = −
i
(2π )3 2
∫ d3r e−isrVint (t ) ψ0 (r)
(50)
with s = p or s = vp (t ) in the velocity and the length gauge,
respectively. The phase in (49) can be scaled using the
dimensionless parameters (28) and (32)
zF
2
S 0 ( p , φ) = − K 0 φ +
∫φ
2
+∞
(q + a(φ′)) dφ′,
(51)
where we introduce dimensionless time φ = ωt , momentum
q = p pF and vector potential a = A pF with pF = E0 ω .
Saddle points tsα satisfy the equation
S0′ ≡
5. Representation in terms of trajectories
∂S 0
2
= 0, → γ 2 + ( q + a( ts)) = 0.
∂t
(52)
Clearly, the solutions of (52) are always complex,
ts = t0 + iτ0 . If the prefactor (50) is not singular at t = ts ,
application of the SPM is straightforward and gives for the
amplitude
The explicit analytic expressions given in section 2.1 are
useful for the numerical calculation of photoionization spectra
in quasimonochromatic laser pulses. They have been extensively applied for the theoretical investigation of different
ionization regimes as well as for the analysis of experimental
data (see, e.g., [3, 89, 90] and references therein). It is,
however, more common to use approximations for the
amplitudes (1) and (20) and the respective probabilities
obtained by the SPM. There are three good reasons for this.
First, application of the SPM allows one to obtain much
simpler analytic expressions than given by (21) and (23).
With compact saddle-point formulas, the qualitative investigation of the main features of photoelectron spectra is
straightforward. Second, the Bessel expansions assume a
quasimonochromatic ﬁeld and cannot be applied in the case
of the now routinely used few-cycle laser pulses where pulse
shape effects are signiﬁcant. Third, the saddle-point result
allows an interpretation in terms of complex classical trajectories satisfying Newtonʼs equation in complex time and
space. This interpretation has appeared to be exceptionally
fruitful for generalization of the theory and for heuristic
prediction of new phenomena. In addition, the domains of the
parameters where the Keldysh theory itself and its saddlepoint version are applicable essentially overlap, so the saddlepoint analysis almost does not narrow the applicability of the
theory, and the respective SPM results are usually quantitatively accurate, if the Keldysh approximation applies in
principle.
MK (p) ≈
∑
α
2π
iS0′′ ( tsα )
P ( p, tsα )e−iS0( p, tsα ) ,
(53)
where the sum is taken over all the solutions of (52) in the
upper complex half-plane. The reason for choosing an integration contour which always goes through the upper halfplane is the sign of the ionization potential, I p > 0 . For the
symmetric solutions of (52) with τ0 < 0 , the imaginary part of
the phase at a saddle point is positive, Im S0 (p, ts ) > 0 , so
that the corresponding contributions are exponentially large.
If the prefactor is singular at t = ts , a modiﬁed version of the
SPM should be applied [6, 16]. The respective expressions
are given in appendix A. As we show in section 8.1, for
practical calculations only one special case is of interest,
namely, when the initial state wave function corresponds to
an (l,m)-state in an SR potential and the length gauge is used.
Then the amplitude takes the form [16] (see appendix A)
MK (p) ≈
∑ ( p, tsα )e−iS ( p,t
0
sα )
,
α
 = 2Cκl
iκ
S0′′ ( tsα )
Ylm ( n v),
(54)
where Ylm is a spherical function and its complex argument
n v = vp (ts ) κ is a ‘unit’ vector along the complex velocity
taken at t = ts .
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J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204001
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Below we consider the simplest and simultaneously frequently used case of a linearly polarized monochromatic ﬁeld
E (t ) = E 0 cos ωt
(
)
ωts = arcsin qx ± i γ 2 + q⊥2 ,
{
{ −2K ⎡⎣ b (γ ) q
0
(
1
b 2 (γ ) =
1
arcsinhγ
γ2
2
x
× 1 + ( − 1)n + l + m cos 2qx z F 1 + γ 2
)}.
(58)
1
3 3
γ−
γ + ...,
3
10
1
1
3 3
b2 ≈ − γ +
γ + ...,
γ
6
40
(56)
+ b2 (γ ) q⊥2 ⎤⎦
⎤
⎥,
2 ⎥
1+γ ⎦
γ
with asymptotics
b1 ≈
where qx and q⊥ are the dimensionless momentum components parallel and perpendicular to the linear polarization
direction. For elliptically polarized ﬁelds (except the special
cases of emission along and perpendicular to the major
polarization axis considered in [92]), bichromatic ﬁelds and
few-cycle pulses, let alone more complex cases, there are, in
general, no closed analytic expressions for saddle points,
although the numeric solution of the saddle-point
equation (52) presents no difﬁculty. Analysis of saddle-point
trajectories in the complex time plane (see examples in
[6, 92]) shows that they never merge, so that the SPM is
always applicable in its simplest form when the expansion of
the phase is truncated after the quadratic term proportional to
the second derivative of the action (45). This is in contrast
with the case of rescattering and high harmonic generation
(HHG), when a pair of saddle points merge at the classical
boundary (e.g. at ε ≈ 10UP for backscattering in a linearly
polarized ﬁeld and at Ω ≈ 3.17UP + I p for HHG with Ω
being the harmonic photon frequency) and the SPM requires a
modiﬁcation [93, 94]. The fact that stationary points do not
merge means, in particular, that there is no cut-off in the
direct ionization spectra, although the ‘classical direct ionization cut-off’ at ε ≈ 2UP is often mentioned in the literature.
Inspecting a typical direct photoelectron spectrum (see e.g.
the spectra shown in the ﬁgures of sections 7.1 and 8.1)
calculated from (53), one notices that nothing special happens
at this energy. The slope of the spectrum changes abruptly
only when the probability of rescattering exceeds that of
direct ionization. The position of this point depends on the
relative magnitude of rescattering and is not universal. For
linear polarization it is usually between 2 and 5 UP.
In the ﬁeld (55) the imaginary part of (56) is minimal for
p = 0. This corresponds to the maximum of probability. The
spectrum drops quickly with increasing the momentum.
Expanding (56) in a series with respect to qx , q⊥, one obtains a
simple analytic expression for the momentum distribution (18)
[4, 13, 16]
dwn
= w0 qq⊥2 m exp
dOp
1
γ2
(55)
when there are two solutions for (52) per laser period in the
upper complex half-plane
⎡
⎢ arcsinhγ −
⎢
⎣
b1 (γ ) =
b1 ≈
γ ≪ 1,
(59)
1
ln (2γ )
,γ≫1
[ln (2γ ) − 1] , b2 ≈
γ2
γ2
(60)
and
w0 =
Cκ2l ωγ −2 m
π
2
2
1+γ (
2 l + 1 (l + m ) !
exp { −2K0 fK (γ ) },
2
2 m m ! ) (l − m ) !
(61)
Cκl is an asymptotic coefﬁcient of the bound state wave function
(A.2) and the Keldysh function fK (γ ) is given by (34). The second term in the parentheses of (57) describes interference between
contributions of the two stationary points. The presence of interference effects in spectra of strong ﬁeld ionization was noted for
the ﬁrst time in [13]. Integrating the distribution (57) over the angle
and omitting the interference term, one obtains the rate of ionization with absorption of n laser photons and the total rate of
ionization for the (l,m)-state in an SR potential
4Cκ2l β1 2 ⎛ 2 ⎞
⎜ ⎟
πK03 2 ⎝ zF ⎠
|m|
wSRn = I p
2 l + 1 (l + m ) !
(n − n th)|m|
2 (l − m ) !
m
2 m !)
(
× e−2K0 fK (γ ) − 2c1(γ )( n − n th ) 
(
)
β [ n − n th ] ,
∞
wSR =
∑
wSRn,
(62)
n = Nmin
β = 2γ 1 + γ 2 ,
c1 (γ ) = γ 2b1 (γ ),
2
2
x
the function  (x ) = e−x ∫ (1 − t 2 /x 2 )|m|et dt turns to the
0
Dawson integral [95] for m = 0, n th = K0 (1 + γ −2 2) is the
ionization threshold and Nmin = [n th ] + 1 is the minimal number
of photons required for ionization (17). In the tunneling limit, this
sum can be replaced by the integral which gives a simple tunneling
formula for the rate of ionization from an SR potential [4, 13]
where
3F (2 l + 1)(l + m ) ! m + 1
F
π 22 m m ! (l − m ) !
⎧
γ2 ⎞ ⎫
2 ⎛
× exp ⎨ −
⎜1 −
⎟ ⎬.
10 ⎠ ⎭
⎩ 3F ⎝
wSR (γ → 0) = 2I p Cκ2l
}
(57)
⎪
⎪
⎪
⎪
(63)
Expressions for momentum distributions and rates in circularly and elliptically polarized ﬁelds and for arbitrary values
of photoelectron momenta can be found in [4, 84, 96–98].
Here
11
J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204001
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motion, in contrast to the trajectories used in Feynman representation of quantum mechanics [103]. For ionization from an
SR potential ITM gives no new result, but provides an
appealing ‘almost’ classical picture. The trajectory representation is, however, very efﬁcient in accounting for semiclassical
effects beyond the Keldysh approximation. This includes, in
particular, the Coulomb interaction and spin effects.
In order to apply the ITM for the calculation of the
Coulomb-corrected ionization amplitude, it is useful to
rewrite the action via the Lagrangian function
5.2. Representation via classical trajectories and the ITM
The saddle-point result of the previous section can be presented in terms of classical trajectories propagating in complex time and space. To this end we note that the value
−S0 (p) in (53), (54) is a classical action expressed as a
function of the detection time td and ﬁnal momentum p (in
order to introduce the time instant td when the electron arrives
at a detector we replace in the upper integration limit in (45),
td → +∞). The corresponding electron trajectory is:
rp (t ) =
∂S 0
=
∂p
∫t
t
()
vp t ′ dt ′
s
= p ( t − ts ) +
∫t
t
()
A t ′ dt ′.
s
(64)
r¨p (t ) = −E (t ),
∫t
− ipτ0 + i
E0
ω2
2
ω
= Im rpm (t ) = 0.
∫t
td
s
dt ′
1 2
v p + v˙p rp − I p
2
}
≡F0 ⎡⎣ rp (t ), td ⎤⎦ + W0 ⎡⎣ rp (t ), td ⎤⎦ ,
(65)
dt ′=
(70)
where W0 is the reduced action expressed via the Lagrange
function
(66)
W0 ⎡⎣ rp (t ), td ⎤⎦ =
∫t
=
∫t
td
s
{
( ) ( ) } dt ′
1 2
v p − E t ′ rp t ′ − I p
2
td
L 0dt + ε0 ( td − ts ).
(71)
s
with ε0 = −I p being the initial electron energy and
td
F0 ⎡⎣ rp (t ), td ⎤⎦ = −rpvp = −rp ( td ) p.
(72)
ts
(67)
Interpretation of the function (71) as a reduced action is
connected to the fact that the latter is deﬁned in classical
mechanics as W = S + εt [99, 100]6. The function (72) is
important because the trajectory rp (t ) has, in general, an
imaginary part. The ITM representation of the Keldysh
ionization amplitude reads then as
where the last two terms in the r.h.s. give a constant imaginary part. There is no physical controversy here, as a trajectory is not an observable in the considered problem. The
most probable ﬁnal momentum pm corresponds to the minimum of the imaginary part of the action
d
( Im S0 )
dp
+
ts
}
{
1 2
vp + Ip
2
td
( cos ωt − cos ωt0 cosh ωτ0 )
sin ωt0 sinh ωτ0,
s
= − rpvp
The detection time td is real. Then the electron velocity
vp (td ) = p + A (td ) at the detector is real too. However, the
trajectory (64) can have an imaginary part even in real time.
For example, in the linear ﬁeld (55) the trajectory (64) is
(assuming time t is real)
E0
{
td
− S0 ( p , t d ) = −
with initial and ﬁnal conditions
rp ( ts ) = 0, v2p ( ts ) = −2I p ≡ − κ 2, vp ( td ) = p.
(69)
Then
It starts from the origin at t = ts with an initial velocity
vp (ts ) = p + A (ts ) which is a purely imaginary value,
according to the saddle-point equation (52). Thus the exponential part of the transition amplitude is deﬁned by a classical action calculated along a trajectory which satisﬁes
Newtonʼs equation in the laser ﬁeld
rp (t ) = p ( t − t0 ) +
1 2
v p − E (t ) rp.
2
L0 =
MK ( p, td ) ≈
(68)
∑ ( p, tsα )e iW ⎡⎣ r
0
⎤
⎡
⎤
pα, td ⎦ + iF0⎣ rpα, td ⎦
(73)
α
p = pm
with the prefactor deﬁned in equation (54). Under proper
calculations, the dependence of the amplitude (73) on the
detection time td reduces to an inessential phase factor.
Concluding this section, note that the ITM is conceptually
close to the Landau–Dykhne method [40, 104] developed for
the calculation of semiclassical transition amplitudes and
matrix elements.
This means that the most probable trajectory is real in
real time.
Equations (64)–(66) introduce the ITM in the theory of
strong ﬁeld ionization [15, 43, 44]. In contrast to standard
classical mechanics where Newtonʼs equation is supplied by 2N
initial conditions (N is the dimensionality of the problem, N = 3
in the considered case), the ITM equations require 2N + 1
conditions (66). The reason is that in the ITM the initial time
instant ts can not be arbitrary chosen, but is found from the
saddle-point equation. These trajectories are often referred to as
‘quantum trajectories’ or ‘quantum orbits’ [101, 102]. We ﬁnd
the term ‘classical complex trajectories’ to be more appropriate,
for it underlines that they do satisfy the classical equations of
6
This interpretation is, however, partially misleading. Reduced action
usually appears in descriptions of conservative systems where it makes sense
to extract the term εt containing the integral of motion ε. In our case energy is
not conserved. An alternative interpretation treats W0 as a sum of two actions.
One is the atomic action in the absence of the laser ﬁeld the electron has
before it is detached at t = ts . Another is the Volkov action. The author is
grateful to B M Karnakov for his explanation of this vague point.
12
J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204001
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6. Modiﬁcations
calculated or postulated. Attempts to develop a regular procedure for the calculation of such a solution face tremendous
difﬁculties. Therefore a heuristic approach looks admissible.
Let us assume that the laser ﬁeld distorts the Coulomb continuum adiabatically, so that the quantum state ‘follows’ the
instant value of the electron velocity. Then a continuum
Coulomb state Ψk( − ) [40]
In its original formulation, the Keldysh theory does not
account for the presence of excited bound atomic states and
the electron–ion interaction in the continuum. Both aspects
are particularly important for atoms and molecules (in contrast to negative ions, where the ﬁnal state interaction is less
signiﬁcant although also causes observable effects). Therefore, modiﬁcations of the Keldysh theory mostly attempt to
account for these two features inherent in atomic systems. A
rigorous theory should incorporate both factors on the same
footing, for they have a common origin. In practice, due to the
approximate and nonperturbative nature of theoretical
approaches, bound states and the Coulomb interaction in the
continuum are usually treated using different methods. Currently, much greater progress has been achieved in describing
the ﬁnal state interaction than in the incorporation of bound
states into the theory.
In the simplest way, the ﬁnal state interaction can be
considered perturbatively in the ﬁrst Born approximation. The
wave function (42) describes photoelectrons emitted by the
atom without a further interaction with the residual ion. This
mechanism of ionization described by the Keldysh model is
known as direct ionization. Considering the electron–ion
interaction as a small perturbation, one can calculate a correction to the wave function (42) induced by scattering of
direct photoelectrons driven by the laser ﬁeld on the parent
ion. This is the basic idea of the rescattering or recollsion
scenario of strong ﬁeld ionization [61, 62]. The respective
rescattering correction to the amplitude (1) reeds
∫
M1 (p) = − d3q
Ψ k( − ) (r) =
+∞
MCV (p) = − i
∫−∞ dt1
Ψ p(CV ) Vint (t ) Ψ0 dt .
⎛
i
Ψ p(CV ) (r, t ) = Ψ p( − ) (r) exp ⎜ iA(t )r −
⎝
2
(76)
t
⎞
∫−∞ v 2p (t′)dt′⎟⎠.
(77)
It was proposed for description of strong ﬁeld processes in
[106, 107] and is known in the literature as the Coulomb–Volkov (CV) wave function. The corresponding
expression (76) for the ionization amplitude was adopted by
Dorr, Shakeshaft and Potvliege [108, 109] and by Basile with
coauthors [110]. Since then it has been widely used in the
theory of strong ﬁeld ionization and laser-assisted bremsstrachlung. We will refer this approach as the CVA.
The ﬁrst advance of the CVA was made through its
application in the description of the Coulomb asymmetry of
photoelectron distributions in elliptically polarized ﬁelds
[110, 111]. A simple examination of the symmetry properties
of the Keldysh ionization amplitude (1) shows that in
monochromatic ﬁelds, angular distributions possess four-fold
symmetry, independent of the ﬁeld polarization, while
experiments show that in elliptically polarized ﬁelds angular
distributions only obey inverse symmetry [59, 112]. This
symmetry violation was, from the beginning, attributed to the
inﬂuence of the Coulomb ﬁeld on the outgoing electron and is
now known as Coulomb asymmetry [67] (see also the discussion in sections 6.2 and 6.3). Later on the CVA technique
was applied in the calculation of photoelectron momentum
distributions in linearly polarized ﬁelds [113]. It was shown
there that the CVA provides a signiﬁcant improvement in the
description of photoelectron momentum distributions compared to the SFA, particularly for low energies and relatively
low intensities when γ ⩾ 1.
Typical results of CVA calculations are shown in ﬁgure 1
where two-dimensional (in the (k z, k ρ ) plane where kz is the
parallel and k ρ is the transversal with respect to the ﬁeld
polarization component of the photoelectron momentum) and
transversal momentum distributions, calculated using the
standard Keldysh approach, the CVA and by solving the
TDSE numerically, are presented. It is clearly seen from the
dt Ψp U Ψq (t )
Ψq Vint Ψ0 ( t1).
∫−∞
Here the notation Ψp(CV ) is used for the function
t
×
(75)
where 1F1 is the conﬂuent hypergeometric function, can be
adopted for approximate description of the laser-distorted
continuum by replacing k → vp (t ). Instead of (1) one then
obtains
+∞
∫−∞
⎛
1
i⎞
eπ 2k Γ ⎜ 1 + ⎟
32
⎝
k⎠
(2π )
⎛
⎞
i
× eikr 1F1 ⎜ − , 1, − i(kr + kr) ⎟ ,
⎝
⎠
k
(74)
It describes the high-energy part of photoionization spectra
(see [7, 9] for reviews of the SFA with rescattering). For
negative ions the amplitude (74) does indeed give a small
correction to the total probability of ionization (although in
spectra it dominates at high energies), so that the perturbative
account is justiﬁed and yields qualitatively correct results (see
also sections 7.1 and 8.1). In the presence of the Coulomb
interaction, the perturbative correction to the probability
appears in the direct part of the spectrum comparable to or
even dominating the Keldysh probability. Even then the SFA
with rescattering is able to reproduce qualitatively some
effects in direct ionization spectra (see, e.g. [105] where the
low-energy structure is described by this method). However,
for the development of a consistent theory of ionization in the
presence of the Coulomb ﬁeld nonperturbative methods are
obviously required. Here we describe several signiﬁcant
modiﬁcations of the theory made in the nonperturbative spirit.
6.1. Coulomb–Volkov approximation (CVA)
In order to improve the Keldysh amplitude (1) one could try
to ﬁnd a continuum wave function approximately describing
both the laser and the Coulomb ﬁeld. Such a function can be
13
J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204001
Review Article
Figure 1. Left panel: photoelectron momentum distributions in cylindrical coordinates (k z, k ρ ) calculated for the case of ionization of
hydrogen by a six-cycle laser pulse (of duration τ = 151 a.u.) with the carrier frequency ω = 0.25 a.u. and the ﬁeld amplitude E0 = 0.05 a.u.
obtained from the SFA, the CVA and from exact numerical solution of the TDSE. Right panel: normalized transversal momentum
distributions calculated from the SFA (dashed line), the CVA (red solid line) and the TDSE (black solid line) for the following parameters:
E0 = 0.05, ω = 0.25 a.u. (a), E0 = 0.0377, ω = 0.05 a.u. (b) and E0 = 0.053, ω = 0.05 a.u. (c). Corresponding values of the Keldysh
parameter are shown in the ﬁgures. Reprinted with permission from [113]. Copyright (2008) by the American Physical Society.
plots that for low values of photoelectron momenta k ⩽ 0.3,
the application of the CVA yields a very good quantitative
agreement with the TDSE, while the SFA result even disagrees qualitatively. The transversal distributions in the right
panel clearly show the cusp observed in experiments and not
reproduced by the SFA [70, 71].
Agreement between the CVA and the TDSE becomes
less accurate for lower frequencies and higher intensities, i.e.
with decreasing γ (see, e.g. ﬁgures 4 and 5 in [113]). Also, the
CVA does not appear to give any visible improvement for
higher photoelectron energies (in the direct part of the ionization spectrum) and does not reproduce the rescattering
plateau.
The main problem of the CVA approach is that the
function (77) is not a solution of any form of the Schrödinger
equation, so that there is no regular way to estimate the
accuracy of the approximation and formulate conditions
restricting its applicability. As a consequence, there is no clear
answer to the question why some Coulomb effects are
quantitatively reproduced while other do not even appear on a
qualitative level. Some insight into the applicability conditions of the CVA was achieved by Kornev and Zon
[114, 115] who examined the accuracy of the CV functions
using a nonstationary generalization of the Siegert theorem
[116]. The latter simply employs the fact that for exact
solutions of the Schrödinger equation the matrix elements of
momentum and coordinate are connected as
f p i = ∂t f r i
fulﬁlled with reasonable accuracy. This observation agrees
with the above-discussed results of [113] where the accuracy
of the result is higher for higher frequencies.
However, to the best of our knowledge, there is no
qualitative understanding of these features of the CV functions. The question of the applicability of these functions in
the description of strong ﬁeld processes remains open. It
could probably be solved by applying the saddle-point analysis to the amplitude (76).
6.2. CCSFA
Another simple and physically transparent approach which
allows the approximate inclusion of the Coulomb ﬁeld into
the theory of nonlinear ionization is based on the ITM.
Instead of correcting the ﬁnal state wave function one can
modify the transition amplitude, which is technically much
easier. Indeed, if the amplitude of ionization can be expressed
via classical complex trajectories (as shown in section 5.2), no
knowledge of the full wave function is needed; it is sufﬁcient
to ﬁnd all the trajectories corresponding to desirable initial
and ﬁnal conditions and satisfying Newtonʼs equation in the
presence of the laser and the Coulomb ﬁelds. Simplicity of the
ITM has allowed to extend it on arbitrary values of γ and to
apply for description of several experimentally observed
features of strong ﬁeld ionization.
6.2.1. The CCSFA algorithm. As described in section 5.2, the
(78)
ITM allows one to represent the ionization amplitude in the
form (73) where the action in the exponent is calculated along
classical trajectories satisfying Newtonʼs equation (65) for an
electron in the laser ﬁeld with the initial conditions (66). If
For two CV functions this relation is in general not satisﬁed.
It was shown in [114, 115], however, that in relatively weak
and high-frequency ﬁelds, E0 < ω , the condition (78) is
14
J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204001
Review Article
there is an extra force acting on the electron, one can account
for it by correcting the functions W0 and F0. When this force
is relatively small, the respective trajectories and action can be
found perturbatively
r (p, t ) = r0 (p, t ) + r1(p, t ) + ...,
W (p) = W0 (p) + W1 (p) + ....
Then the Coulomb-corrected ionization amplitude is
α
(79)
td
W=
( ) ∫
F ( p, t ) ≈ F ( p′, t ) ± iν
(80)
d
.
}
td
dt , F = −rv
. (85)
ts′
0
td
ts
Z
dt ,
r0 (p, t )
(86)
d
Here
Z
Z
≡
κ
2I p
ν=
(87)
is the effective principle quantum number, ν = 0 in the case
of negative ions. The term W0 (p′, td ) is the Coulomb-free
action calculated along the new trajectory and the second term
is the Coulomb action calculated along the Coulomb-free
trajectory. The correction to ±iν is a constant close in
absolute value to unity.
The Coulomb integral
WC =
∫t
td
s
Z
dt
r0 (p, t )
(81)
which assumes that the electron is already far from the atom,
so that the Coulomb force is small with respect to the laser
force, but the time passed after ts is still a small fraction of the
laser period. Then the Coulomb-free trajectory (64) can be
expanded in the series with respect to t − ts :
(82)
(iii) Photoelectron momentum is no longer conserved, so that
its value p′ entering the saddle-point equation (52) is
different from the one measured at a detector. This initial
drift momentum is to be found from
r0 (t ) ≈ ( p + A ( ts ) ) ( t − ts ) −
1
E ( ts )( t − ts)2 ,
2
r 20 (t ) ≈ − κ 2( t − ts)2 − ( p + A ( ts ) ) E ( ts )( t − ts)3 .
v ( td ) = p′ + v1( td ) = p.
(88)
is logarithmically divergent at the lower limit when
r0 (p, t → ts ) → 0 and requires regularization. This is performed by replacing ts → t* (see ﬁgure 2, right panel) and
matching the result of integration to the asymptotic of the
atomic bound state wave function [15, 44, 118]. The
matching point t* satisﬁes the condition
νγ
≪ ω2 ts − t * 2 ≪ 1.
(89)
2
K0 1 + γ
with the correction r1 to be found from Newtonʼs
equation
r03
{
1 2
Z
v − E (t ) r + − I p
2
r
W ( p, td ) ≈ W0 p′, td +
vp (t ) = p + A (t ) → v (p, t ) = v0 (p, t ) + v1(p, t ) ,
r¨1 = −
s′α
Taking into account that the trajectories (81) are found
perturbatively, we should only keep contributions linear with
respect to the charge Z in (84). The ﬁrst-order expansion of
the action (85) yields (see appendix B)
(i) For ionization of a level with quantum numbers (l,m) and
ionization potential Ip we calculate the respective SR
Keldysh amplitude and present it in the form (73). For
each ﬁnal momentum p , there are several (typically two
from each laser period) Coulomb-free trajectories to be
found from (65) with initial conditions (66).
(ii) A Coulomb-free trajectory is replaced by a corrected one:
Z r0
∫t ′
s
so the effect of these corrections on the photoelectron
momentum distribution can be signiﬁcant. As we will see
below this is exactly the case. In order to develop the
perturbation theory we need to express the action as a
function of coordinate, because this coordinate will then be
corrected. This is the reason why action (71) but not (45)
should be used as a zero-order approximation. Appendix B
illustrates that starting from the form (45) one obtains a
meaningless result.
The procedure of calculating the Coulomb-corrected
ionization amplitude is the following [66, 117, 118]:
rp (t ) → r (p, t ) = r0 (p, t ) + r1(p, t )
(p′, t ) exp (iW [r (p, t) ] + iF [r (p, t) ])(84)
with
Here we introduce a new notation r0 (p, t ) ≡ rp (t ) for a
trajectory (64) in the laser ﬁeld. This approach is known as a
perturbation theory for the action. For applicability, it
requires that the corrections W1 and F1 are small in
absolute value compared to the leading terms. At the same
time, a correction to the action can be numerically large:
W0 ≫ W1 ≫ 1,
∑
M (p) ≈
(83)
(90)
(91)
Replacing in (71), (88) the lower integration limit by t* we
obtain the following asymptotic
(iv) A new saddle point ts′ = ts (p′) is calculated from the
same saddle-point equation (52) with p′ (found from
(83)) instead of p.
(v) The Coulomb potential energy UC = −Z r is added to
action (71).
W ( t * → ts ) ≈ κ 2t * + iν ln ω ( t * − ts ) + ...
(92)
It follows from (91) that the distance between the electron and
the atom at the matching time instant is
15
J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204001
Review Article
Figure 2. Left panel: complex time plane with the saddle point ts = t0 + iτ0 and different integration paths connecting ts and td shown by a
solid and two dashed lines. Right panel: an integration path (dashed line) corresponding to the matching at t*. The circle characterizes the
radius |ts − t * | of the matching region.
r * ≡ r0 ( p, t * ) = iκ ( t * − ts ).
(93)
g=
The phase of the atomic wave function in this part of space,
κ|r * | ≫ 1, is given by its semiclassical asymptotic
Wat ( r * ) =
∫
r*
p (r )dr ≈ iκr * − iν ln κr *
(94)
()
(i) At the upper integration limit td → ∞ the action (86)
grows as
⎧
⎫
Z
iν ⎪
⎨
⎬ dt. (95)
+
⎪
2
t − ts ⎪
⎩ r0 (p, t )
⎭
td ⎪
2
p′
Z
W∼
td +
ln td
2
p′
A similar procedure allows one to obtain a regularized
expression for the Coulomb correction v1, by matching a
solution of (81) with a trajectory in the Coulomb ﬁeld of the
atomic core calculated in the domain r ≃ r * [120]. The
regularized result has the form (for the linearly polarized ﬁeld
(55))
v1 =
⎧
⎫
Z r (t )
⎨ − 30
+ f ( t − t s ) ⎬ dt
s
r0 (t )
⎩
⎭
+ iZF g (p, γ ) ln ( td ),
∫t
(100)
For a ﬁxed value of the ﬁnal momentum p the Coulombcorrected momenta p′α corresponding to different trajectories indexed by α are different. As a result, a phase
difference between any two contributions becomes
proportional to the time of observation td which is
apparently unphysical. This effect results from the
perturbative expansion of the action: it is clear that the
exact action (85) does not suffer this problem. Therefore,
although the Coulomb-corrected trajectories can be
found perturbatively, the action must be calculated from
(85) without expansions, particularly if interference
effects are of interest. The actual calculations of
[68, 75, 117] have been performed by solving the full
Newtonʼs equation numerically.
(ii) Coulomb-free and corrected trajectories are, in general,
complex: velocity vp (t ) is real for real t, while the
td
(96)
where
⎧ q+a
g ( q, φs ) ⎫
⎬.
f = iZF ⎨
−
2
( t − ts ) ⎭
⎩ ω( t − ts)
(99)
Equations (95)–(99) deﬁne the action in (85), (86) and thus
the ionization amplitude.
In practice, calculations along the described algorithm
encounter four serious difﬁculties:
is ﬁnite. The regularized action (88) can be presented in a
form convenient for calculations [15]
s
()
v ( ts ) = p + A ts′ − v1 ≡ p′ + A ts′ .
⎛ ω⎞
Wat ( r * ) + W ( t * ) ≈ κ 2ts + iν ln ⎜ 2 ⎟
⎝ iκ ⎠
∫t
(98)
Here q = p pF , a = A(ts ) pF and pF = E0 ω and the unit
vector e along the polarization direction. The integral in (96)
converges and its value does not depend on the integration
path provided the latter does not intersect cuts of the function
r0 r03. The obtained value of v1 corrects the initial complex
photoelectron velocity so that
with p (r ) = 2( − I p + Z r ) . Comparing (92) and (94) we
ﬁnd that the terms containing t* and r* do cancel, so that the
action
WC = −iν ln ( 2iK0ω t d ) +
⎤
1⎡
3
⎢ e + 2 (q + a) (e (q + a) ) ⎥ cos ωts
2⎣
⎦
γ
(97)
and
16
J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204001
Review Article
Figure 3. Ionization rate (in atomic units) versus laser intensity (in units W cm−2) calculated from (108) (solid line), without the correction (106)
(dashed line) and from the tunneling formula (rate (63) with the correction Q1) (dotted line). The gray line in the left panel shows the rate with only
Q1 accounted for. Rates obtained numerically using the Floquet method [124] and from the solution of the TDSE [125] are shown using dots and
triangles, respectively. Left panel: hydrogen ground state in the ﬁeld of a Nd laser with λ = 1064 nm. The correction QC ≈ 103. Right panel: Xe17+
(4p0-electron with I p ≈ 434 eV) in the ﬁeld of an XUV laser with λ = 13.3 nm, corresponding to the parameters of the experiment [126]. The
correction is QC ≈ 10 9 . The values of the Keldysh parameter are shown at the upper horizontal axis. Reprinted with permission from [118].
trajectory rp (t ) has a constant imaginary part (67).
Complex coordinates in the r.h.s. of Newtonʼs equation
make numerical solution difﬁcult. In practice, in most
cases the imaginary part of a trajectory can be omitted
without introducing a signiﬁcant error, for the following
reason: the most probable trajectory is always real,
therefore imaginary parts of trajectories corresponding to
relatively high probabilities of ionization remain small.
Most of the calculations along the CCSFA (including
those of [68, 75, 77, 117, 118]) have ignored these
imaginary parts. Account of the complexness of
trajectories can also be important at the initial stage of
motion corresponding to the vertical part of the
integration path of ﬁgure 6(a), known as sub-barrier
motion (see also section 8.2). In particular, it has been
shown in [121] that this account corrects the phase
difference between the two dominant trajectories by π for
emission of photoelectrons along the polarization direction. This shifts the interference pattern replacing minima
by maxima and vice versa, in agreement with the results
of ab initio numerical TDSE solutions.
(iii) Unlike the laser-induced action (71), the the full actions
(85) and(86) are not analytic functions in the complex
time plane. An analytic function r 20 (p, t ) has, in general,
an inﬁnite number of ﬁrst-order zeros, which generate
branching points and cuts of the functions 1 r0 and r0 r03.
A proper integration path in (95) and (96) must connect ts
and td without intersecting the cuts. In a linearly
polarized monochromatic ﬁeld roots of the equation
r 20 ( tn ) = 0
always is. Then the Coulomb potential energy has no
branching points but only ﬁrst-order poles. The presence
of even a small lateral momentum component converts
these poles into pairs of branching points which move
away one from another with increasing p⊥. Cuts are
deﬁned as lines in the complex time plane where r 20 < 0 .
Thus, integrating between ts and td one should ensure that
no cuts are intersected [120]. The choice of integration
paths and their relation to the position of the exit point
from the barrier is discussed in section 8.2.
(iv) The Coulomb ﬁeld generates new trajectories. In a
linearly polarized ﬁeld, and with no Coulomb interaction
accounted for, there are two trajectories per laser period
bringing the electron to a given ﬁnal momentum p . They
are known in the literature as the short and the long orbit.
When the Coulomb ﬁeld is accounted for, these two
Coulomb-free trajectories survive, experiencing only a
smooth deformation (see, e.g. Figure 3 of [117] for
illustration). In addition, another two trajectories can
emerge, where the lateral momentum changes its sign
once. The new classes of trajectories have their classical
cut-offs, i.e. they exist only in some part of the
momentum space. The boundary where they disappear
can be analytically found for a model case [76]. As is
typical for classical boundaries, the density of trajectories
has an integrable divergence there, the classical probability is divergent as well and the probability calculated
quantum mechanically reaches a sharp maximum. This
effect generates the recently observed low-energy
structure [72, 74–76]. In the above-described perturbative
algorithm of calculation, these new trajectories are
missing. In order to surmount this difﬁculty one should
use the ‘shooting’ method instead of a perturbative
calculation [75, 77]. The method works as follows: in the
(101)
come in pairs which merge for electron momenta parallel
to the polarization axis. In this special case tn are secondorder zeros of the function r20 like the saddle point ts
17
J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204001
Review Article
momentum space a sufﬁciently large number (up to
several million) of photoelectron momenta are randomly
selected, so that they cover the momentum space with
sufﬁcient density. For each momentum all the corresponding Coulomb-free trajectories and initial conditions
(66) are determined. With these initial conditions new
trajectories are calculated solving the full Newtonʼs
equation and the respective new ﬁnal momenta are
determined. The ﬁnal momentum space is covered by a
grid whose cell size is adjusted in such a way that about
ten or more new trajectories arrive in each cell on
average. The momentum distribution is then calculated as
∫0
τ
dτ ′
cosh τ0 − cosh τ ′
⎡ ( cosh τ0 + 1) ( cosh τ − 1) ⎤
2
⎥
arctanh ⎢
=
⎥⎦
⎢⎣
sinh τ0
sinh τ0 sinh τ
we arrive at an explicit result
⎛2⎞
Im ΔW = − ν ln ⎜ ⎟ + ν (ln 2γ − 1) .
⎝F⎠
This correction leads to appearance of the Coulomb factor in
the ionization rate
2
w ( pmn ) ∼
∑ke
iWk + iFk
,
QC = e−2 Im ΔW =
(102)
k
6.2.2. Examples of applications. Total ionization rate.
Coulomb corrections change the imaginary part of the
action and therefore the total rate of ionization. In order to
calculate a correction to the rate it is sufﬁcient to consider the
most probable trajectory. In the monochromatic linearly
polarized ﬁeld this trajectory corresponds to p = 0 . A
perturbative correction to the action (86) reads
()
ΔW ≡ W (p = 0) − W0 (p = 0) = W0 p′ − W0 (0)
∫t
td
s
⎧
⎫
⎪
Z
iν ⎪
⎨
⎬ dt . (103)
+
2
⎪
t − ts ⎪
⎩ r0 (p = 0, t )
⎭
2ν
Here p′ is the initial drift momentum found from (83). It
cannot be calculated analytically for arbitrary γ, while for
γ ≫ 1 using the Kapitza averaging method [99, 123] one
obtains [66, 118]
p′2 ≈ 2ων ≪ I p.
⎛ 2 ⎞2ν ⎛ 2γ ⎞−2ν
⎜
⎟ ⎜
⎟
≡ Q1Q2 .
⎝F⎠ ⎝ e ⎠
(106)
Here F is the reduced ﬁeld (30). Its value, typical for
experiments on nonlinear ionization, is F ≃ 10−2 ÷ 10−1. As
a result the correction Q1 is always numerically large and
enhances the rate of ionization by several orders in magnitude
—this effect is well documented in experiments [4, 65]. This
correction originates from the presence of the Coulomb
potential energy in the action and was ﬁrst calculated by
Perelomov and Popov [15]. The respective expression is valid
at arbitrary γ. In the tunneling limit the effect can be
interpreted as a consequence of the lowering of the potential
barrier the electron tunnels through. The correction Q2
calculated in [66] originates from the Coulomb deceleration
of the photoelectron: in order to arrive to the detector with the
minimal possible momentum p = 0 , the electron has to start
having a momentum along the ﬁeld polarization satisfying
(104). The price to pay for this is the reduction of the
ionization probability by the factor Q2 ≪ 1. The respective
expression is derived assuming the multiphoton regime. This
result can be extrapolated into the tunneling domain by
replacing γ → γ + e 2 in (106). Although in the multiphoton
limit Q2 ≪ 1, the net correction remains numerically large:
where pmn is the momentum corresponding to the center
of the (m,n)th cell (for a two-dimensional distribution),
the sum is taken over all trajectories rk (t ) arriving in this
cell. A detailed description of the shooting algorithm is
given in [75, 122].
With these four extensions the CCSFA algorithm is ready
for application.
− iν ln ( 2iK 0 ω td ) +
(105)
QC ≈ ( 2eK0) , γ ≫ 1
(107)
The result (106) allows one to obtain a quantitatively
correct expression for the total ionization rate of atoms
[66, 118]:
(104)
w = QC wSR,
(108)
Then from (57), (58)
where wSR is the ionization rate of a level with the same Ip, l
and m in an SR potential (62). In the tunneling limit (108)
reproduces the well known PPT rate [13, 15]. The high
quantitative accuracy of the rate (108) has been checked by
comparing it with results of exact numerical TDSE solutions
in the single-electron approximation. Two examples of such
comparisons are given in ﬁgure 3.
Coulomb asymmetry in elliptically polarized ﬁelds.
Violation of the four-fold symmetry of angular distributions
in a ﬁeld with elliptical polarization is an example of a
qualitative Coulomb effect. For a sufﬁciently long laser pulse
such that the carrier-envelope phase effects are not important,
the Keldysh amplitude (1) predicts angular distributions
symmetric with respect to both the major and the minor
⎛ ⎞2
p′
Im W0 p′ − W0 (0) = K0 b1 (γ ) ⎜⎜ ⎟⎟ ≈ ν (ln 2γ − 1) .
p
⎝ F⎠
( ()
)
The Coulomb-free trajectory (64) corresponding p = 0 has
the form
r0 (p = 0, t ) =
ωts =
E0
ω2
( sin (ωt ) − sin ( ωts ) ),
π
π
+ iarcsinhγ ≡ + iωτ0.
2
2
A contribution into the imaginary part of the integral comes
only from the vertical part of the integration path shown on
ﬁgure 6(a). Using the integral
18
J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204001
Review Article
Figure 4. Normalized angular distributions in the polarization plane, evaluated by different methods: the standard Keldysh approximation
(green), the CCSFA (red) and the ab initio TDSE solution (black) for the ground state of hydrogen (a),(b), neon (c) and argon (d). In the latter
case the data recorded in [112] are shown using black circles while the TDSE result is not shown. The laser intensity, frequency, ellipticity
and the photoelectron energy are, respectively: (a) 1.0 · 1014 W cm−2, 1.55 eV, 0.5 and 8.2 eV, (b) 1.6 · 1014 W cm−2, 3.1 eV, 0.5 and 2.6 eV,
(c) 2.0 · 1014 W cm−2, 1.55 eV, 0.36 and 7.1 eV(d) 0.6 · 1014 W cm −2 , 1.55 eV, 0.36 and 2.8 eV. For neon and argon the ionization
probability is averaged over the magnetic quantum numbers m = 0, ± 1 within the p-shell. The orientation of the polarization ellipse is
shown in the insert with the rotation direction of the electric ﬁeld vector indicated by an arrow. Reprinted with permission from [117].
Copyright © 2008 Taylor & Francis.
polarization axes. Experimental data clearly show that the
distributions possess only central symmetry [59, 112].
According to the Keldysh theory or the SFA, at intermediate
ellipticity (say, 0.2 < ξ < 0.8) the most probable photoelectron momentum is directed along the minor polarization axis
[14]. In the tunneling regime this can easily be explained: the
electron appears (with the highest probability when the ﬁeld is
maximal) at the tunnel exit with a zero velocity, vp (tm ) = 0.
At this time instant tm the vector potential is minimal in
absolute value and directed along the minor axis. Thus the
momentum p = vp − A(tm ) coincides with the vector
potential.
In elliptically polarized ﬁelds, realization of the CCSFA
algorithm is particularly simple, because only one trajectory
gives a dominant contribution to the probability, so that one
needs not care about a proper account of interference and the
shooting algorithm is also not required: a calculation can be
performed in a straightforward manner along (84)–(99). The
results shown in ﬁgure 4 demonstrate good quantitative
agreement with exact TDSE calculations. Agreement with the
data is acceptable, but less accurate. This discrepancy can be
attributed either to multielectron effects or to uncertainties in
experimental parameters (mostly in the laser intensity).
Coulomb-induced trajectories. With the Coulomb ﬁeld
accounted for, new complex classical trajectories corresponding to a given value of the ﬁnal momentum emerge. In a
linearly polarized ﬁeld it can be up to four trajectories per
cycle, instead of the two Coulomb-free trajectories. Figure 5
shows an example. All trajectories can be grouped into four
types. The ﬁrst two types are the so-called short and long
trajectories existing in the Keldysh theory. The short
trajectory (type I) fulﬁlls the condition bz pz > 0 and
px px′ > 0 , where bz is a z-projection of the tunnel exit position
(see section 8.2 for a deﬁnition of this value) and pz is the
photoelectron momentum projection on the polarization axis.
This means that the ejected electron moves from the tunnel
exit directly towards the detector. Type II is a long trajectory
obeying bz pz < 0 and px px′ > 0 . Here, the electron starts from
the tunnel exit that points away from the detector but ends up
with a parallel momentum in the opposite direction because of
the drift it acquires from the laser ﬁeld at the time of
ionization. The lateral initial momentum px′ of type I and type
19
J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204001
Review Article
Figure 5. (a) Partial photoelectron momentum distributions in the (pz , px ) plane (with z chosen along the linear polarization axis) due to
trajectories of types I–IV calculating by the shooting method [75, 122]. A caustic forming the LES is seen on the panel III. (b) Dominant
trajectories in the zx plane contributing to the ﬁnal momentum p = ( − 0.22, 0.1) close to the LES position. Open circles indicate the end of
the laser pulse. The scales in x and z are different. Reprinted with permission from [75]. Copyright (2010) by the American Physical Society.
II trajectories is already in the same direction as the ﬁnal
momentum at the detector. This is always the case in a
Coulomb-free theory where p = p′. Type III is classiﬁed via
bz pz < 0 and px px′ < 0 . The corresponding electrons start on
the opposite tunnel exit, as type II electrons do, but also have
an initial lateral momentum pointing in the ‘wrong’ direction.
Due to the Coulomb force their lateral momentum is
ultimately reversed. The remaining class of type IV
trajectories obey bz pz > 0 and px px′ < 0 , i.e. although the
tunnel exit already points towards the detector, the initially
incorrect lateral momentum is reversed by the Coulomb ﬁeld.
The new classes of trajectories generate at least two
signiﬁcant effects in photoelectron momentum distributions.
First, trajectories of type III form a caustic (a sort of classical
boundary, where the density of trajectories diverges) which
induces the low-energy structure [75]. Second, interference
between type III from one side and types I and II from
another, modulates momentum distributions forming the
structures known as side lobes [77].
Coulomb ﬁeld [127, 128]. The method combines two techniques. The ﬁrst is based on the partitioning of the coordinate
space into inner and outer regions, such that the electron
dynamics is dominated by the electron–ion interaction in the
ﬁrst, and by the laser ﬁeld in the second region. The ARM
method allows one to formulate equations for the two (inner
and outer) wave functions matched at the boundary [128].
The second technique known as EVA [127] allows one to
construct an approximate Coulomb-corrected Greenʼs function in the outer region needed to deﬁne the amplitude of
ionization. Both the ARM and EVA methods explore ideas
similar to those used in CCSFA, but independently developed
and differently realized. The perturbation theory for the action
and the matching with the atomic wave function described in
section 6.2 are close (but not identical) to the eikonal calculation of the Coulomb correction and to the space partitioning,
respectively. Here we describe the algorithm which combines
the ARM and EVA and illustrate its application using several
examples.
6.3.1. Ionization amplitude in the EVA and ARM
approximation. Within the ARM the coordinate space is
6.3. Eikonal–Volkov approximation (EVA) and the analytical Rmatrix (ARM) method
partitioned into inner (+) and the outer (-) regions separated
by a spherical boundary of radius a centered at the atom. The
objective is to ﬁnd an approximate wave function (in contrast
to the CCSFA where the ionization amplitude is constructed
Another method of the inclusion of Coulomb effects suggests
a regular procedure for calculation of an approximate wave
function in the presence of a strong laser ﬁeld and the
20
J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204001
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without knowledge of the Coulomb-corrected wave function)
developing from the initial bound state Ψ0 (r, t ) under the
action of a laser pulse. The Hamiltonian operator
H=−
Δ
+ U (r ) + Vint (r , t )
2
obtained, if we account for the Coulomb interaction
correcting the phase of the Volkov function. In the outer
region the electron does not approach the nucleus and the
Coulomb ﬁeld can be considered to be small compared to the
laser ﬁeld. In addition, the laser-induced velocity (6) of the
electron remains relatively high most of the time, so that
electron trajectories are smoothly deformed by Coulomb
attraction. Under such conditions the Coulomb correction to
the phase of the wave function can be evaluated in the eikonal
approximation along a trajectory found in the absence of the
Coulomb ﬁeld [127, 128]
(109)
is not Hermitian if only part of the position space is
considered. Hermiticity can be restored by adding the Bloch
operator [129] to the inner and outer Hamiltonians:
⎛ d
b⎞
+ ⎟ (110)
H → H (±) = H + L(±) , L(±) = ± δ (r − a) ⎜
⎝ dr
r⎠
ΨEVA (p, r , t ) = Ψp (r , t )eiWC (r, p, t ) ,
with b being an arbitrary constant which escapes the ﬁnal
result. The time-dependent Schrödinger equation (36) then
reads
( i ∂ ∂t − H ) Ψ ( r , t ) = − L
(±)
(±)
Ψ (r , t )
r=a
.
where Ψp (r, t ) is the Volkov function (3) and the Coulomb
phase is given by
(111)
WC (r , p, t ) =
Because of the delta-function entering the Bloch operator, the
wave function in the r.h.s. of (111) is taken at the boundary
r = a. Then, using the respective Greenʼs function one can
present these equations in the integral form (see (39))
∫t
td
U [ rL (r , p, t , t′) ] dt′ .
t
( )
× L(±) Ψ r′, t ′
.
rL (r , p, t , t′) = r +
(112)
MARM (p) = − i
t
∫−∞ dt′ ∫ d3r1G (−) ( r, t, r1, t1)
× L(−) ψ0 ( r1)
r=a
e−iε0 t1 .
∫t
t′
vp ( t1)dt1
(116)
and is shifted by r with respect to the trajectory (64) of the
ITM. The wave function (114) with the phase (115) is known
in the literature as the EVA solution [127, 128]. The EVA
functions can be used for construction of the outer Greenʼs
function in (113). Projecting Ψ1( − ) (r, td → +∞) onto a plane
wave with the momentum p , one obtains for the amplitude of
ionization in the ARM approximation
r=a
Equations (112) can be approximately solved by applying an
iteration procedure. Replacing the inner wave function by the
one of the unperturbed bound state, Ψ0( + ) (r, t ) = ψ0 (r)e−iε0 t
(where the ground state energy ε0 may include a complex
Stark shift) we obtain for the ﬁrst approximation in the outer
region
Ψ1( − ) (r , t ) = −
(115)
The argument of the Coulomb potential U (r) in (115) is a
trajectory in the laser ﬁeld which starts at the time instant t at
the spatial point r :
∫−∞ dt′ ∫ d3r′G (±) (r, t, r′, t′)
Ψ (±) (r , t ) = −
(114)
td
td
∫−∞ dt ∫ d3rΨp* (r, t )e−i ∫
t
⎛ d
b⎞
× δ (r − a ) ⎜
+ ⎟ Ψin (r , t ).
⎝ dr
r⎠
(113)
Substituting this Ψ1( − ) into (112) for the inner part and using
there the atomic Greenʼs function, one obtains the ﬁrst
correction Ψ1( + ) to the atomic wave function in the inner
region, and so on.
If the radius of the inner region is sufﬁciently small, the
series resulting from this iterative procedure is an expansion
with respect to the number of hard recollisions experienced by
the electron: the function Ψ1( − ) corresponds to no recollision
(direct electrons), Ψ2( − ) describes electrons experiencing a
single hard recollision, etc. After a hard recollision an electron
usually acquires a large drift momentum, so a second hard
recollision is very unlikely, let alone a third. Thus the series
should converge quickly.
In order to perform calculations within this algorithm, an
analytic expression for the outer Greenʼs function
G (−) (r, t , r1, t1 ) is required. The simplest realization corresponding to the Volkov Greenʼs function (41) will reproduce
the ZRP version of the Coulomb-free SFA for direct
ionization and (if the next terms Ψ1( + ) and Ψ2( − ) are found)
rescattering. A more accurate (and desirable) result can be
⎡
⎤
U ⎢ rL t ′ ⎥ dt ′
⎣
⎦
()
(117)
Because of the delta-function, spatial integration is conﬁned
to the sphere of radius a. This radius should be small enough
that the Coulomb and laser ﬁelds are of the same order there.
Under this condition, integration over the R-matrix sphere
and application of the SPM yield the optimal trajectory rL
which is identical to the Coulomb-free trajectory of ITM (64).
The amplitude (117) has to be compared to the Keldysh
(1) and the CCSFA (84) amplitudes. Omitting the Coulomb
integral in the phase and replacing the inner wavefunction by
the unperturbed initial atomic function ψ0 (r)e iIp t we obtain
the amplitude (44) for the ZRP. With the Coulomb ﬁeld
accounted for, the amplitude (117) is different from the
CCSFA result (84). Indeed, the Coulomb integral WC is
introduced into the phase before the application of the SPM
and the phase itself is calculated along trajectories unperturbed by the laser ﬁeld, in contrast to CCSFA where the
trajectories are also corrected. Exact numerical evaluation of
the time integral (117) is cumbersome, while the saddle-point
analysis remains efﬁcient if the Coulomb ﬁeld is considered
as a perturbation. The saddle-point equation, which now
21
J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204001
Review Article
includes the Coulomb term, reads
−
for the full Coulomb factor QC may appear different from
(106)), but this has not been attempted yet.
Within the method, the Coulomb action can be calculated
for arbitrary values of the ﬁnal momentum p with a result
identical to (95). Trajectories corresponding to arbitrary ﬁnal
momenta are complex, so the action WC inﬂuences both the
probability of ionization and the interference pattern of
photoionization spectra. In order to calculate the Coulomb
action along a complex trajectory, an analytic continuation
of
the
Coulomb
potential
in
the
form
1
∂
( S0 + WC ) = 2 v2p ( ts ) + I p + U (a)
∂t
+
∫t
td
U vp (t′)dt′ = 0.
(118)
s
The last two terms are the Coulomb potential energy at the
boundary separating the inner and the outer regions and the
work done by the Coulomb ﬁeld along the Coulomb-free
trajectory. The Coulomb phase is matched to that of the
atomic wave function using a technique similar to that
described in section 6.2.1. After this, (118) is solved
perturbatively, taking a Coulomb-free solution for the
zeroth-order approximation. As a result, the stationary point
ts (p) takes the form
U (r ) = − Z r = − Z r 2 has been used [121, 132, 133].
A very important advantage of the method, compared to
the CCSFA, it that it allows a straightforward generalization
to many-electron systems. As the outer wave function is
expressed via the inner one, a multiparticle structure of the
ts (p) = ts0 (p) + Δts (p),
(119) latter can be accounted for in the amplitude of ionization. This
allows the study of multichannel ionization of complex atoms
where the Coulomb correction is given by [130, 131]
and molecules. The corresponding formalism is based on the
introduction of direct and indirect (driven by electron–elecW ′ ( t s0 )
Δts (p) = − C′′
.
(120) tron correlations) ionization amplitudes for each ﬁnal ionic
S0 ( t s0 )
state and appears to be rather involved technically , so we
This value is, in general, complex and its real part was direct the interested reader to [134]. A rather unexpected
interpreted as an ionization delay appearing due to decelera- result of the calculations performed there is that in the
tion of the electron during its sub-barrier motion [130, 131] nonadiabatic regime γ > 1 correlation-driven channels of
ionization of molecules can considerably dominate over the
(see also the discussion in section 8.2).
These equations summarize the application of the ARM direct channel. In connection to this observation, ab initio
and EVA methods to the problem of strong ﬁeld ionization. numerical solutions of the TDSE for lithium in a strong laser
The logic of this theory is close to that of the CCSFA. ﬁeld [135] show that core excitations are only important at
Differences originate mostly from the slightly different relatively high frequencies, K0 ≃ 1, when ionization proceeds
in the few-photon regime, while in the multiquantum regime
mathematical realizations of the same idea, namely:
the ionization dynamics remain essentially single-electron, if
(i) After matching, the Coulomb phases WC of both methods the laser frequency does not hit a resonance. Whether or not
coincide, but not the corrections to the ﬁnal photoelec- this qualitative difference reﬂects the difference between an
tron momentum.
atom and a molecule or between the applied theoretical
(ii) The Coulomb corrections to the stationary point given by approaches, remains an open question.
ts′ = ts (p′) in the CCSFA and by (120) are also different,
As was already discussed in section 6.2.2, a natural
although both are linear with respect to the atomic application for Coulomb-corrected theories is ionization in
residual charge Z.
intense elliptically polarized ﬁelds, where the Coulomb
(iii) The ARM method allows one to incorporate rescattering asymmetry is a pronounced qualitative effect. In short pulses,
into the Coulomb-corrected theory, while the CCSFA in there is no qualitative difference between elliptical (with ξ ≃ 1)
its present form only describes direct ionization.
and circular polarization, because of the pulse envelope effect.
The predictions of the CCSFA and ARM methods seem Recently, a series of high-precision experiments has been
to be close but not identical. If one of the two versions of the performed where ionizations of atoms by short, near circularly
theory suffers logical inconsistencies or both do not precisely polarized pulses have been used for veriﬁcation of the so-called
attoclock setup [136–140]. The attoclock idea is, put simply,
achieve the ultimate goal, there remains an open question.
the following: in a circularly polarized ﬁeld the direction of the
6.3.2. Applications. The high quantitative accuracy of the most probable photoelectron momentum is determined by the
ARM and EVA methods has now been proven by several vector potential of the wave at the instant of ionization tm ,
applications. First, in the simplest realization, when the when the ﬁeld is maximal (see also the discussion in
Coulomb action is only calculated along the most probable section 6.2.2 and 8.2). Therefore, measuring a photoelectron
trajectory, the method naturally reproduces the Coulomb momentum distribution in such a ﬁeld, one can (provided the
factor Q1 or, in other words, the rate of ionization obtained by time dependence of the electric ﬁeld vector E(t ) is known)
Perelomov and Popov [15]. Calculating the corrections to the extract the instant of ionization with a sub-cycle (attosecond)
stationary point (120) one should obtain the factors Q2 and accuracy. In order to properly perform such an extraction one,
QC from (106) and (107) (to be more precise, due to however, needs to account for the Coulomb distortion of
quantitative differences between the AMR and EVA methods photoelectron trajectories. Both the CCSFA and the ARM
on the one hand and the CCSFA on the other, an expression methods are perfectly suited for such calculations. The ﬁrst has
22
J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204001
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the eigenvalue problem
not yet been applied to the problem, while application of the
second has shown an excellent agreement with the results of
numerical solutions of the TDSE [130, 131] performed by
three different methods: relative deviations in the value of the
offset angle calculated analytically from the ARM method and
numerically do not exceed 2%, while the predictions of
different numerical methods also differ one from another at this
level. Moreover, it has been shown that just a 0.1% deviation
from the peak intensity shifts the results to a value exceeding
the difference between them.
⎤
⎡ ∂
Δ
− U (r ) − E (t ) r⎥ Φϵ (r , t ) = 0
⎢⎣ i + ϵ +
⎦
2
∂t
(121)
has to be found. Here ϵ = ϵ′ − iϵ′′ is a complex quasienergy.
Its imaginary part determines the total ionization rate,
Γ = 2ϵ′′. For an SR potential U(r) this equation can be solved
by matching a solution for a free electron in the laser ﬁeld
E(t ) valid at r > rc , where rc is the effective radius of the
potential (rc = 0 for the ZRP), with a laser-free solution at
r < rc . The approximation of the ﬁeld-free wave function at
r < rc is similar to the one made in the Keldysh model but not
exactly identical to it. The difference which allows an
increase of the accuracy of the solution, compared to the
Keldysh ionization amplitude, consists in the matching of the
inner and the outer wave functions at r = rc (similarly to the
ARM method of section 6.3). For the ZRP, the atomic force
acts at the singular point r = 0 where it is divergent, then the
respective solution of (121) becomes exact. For an SR
potential with rc ≠ 0 (modeling bound states of negative ions
with nonzero angular momentum), the solution is approximate but still highly accurate. The procedure of calculation of
the wave function in the two spatial regions and their
matching at r = rc is described in [29]. When Φϵ (r, t ) is
found, its Fourier expansion (at r → ∞) determines the
amplitude of ionization with absorption of n photons. For a
system bound by zero-range forces and subject to a linearly
polarized ﬁeld
7. Generalized and complementary approaches
In the theory of strong ﬁeld ionization there are several efﬁcient analytic approaches which combine relative simplicity
with quantitative accuracy and stay conceptually close to the
Keldysh method. Here we sketch two analytic methods
known for their successful application to the problem. The
method of QQES [29, 141, 142] provides an exact solution of
the photodetachment problem for the ZRP and a quantitatively accurate solution for an SR potential treated within the
effective range approximation. As far as detachment of
negative ions is concerned, the QQES method includes Keldysh theory as a limiting case. It is instructive to trace how the
Keldysh result follows from this more general approach.
Another method, developed by Berson [143, 144], is based on
the semiclassical description of electron motion in the Coulomb ﬁeld of the atom and perturbatively accounts for laserinduced corrections to the electron action. It can be viewed as
an approach complementary to Coulomb-corrected versions
of the Keldysh theory.
∞
 n (p) = in
∑
( − 1)k fk
k =−∞
∞
×
⎛ E 0 k n cos θ ⎞
⎛ zF ⎞
⎟J
⎟,
n + 2 s − 2k ⎜
⎝
⎠
8 ⎠
ω2
∑ Js ⎜⎝
s =−∞
(122)
where θ is the angle between the photoelectron momentum p
and the laser polarization and k n = 2(ϵ + nω − UP ) is the
complex quantity which turns into the absolute value of the
photoelectron momentum when we take ϵ = − I p . Coefﬁcients fk determine the Fourier expansion of the function
Φϵ (r, t ) at r → 0
7.1. The quasistationary quasienergy states (QQES) method
The concept of quasienergy states (QES) of a quantum system
subject to a periodic external ﬁeld is based on the Floquet
theorem [145] and was introduced by Shirley [146], Zeldovich [147] and Ritus [148]. The term quasienergy (introduced
by Zeldovich and Ritus) relates to a new quantum number—
the Floquet index—which is conserved in a periodic ﬁeld.
There is an obvious analogy with conservation of quasimomentum in a spatially periodic potential. In order to apply the
QES formalism to the problem of laser-induced ionization, a
continuum of quasienergies has to be considered, because
there are no true bound states in the system. This complicates
calculations enormously. To avoid this difﬁculty, a generalization of the QES formalism making it suitable for treatment
of problems involving bound–free transitions was suggested
in the mid−1970s [149, 150] by introduction of complex
quasienergies and the respective QQES. For a complete
description of the method we direct the reader to the
review [29].
According to the QQES method, in order to describe
ionization of a bound state in the potential U(r) a solution of
Φϵ (r → 0, t ) =
⎛1
⎞
⎜
− 1⎟ fε (t ), fε (t ) =
⎝r
⎠
∞
∑ fk e−2ikωt .
(123)
k=0
Absence of odd Fourier components in the series (123)
reﬂects a speciﬁc symmetry of the wave function at the origin:
equivalent wave packets of the detached electron return there
two times per laser period. Generalizations on arbitrary
elliptical polarization and an SR potential are described in
[29, 151].
In order to obtain the result of the Keldysh theory from
(122) we replace the quasienergy by its ﬁeld-free value −I p ,
then kn in (122) coincides with pn from (19). This means that
we disregard the Stark shift and the width of the bound state.
Second, the Fourier coefﬁcients fk are approximated by
fk = δ k,0
2π ,
(124)
i.e. 2ω-periodic oscillations of the wave function at the origin
23
J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204001
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Figure 6. Left panel: photoelectron spectra along the polarization direction as functions of electron energy En scaled by Up for H− (s-state;
thin lines) and F− (p-state; thick lines) for E0 = 0.2835 and ω = 0.203 (with the photon energy scaled by the ionization potential Ip and the
electric ﬁeld amplitude by (2I p )3 2 ). Solid lines: exact QQES results; dashed lines: Keldysh theory. Solid circles mark the positions of ATI
peaks. The rates are shown in scaled units (s.u.): results for F − are multiplied by the factor 6.24 in order that the maximum rates are the same
for both ions. From [151]. Copyright (2003) by APS. Right panel: photoelectron spectra along the polarization direction for ionization of Br −
in a laser ﬁeld with the wavelength 1300 nm and intensity 6.5 · 1013 W cm−2. The data are shown by circles, including several points with
error bars; the Keldysh result is shown by the dashed line, the SFA result with rescattering by the solid line. The theoretical distributions are
normalized to the data at the maximum of the signal. Reprinted with permission from [83]. Copyright (2010) by the American Physical
Society.
corresponding to ionization in a ﬁeld with relatively low
intensity and high frequency, was developed by Berson
[143, 144]. The term in the action describing interaction
between the electron and the laser wave is
induced by the photoelectron wave packets returning to the
core each laser half-period are neglected. Thus approximation
(124) does not describe rescattering7. After these simpliﬁcations the amplitude (122) turns (21) with accuracy into a
factor reﬂecting different deﬁnitions of the ionization
amplitudes.
The left panel of ﬁgure 6 shows comparisons between the
photodetachment spectra of H− and F− ions calculated using
the QQES approach and the Keldysh model. As expected, the
Keldysh result does not reproduce the rescattering plateau
which dominates in the spectrum for energies ε > (4 ÷ 5) UP .
For lower energies, where the direct ionization mechanism
dominates, the agreement is excellent. The right panel of this
ﬁgure shows a comparison of the photodetachment spectra of
Br − calculated along the Keldysh model with the data
recorder in [83]. This comparison proves that for SR potentials, Keldysh theory yields quantitatively accurate results.
For a review of the further development of the QQES
approach, including its generalizations for atoms and molecules, we direct the reader to [152–156] and references
therein.
Wint = −
∫ E (t ) rC (t )dt
(125)
with rC (t ) being the electron trajectory in the atom. Using
classical trajectories for the description of atomic motion in
bound states requires semiclassical conditions, therefore (125)
assumes a highly excited (Rydberg) initial state. The amplitude of ionization with absorption of n laser photons
Mn ∼
∫ dt ei[W
int(t ) + nωt
] ∼ Jn (B),
(126)
where Jn is the Bessel function and B has to be found from
(125). To this end the method suggested by Kramers for
calculation of single-photon ionization [157] can be used. In a
Rydberg state the electron moves almost along a parabola.
Taking a parabolic trajectory from a parametric solution of the
respective classical problem [99]
⎛
1 ⎞2
1⎛
1 ⎞2
xC (η) = ⎜ l + ⎟ η , zC (η) = ⎜ l + ⎟ 1 − η2 ,
⎝
2⎠
2⎝
2⎠
3 ⎛
2
1⎛
1⎞
η ⎞
t = ⎜l + ⎟ η ⎜1 + ⎟
⎝
⎠
2
2
3⎠
⎝
(
7.2. Semiclassical theory of ionization
The EVA and CCSFA approaches are based on a perturbative
account of the Coulomb interaction in the classical action of a
photoelectron in the ﬁeld of a strong laser wave. In both
methods the laser-induced action (45) is taken as a zeroth
approximation. It makes sense to consider the opposite limit
when the Coulomb ﬁeld is treated as the main interaction,
while the effect of the laser ﬁeld is accounted for within the
perturbation theory for the action. Such an approach,
)
one can calculate the integral (125) and extract the coefﬁcient
B at sin ωt . For a linearly polarized ﬁeld with the polarization
direction speciﬁed by angles θ and φ respective to the axes z
and x this calculation gives [143]
⎛ 2 ⎞5 3
B = πE 0 ⎜ ⎟
Ai′ 2 (u) cos2θ + uAi2 (u) sin2θ cos2φ (127)
⎝ω⎠
7
An account of the Fourier components fk with k ⩽ (3.17UP + I p ) 2ω
reproduces the standard rescattering plateau predicted by the generalized SFA.
where u = (l + 1 2)2 (ω 2)2 3 and Ai(u) is the Airy function.
24
J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204001
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Note that the ionization potential I p ≪ 1 does not enter this
expression. It is assumed that B ≪ 1, so the Bessel function
can be replaced by its expansion for small arguments. Then,
after integration over the angles θ and φ and summation over
the orbital quantum number l ⩽ N − 1 (N ≫ 1 is the principal quantum number, I p = 1 2N 2 ) one obtains for the rate of
n-photon ionization [143]
wn ( E 0, ω , N ) =
E 02n Tn
This is not so for the Coulomb potential, as its Fourier
transform 〈p|U |q〉 present in the amplitude (74) diverges for
small momentum transfers, |p − q| ≪ p. As a result, at low
photoelectron energies εp ⩽ UP the amplitude M1 is not a
correction.
A gauge transformation leads to some exchange between
the terms MK and M1. Adopting upper indexes l and v for the
length and the velocity gauge, respectively, we can write
(128)
N 5ω(10n + 2) 3
M l = MKl + M1l = eiΦM v = eiΦ ( MKv + M1v ),
where Tn is an n-dependent number. The respective expression for circular polarization can be found in [143].
For n = 1 this result recovers the Kramers’ formula for
single-photon ionization. However, at n ≫ 1 there is no
matching to the multiphoton limit of the CCSFA. The latter
follows from (108) at γ ≫ 1 [118]:
but it is generally
MKl ≠ eiΦMKv ,
E 02n( 2eK0)
ω2n + 2ν − 3 2
.
M1l ≠ eiΦM1v .
Here Φ is a phase generated by the transformation,
| exp (iΦ ) | = 1. For an SR potential |M1 | ≪ |MK | for direct
ionization, so it cannot change MK signiﬁcantly on transformation, as a result the theory is only weakly gauge-noninvariant, while for the Coulomb potential it is strongly
noninvariant.
Although the gauge invariance problem in strong ﬁeld
ionization has been addressed in a number of papers (see
[159–164] and references therein), there are still a lot of
misunderstandings and erroneous statements. Here we consider three viewpoints commonly accepted in the literature:
2ν
wn ( E 0, ω , ν ) ≈ T˜n
(130)
(129)
Here T˜n is another n-dependent coefﬁcient [118]. The power
dependence of the rates (128) and (129) on the laser frequency is essentially different. The reason for this disagreement is that the applicability domains of the two theories do
not overlap. Equation (125) implies that the characteristic size
of the electron orbit in the atom is larger than the quiver
amplitude in the laser ﬁeld. The technique of Coulomb corrections used in the CCSFA assumes the opposite condition:
the size of the initial bound state is negligible compare to all
other spatial scales of the problem.
The amplitude (127) is proportional to the parameter
E0 ω5 3 which is supposed to be small in the theory, indicating that it is applicable in ﬁelds with low intensities and
relatively high frequencies. The same parameter appears in
semiclassical calculations of WKB matrix elements in the
Coulomb ﬁeld [56, 158]. In this context, the border between
semiclassical methods, in essence close to the one described
above and strong ﬁeld Keldysh-like theories, is deﬁned by the
condition E0 ≃ ω5 3.
(i) The problem of gauge invariance is simply misundersood. Under a proper treatment of the full Hamiltonian of
the system ‘atom+ﬁeld’, the velocity and the length
gauge versions of the SFA ionization amplitude are
equivalent.
(ii) There is a preferable gauge, which has to be used in
calculations.
(iii) The theory is essentially noninvariant and none of the of
the interaction operator forms (2), (4) provide a correct
result. An essential modiﬁcation of the theory is needed
in order to restore the property of gauge invariance.
Below we comment these statements and argue in favor
of the third, although the second one is also partially correct.
The ﬁrst statement is fully erroneous.
Under a certain choice of partition of the full Hamiltonian
of the system, the amplitude of ionization (1) appears gaugeinvariant [160, 165]. For example, in the velocity gauge the
Hamiltonian of the problem can be presented in the forms
8. Difﬁculties and open questions
8.1. Gauge problem
The probability distributions following from the amplitude (1)
depend on the gauge chosen for description of the electron–laser coupling. Exact gauge invariance is only achieved for
the special case of the ZRP (see section 2.1). There is nothing
surprising in the fact that a nonperturbative approximate
theory does not possess the property of gauge invariance. The
Keldysh amplitude MK differs from the exact unknown
amplitude M by a term M1 (also unknown) which is in general
not small in absolute value compared to MK. In order to
estimate the magnitude of M1 we may approximate it by the
amplitude of rescattering (74). From what we know about
rescattering, if the atomic potential U(r) is an SR, then
|M1 | 2 ≪ |MK | 2 for the direct part of the ionization spectrum.
H=
1
(p + A (t ) )2 + U (r ) = Hi + Vi = H f + V f , (131)
2
where the partial Hamiltonians Hi, f and Vi, f can be chosen
differently for the initial and the ﬁnal Hilbert space, for
example
Hi = H − E (t ) r , Vi = E (t ) r
(132)
and
Hf =
1
(p + A (t ) )2 , V f = U (r ).
2
(133)
With this partition we obtain for the eigenfunctions of the
25
J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204001
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initial unperturbed Hamiltonian Hi
ψ˜ i (r , t ) = ψi (r)e−iεi t − iA(t ) r ,
that the statements (i) and (ii) are in general erroneous. For
nonzero frequencies there is no analytic benchmark, however
the results of the SFA or the Keldysh model in any gauge can
be compared to numerical solutions of the TDSE. Such
comparisons, presented widely in the literature, show that in
the case of atoms (not to say molecules and more complex
systems), the results of the SFA agree with exact solutions
only qualitatively, while some features of momentum distributions are not reproduced at all (see the discussion in
section 6). Note that there are features of ionization spectra
stable with respect to the gauge transformations. This
includes, for example, the absence of every second ATI peak
for lateral emission in a linearly polarized ﬁeld [166, 167] or a
smooth spectrum with the maximum around UP in a ﬂied with
circular polarization [90].
On the other hand, the predictions of the Keldysh theory
in the length gauge agree quantitatively with exact calculations and data for negative ions (see an example given on
ﬁgure 2). Combining these observations together we can
formulate the following conclusions:
(134)
where ψi are the eigenfunctions of the atomic Hamiltonian.
Taking initial states in the form (134) one obtains the
amplitude of ionization (1) in the length gauge. From the
other side, staring from the length gauge Hamiltonian and
using the partition
Hi = p2 2 + U (r ), Vi = E (t ) r ,
H f = p2 2 + E (t ) r , Vi = U (r )
(135)
one again obtains the length gauge form of the amplitude.
This argumentation ‘proves’ that the theory is gauge-invariant
(on proper partitioning) and that the correct form is given by
MlK. Thus, this treatment of the gauge problem combines the
viewpoints (i) and (ii).
In our opinion, the described arguments do not solve the
problem, because the choice of the gauge is replaced there by
the choice of the partition. Indeed, there is no difﬁculty in
showing that the Hamiltonian Hi from (132) does not contain
any interaction with the laser ﬁeld and therefore is equivalent
to the atomic Hamiltonian obtained from the latter using a
unitary transformation. There is an inﬁnite number of such
Hamiltonians, so that by choosing different partitions one will
arrive at different forms of the ionization amplitude. Thus the
technique of partitioning simply redeﬁnes the problem without solving it.
It can be unambiguously shown that for atoms both the
velocity and the length forms yield incorrect (and different)
results. To this end we consider the static limit γ → 0 and
calculate total ionization rates using the saddle-point expressions of section 5 and appendix A for the ground state of
hydrogen (ν = 1, l = m = 0 ). In the velocity gauge there is
no divergency in the prefactor P and one obtains from (53)
after integration over photoelectron momenta
wv = Cv
4
I p ⎛ ω ⎞ −2 3F
⎜⎜ ⎟⎟ e
.
F 9/2 ⎝ I p ⎠
(i) When the Coulomb interaction is present, the Keldysh
model and the SFA do not possess gauge invariance and
neither the length (2) nor the velocity gauge (4) give a
correct result.
(ii) However, in the saddle-point expression for the amplitude, only the prefactor  (p, ts ) is gauge-dependent,
while the exponential factor e−iS0(p, ts ) is invariant. As the
exponential factor is much more sensitive to atom and
laser parameters than the prefactor, incorrectness of the
latter usually causes secondary-order effects. This
explains why, being gauge-noninvariant, the Keldysh
theory can be used for reliable calculations of the
photoionization spectra of atoms.
(iii) In order to describe the ionization of atoms correctly, one
must not apply the amplitude (1) for a corresponding
atomic state in any gauge. Instead, the respective
amplitude for a state with the same I p, l, m in an SR
potential should be calculated and then corrected
applying the techniques described in section 6.
(136)
The same calculation performed in the length gauge involves
(A.8) and gives
w l = Cl I p F e−2 3F .
The next question which now arises naturally is: what
gauge should be used for the calculation of the respective
amplitude in an SR potential (if it is not the ZRP). The results
of the numerical study [159] show that the length gauge (2)
should be chosen. Indeed, comparisons with exact TDSE
solutions demonstrate that with increasing radius rc of an SR
potential, the disagreement between them and predictions of
the velocity gauge version of SFA grows fast, so that already
for rc = 2 a.u. the velocity gauge gives an essentially erroneous result, while the length gauge version of the SFA stays
in good agreement with the numerical data8. This conclusion
can be supported by comparisons with multiple experimental
results on detachment of negative ions. The example shown in
ﬁgure 7 also provides a clear argument in favor of the length
(137)
Here Cv, l are numeric coefﬁcients. The quasistatic limit for the
ionization rate of hydrogen in a linearly polarized ﬁeld follows from (108) for v = 1, l = m = 0 and γ = 0 :
wst = 8I p
3 −2 3F
e
.
πF
(138)
It differs from the rate of ionization in a constant electric ﬁeld
[40] by the factor 3F π appearing due to the averaging over
the laser period.
Comparing these three results we ﬁnd that the prefactors
in (136) and (137) are essentially incorrect (the velocity gauge
result even turns to zero as ω4!). This constrains one to
conclude that none of the gauges provide a quantitatively
correct description in the case of atoms. This example proves
8
As expected from the general analysis above, with a further increase of the
effective potential size the length gauge result becomes different from the
exact solution as well.
26
J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204001
Review Article
emphasized however, that such arguments can only make
sense within an approximate theory. Another, rather mathematical argument in favor of the length gauge for SR
potentials was found in [151]: the QQES approach is explicitly gauge-invariant, however, the matching procedure
necessary for actual calculation of the eigenfunction Φϵ (r, t )
can only be performed in the length gauge representation.
8.2. Tunnel exit, Keldysh tunneling time and the simple-man
model (SMM)
An invaluable insight into strong ﬁeld physics including
recollision has been made by the SMM. The scheme of this
semiclassical method [61, 168–170] is remarkably clear: the
process of ionization is considered as a sequence of either two
or three steps. The ﬁrst step is quantum-mechanical tunneling
of the electron through the potential barrier created by the
laser ﬁeld. The tunneling probability can be calculated at any
time instant using the quasistatic rate (63) corrected by the
Coulomb factor (106) (i.e. the tunneling limit of the PPT rate)
where the electric ﬁeld E(t0 ) at the time instant t0 is substituted. The electron appears at the end of the barrier, the
point known as the tunnel exit. Considering the laser ﬁeld to
be quasistatic and ignoring the Coulomb distortion of the
barrier, we obtain for the exit point:
b 0 ≡ b (γ → 0 ) = −
I p E ( t0 )
E 2 ( t0 )
(139)
The time interval the electron needs to cover this distance
moving with the atomic velocity vat = 2I p
Figure 7. Comparison between the photoelectron spectra recorded for the
case of photodetachment of F− in a linearly polarized ﬁeld of the
wavelength 1400 nm and intensity 1.7 · 1013 W cm−2 for emission along
(circles) and perpendicular (crosses) to the polarization axis. The predictions
of the SFA with rescattering are shown by solid and dashed lines
correspondingly. The upper panel shows the theoretical results obtained in
the length gauge and the lower panel in the velocity gauge. The
experimental data are identical for both panels. Reprinted with permission
from [82]. Copyright (2007) by the American Physical Society.
Δt K ≃ b0 vat ≃ γ ω
(140)
is known as the Keldysh tunneling time. In the tunneling limit
it is equal to the imaginary part τ0 of the saddle point ts. This
value gives an estimate of the time necessary for establishing
of the static-ﬁeld ionization rate when the electric ﬁeld turns
on instantly [174]. The second step is a classical motion of the
photoelectron from the tunnel exit to a detector. The third step
comes into play if during this motion the electron revisits the
parent ion—a phenomenon known as recollision. A recollision can result in elastic scattering, recombination with the
emission of a high harmonic photon or inelastic scattering
with excitation or further ionization of the parent ion.
This simple model allows one to investigate the kinematics of strong ﬁeld ionization and recollision and to predict
the positions of classical cut-offs of rescattering spectra and
spectra of high-order harmonics [61, 63, 169]. The model is
particularly ﬂexible for generalizations, namely for inclusion
of the Coulomb force into the classical kinematics of the
photoelectron after its emergence at the tunnel exit, Coulomb
correction of the tunnel exit position and inclusion of the
dipole part of the electron–ion interaction. A number of
effects in the spectra of direct and rescattered photoelectrons
have been described applying the simple-man approach (see,
e.g. [69, 71, 74, 136, 138–140, 171, 172] and references
therein).
It is apparent from analysis of the quantum orbit representations of the SFA [101] or, equivalently, the ITM versions
gauge. Thus the viewpoint (ii) is partially correct: for SR
potentials there is indeed a preferable choice of gauge. Why
does the length gauge work better than the velocity gauge?
The reason for this can be understood through the saddlepoint analysis [16]: in the length gauge the Fourier transform
of the bound state wave function is singular at the saddle
point. Therefore, in this case we need not know the behavior
of the atomic wave function in the whole space, but only at
distances r ≃ 1 κ where it is given by the asymptotic (A.2). In
contrast, the velocity gauge form involves a p -dependent
transform of the wave function which, for its correct calculation, requires knowledge of the wave function in the whole
position space, including the region where it is essentially
perturbed by the laser ﬁeld. The qualitatively erroneous factor
(ω I p )4 in (136) appears from the momentum dependence of
the ground state Fourier transform. In addition, as is noted in
[159], in the length gauge the electron does not interact with
the laser ﬁeld prior to ionization, because the operator of
momentum present in the atomic Hamiltonian is equivalent in
this case to the ﬁeld-free kinetic momentum. It should be
27
J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204001
Review Article
of the Keldysh theory that the SSM follows from these
quantum-mechanical approaches in the limit of tunneling,
γ ≪ 1. Here we address two questions: (1) how should one
modify the SMM equations in order to make them applicable
at arbitrary γ and (2) are the concepts of the barrier exit point
and the tunneling time (as they appear in the Keldysh theory)
physically meaningful?
The answer to the ﬁrst question is straightforward. As the
SMM is based on the same physical assumptions as the
Keldysh model (with the Coulomb ﬁeld included if necessary), correct classical modeling must deal with real parts of
complex classical trajectories9. Therefore the initial time
instant for classical photoelectron motion is given by the real
part of the stationary point satisfying (52), t0 = Re ts . The
tunnel exit point is a real part of the electron position at this
time instant, b = Re [r0 (p, t0 )] given by (64). The initial
photoelectron velocity at the tunnel exit is given by
v0 = p + A(t0 ), etc. In a linearly polarized monochromatic
ﬁeld E (t ) = E 0 sin ωt one obtains
E0
sin ωt0 ( cosh ωτ0 − 1) ,
ω2
v0 ≡ v ( t0 ) = p + A ( t0 ) = v⊥ + ev
b ( t0 ) = −
= v⊥ −
E0
cos ωt0 ( cosh ωτ0 − 1) ,
ω
The second question is more difﬁcult. Both values, b and
ts, emerge in the Keldysh theory after application of the SPM
for calculation of the time integral and associate with the
photoelectron trajectory (64). It is usually assumed that the
integration path connecting ts and td is chosen as shown in
ﬁgure 2(a) by a solid curve: a vertical segment connecting
the stationary point ts with its real part t0 is followed by a
horizontal segment along the real time axis. With such a
path, a seemingly natural separation into the ‘sub-barrier’
(t ∈ [t0 + iτ0, t0 ]) and ‘after-barrier’ motion (t ∈ [t0, td ]) is
introduced. Within this picture, the SMM accounts for the
electron dynamics along the second shoulder of the path,
while the sub-barrier part is reduced to an instant emergence
of the electron at the tunnel exit with a given probability.
There are several alternative treatments of the sub-barrier
dynamics present in the literature, some of them involve
interpretations of the Keldysh tunneling time τ0 as a real
interval of the physical time the electron spends between the
‘moment of ionization’ and the time instant t0 when it starts
its classical motion.
These concepts considering t0 and b as physically
meaningful and even indirectly measurable values can,
however, be seriously criticized. Indeed, the trajectory (64)
and the respective actions (45) and (71) are analytic functions
in the whole complex time plane. This means that the saddlepoint result depends only on the initial and ﬁnal points ts and
td but not on the form of an integration path connecting these
points. Any path different from the standard (a few examples
are shown on ﬁgure 2) can be chosen. As the tunnel exit
position and the time t0 when the electron arrives at the real
time axis depend on the path, this invariance of the Keldysh
ionization amplitude with respect to the path deformation
means that both values b and t0 have no physical meaning in
the sense that there can be no experiment whose results would
allow concluding even indirectly that these values are equal to
this and that. This does not contradict the fact that the
introduction of such values into the theory can be useful for
calculations or for development of pictorial interpretations.
The situation becomes complicated (however, without a
principle change) when the Coulomb interaction is accounted
for via corrections to the classical action (as described in
section 6.2 and 6.3). The full action (85) is no longer an
analytic function, as the functions 1 r0 (p, t ) and 1 r03 (p, t )
have branching points in the complex time plane. In this case
the integration path is not arbitrary but should be chosen so
that it does not intersect cuts [120]. Note, that the Coulomb
corrections to the action (95) and to the photoelectron velocity
(96) are presented in a form independent on the integration
path (provided it does not intersect cuts) and therefore they do
not depend on the position of the tunnel exit as well.
A question naturally arises: how can classical simpleman calculations based on the use of a physically meaningless
(as shown above) concept of the tunnel exit yield good
agreement with experimental data and numerical results?
Indeed, within the SMM, Coulomb-corrected trajectories are
(141)
(142)
where e is a unit vector along the polarization direction. These
values are momentum-dependent (via the time of start t0
connected to the ﬁnal momentum). In the tunneling limit,
ωτ0 ≪ 1, (141) gives (139) and the longitudinal initial velocity vanishes as γ2, so that this justiﬁes the common approach
where it is set equal to zero. For γ ≃ 1, both the tunnel exit
position and the initial velocity differ considerably from their
tunneling values. In the multiphoton limit γ ≫ 1 the exit
(141) becomes ﬁeld-independent, b = 2I p ω. At a ﬁxed
ﬁeld amplitude the barrier gets narrower when ω and γ grow,
although the Keldysh tunneling time remains constant.
Longitudinal initial velocity v|| is, as follows from (142),
an unambiguously deﬁned function of the start time t0. No
longitudinal velocity spread at the instant of ionization takes
place. Therefore, calculations where this spread is heuristically added introduce an error instead of an inprovement.
In contrast, transversal spread is inherent to the classical
model due to the fact that v⊥ does not depend on t0.
Equations (141) and (142) can easily be generalized to arbitrary polarization of the laser pulse; then the transversal initial
velocity v⊥ is perpendicular to the electric ﬁeld vector E(t0 ).
Summarizing this part, when γ is not small, the initial conditions (141) and (142) following from the trajectory representation of the Keldysh model differ substantially from their
tunneling values adopted in the SMM. Already at γ ≃ 1, the
difference can have a much greater inﬂuence on the photoelectron classical kinematics than the Coulomb correction to
the tunnel exit position and the dipole correction to the
electron–ion interaction.
9
Neglecting imaginary parts of the ionization times and trajectories, which
is typical for classical models, makes them less accurate compared to the
Keldysh model and the SFA (see [173] in this context).
28
J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204001
Review Article
usually found by solving Newtonʼs equation in two ﬁelds,
restrictions in the application of the Keldysh model for the
description of molecules in intense laser pulses. These
Zr
r¨ = − E (t ) − 3
(143) restrictions already become apparent for the simplest case of a
r
diatomic molecule with frozen nuclei.
As discussed in section 8.1, the gauge dependence of the
˙
with initial conditions r (t0 ) = b , r (t0 ) = v0 with v0 taken
matrix
element leads to an erroneous pre-exponential factor in
from (142) or v0 = 0 . Obviously the ﬁnal photoelectron
momentum found in this calculation depends on the values of the rate of ionization. For atoms, a practical recipe for
b and t0. However, if the result is correct it can not be obtaining a quantitatively correct result consists in the use of
b -dependent. In order to clarify this seeming controversy we the length gauge version in an SR potential with a subsequent
compare the barrier width with the distance between the Coulomb correction of the exponent. Simple arguments show
electron and the ion at the matching time instant t* used in the that in the application to the ionization of a diatomic moleCoulomb-corrected theory of section 6.2. Combining (89), cule, the length gauge version of the SFA will immediately
(93) and (141) we ﬁnd that, in the tunneling regime, the two lead to a contradictory result [181]. This argumentation is
length parameters, the spatial matching point r* and the bar- particularly transparent for a molecule with a large internuclear separation, 2a ≫ 1 κ . In this case the ground state
rier width b, obey the same inequality
wave
function
in
a
two-center
potential
1 ≪ κr *, κb ≪ 1 γ 2F ,
(144) U (r) = U0 (r − a) + U0 (r + a) can be approximately preso the tunnel exit can be taken as a starting point for the sented as a superposition of the two atomic functions, each
electronʼs motion, as well as any other spatial point from the located at its center,
vicinity of a circle, shown in ﬁgure 2(b). In other words, in
(145)
ψ0 = A ⎡⎣ ψ0 (r − a , t ) ± ψ0 (r + a , t ) ⎤⎦ .
the tunneling regime solutions of (143) are weakly sensitive
to the value of the initial coordinate. In the multiphoton Using this initial state for calculation of the Keldysh ampliregime, the relation is different, r * ≪ E0 ω2 and b ≫ E0 ω2 tude with the interaction operator (2) and the Volkov function
(the diagram in ﬁgure 2(b) corresponds to this limit). Then the (3), we ﬁnd that the ionization potentials of two contributions
integrals in (88) and (96) contain a substantial ‘sub-barrier’ are effectively shifted by ± E (t ) a , because the saddle-point
contribution which is neglected in simple-man calculations. equations read
The results of the classical modeling along (143) become
1
(146)
( p + A( ts))2 ± E (t ) a + I p = 0.
essentially dependent on the choice of b and, generally
2
speaking, incorrect—a fact which has been widely noticed in
the literature and received various interpretations. Thus, the This result is apparently unphysical. The problem is solved by
results of the Coulomb-corrected Keldysh theory, although dressing the bound states according to the rule
they do not depend on the width of the barrier, appear virψ0 (r ± a) → ψ0 (r ± a)e∓iA(t ) a .
(147)
tually identical to those of the SMM in the tunneling regime.
For γ ⩾ 1 predictions of the two approaches can differ With this ansatz the saddle-point equation takes its standard
considerably.
form, and the amplitude reads
Closing this subsection we have to emphasize that the
M (p) = A e−ipa ± e+ipa MK (p),
(148)
conclusion made here about the unphysical nature of the
tunnel exit and the tunneling time as they appear in Keldysh
theory does not mean that the problem of time delays in where MK (p) is the single-center contribution given by (1).
strong ﬁeld processes has no physical content. Experimental The factor in parenthesis describes to some extent the orienand theoretical investigations of time delays in ionization tation dependence of the ionization rate [181]. Comparisons
have raised great interest. The discussion of this problem with the data also support the dressed length gauge verbeyond the Keldysh approximation requires a separate sion [182].
The method of dressed wave functions is, however,
review. For the problem of delays in ionization and in
limited
by the case where the internuclear separation is large,
quantum and classical systems in general, we direct the reader
so
the
wave functions of the two centers almost do not
to [175–177].
overlap. If this condition is not satisﬁed, neither of two versions of the SFA applies: the dressing recipe of the length
8.3. Ionization of molecules and dielectrics
gauge is not valid and the velocity gauge version, though
The physics of molecules driven by intense laser pulses is seemingly not problematic, gives an incorrect result, as it does
currently one of the pivotal parts of strong ﬁeld science. The for atoms. The predictions of the two gauges diverge even
ionization dynamics of even the simplest H2+ molecule is more than for the case of atoms [183]. Thus in the current
qualitatively much richer than that of an atom, due to the status of the theory, a direct application of the Keldysh matrix
effects of the orientation of the molecule with respect to the element for the description of ionization in molecules is
laser ﬁeld and of nuclear motion. The literature devoted to questionable.
In the tunneling domain, quasistatic ionization models
strong ﬁeld molecular phenomena is enormous. For overviews of this research ﬁeld we direct the reader to [178–180]. accounting for the structure of molecular orbitals are widely
The only question we address here is related to the principal used in calculations instead of the Keldysh approach. Another
(
29
)
J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204001
Review Article
possibility for a theoretical breakthrough is connected to the
application of the ZRP model with two or more centers. For a
two-center negative molecular ion, this model was introduced
in [184] and later developed in application to a circularly
polarized ﬁeld [185]. Recently, it was shown [156] that the
QQES approach described in section 7.1 can be generalized to
the case of two ZRP in an arbitrary polarized laser ﬁeld.
Within the QQES method the problem becomes exactly solvable and therefore does not suffer the above-described difﬁculty of the ionization potential shift. This makes the method
applicable at arbitrary internuclear separations. In [156],
general equations determining the complex quasienery were
derived and studied in the low-intensity limit. The question of
how the Keldysh limit emerges from this theory (as it does for
a single ZRP, see section 7.1) is not yet solved. This work is
currently in progress.
The situation becomes even more complicated when
ionization of macroscopic systems is considered. In the work
by Keldysh [1], his method was applied to the description of
inner ionization in dielectrics, i.e. a nonlinear excitation of
electrons from the valence to the conductive band. The
formalism is similar to the one for atoms with the following
difference: now both the initial and ﬁnal states are approximated by Bloch waves where the quasimomentum p is
replaced by the time-dependent velocity vp (t ) = p + A (t )
dressing procedure can be used to eliminate unphysical terms
from the saddle-point equation. For solids with normal density, where the interatomic distance is comparable to the atom
size, this representation is no longer possible and the dressing
method does not apply. On the other hand, because the Bloch
functions are spatially delocalized, ionization described as a
transition between the states (149) looks more similar to a
free–free transition in the presence of an external laser ﬁeld.
In a free space (which corresponds to the choice
up (r) = const ) the gauge invariance property would be
restored. It is unclear, however, how much a periodical spatial
dependence of the Bloch function would inﬂuence this
property. Investigation of simple model examples (not yet
performed to the best of our knowledge) where the spatial
matrix element can be explicitly calculated in both gauges
could be extremely useful for understanding this currently
open question.
In practice, only the tunneling limit of the theory when
the rate of excitation is determined by the standard tunneling
exponent [186] is reliably checked. Note that the dispersion
law (150) has the same form as in relativistic mechanics of
free particles. This formal similarity allows one to apply the
mathematical methods developed in the relativistic theory of
ionization and pair production in superintense laser ﬁelds to
the theory of laser-driven dielectrics. This interesting analogy
has been pointed out in [4].
In contrast to atoms, inner ionization of dielectrics is
tricky for a careful experimental investigation because the
interaction of exited electrons accelerated by the laser wave
with the lattice causes additional effects. Only relatively
recently, thanks to the common use of femtosecond lasers,
direct investigation of multiphoton inner ionization of crystals
has become possible (see, e.g. [187, 188] for details and
references). Studies of the laser–crystal interaction reveal a
number of new phenomena including the orientation dependence of nonlinear ionization, the development of avalanches
in the femtosecond regime, etc. For their description a theory
going far beyond the matrix element (1) with the functions
(149) is required.
Ψ pc, v (r , t ) = u vcp,(tv) (r) exp i ⎡⎣ vp (t ) r
{
−
∫
t
( ( ))
ε c, v vp t ′
⎤⎫
dt ′⎥ ⎬ .
⎦⎭
(149)
Here upc, v are periodical Bloch functions for the conductive
and the valence band and ε c, v are the respective energies
approximated by a simple dispersion law
ε (p) ≈ Δ 1 +
p2
mΔ
(150)
with m and Δ being the effective electron mass and the forbidden band width, respectively10.
In contrast to the Volkov function, (149) is not an exact
solution to the Schrödinger equation so this approximation is
rather similar to the CV ansatz discussed in section 6.1. In
combination with the fact that explicit forms of the Bloch
functions up (r) are rather complicated even for model cases,
this makes the question of gauge dependence harder than it is
for molecules. On the one hand, for solids with a relatively
large interatomic separation the wave functions of the valence
and the conductive bands can be approximately represented as
superpositions of individual atomic functions each localized
around the respective atom. Then a problem similar to the one
discussed above for the case of molecules appears: in the
amplitude of ionization which will have a form of the sum of
partial amplitudes for all possible pairs of atoms, each
amplitude will carry the factor E (t )(R i − R j ) in the exponent
(in the length gauge representation). Here R i, j are the positions of the respective atoms. Then a generalization of the
9. Conclusions and outlook
Perhaps the secret of the enduring 50 year success of the
Keldysh theory is that it precisely fulﬁlls the criterion of
‘making things as simple as possible, but not simpler’.
Although a number of modiﬁcations and generalizations of
the model have been developed over the past few decades,
some of which are described in this paper, none of them—
apart from the SFA with rescattering and the corresponding
simple-man recollision scenario—have achieved the exceptional combination of simplicity and universality inherent to
the Keldysh ionization ansatz. It is remarkable that even the
gauge was properly chosen, although this became apparent
only recently.
In its current form, with the technique of Coulomb corrections, the Keldysh theory describes the ionization of
negative ions and atoms with a quantitative accuracy. For the
10
Note that this approximation does not account for the anisotropy of the
electron effective mass.
30
J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204001
Review Article
V Frolov and O Smirnova who read the manuscript in the
preparatory stage and suggested a number of improvements.
The work was supported by the President Program for
Support of Young Russian Scientists and Leading Research
Schools (grant no. MD-5838.2013.2) and the Russian Foundation for Basic Research (project no. 13–02-00372).
Coulomb-free version of the theory, which implies an SR
electron–atom interaction, the only restriction of its applicability is the relative smallness of the reduced ﬁeld strength,
F ≪ 1. Precisely this restriction often ignored in applications.
When the Coulomb ﬁeld is taken into account, additional
limitation emerge, including K0 ≫ 1. Besides, it is required
that the size of the atomic bound state is small compared to
the electron quiver amplitude in the laser ﬁeld—a restriction
making the theory invalid for the description of ionization
from highly excited Rydberg states. In the application to the
ground state this restriction means, according to (89), that
K0 ≫ ν . The Keldysh parameter can meanwhile be formally
arbitrary.
Looking to the future, we try to outline possible prospects for the further development of the theory. There are
several conceptually simple (although important for applications) improvements waiting to be realized. This includes
extension of the Coulomb-corrected version of the theory into
the relativistic domain (a correction to the total ionization rate
and a spin correction at relativistic conditions have already
been found [4]), trajectory analysis of the CV version of the
theory, examination of yet unexplained features of ATI, like
the recently observed very-low-energy structure [189], further
application to ionization in multimode ﬁelds including XUV
+IR schemes with free-electron lasers, and more. These tasks
relate to a routine extension of the theory.
There are, however, at least three directions for fundamental improvements. They include:
Appendix A. Saddle-point expression for the
prefactor
A saddle-point calculation of the amplitude (1) involves two
steps. First, evaluation of the spatial integral (50) gives a preexponential time- and momentum-dependent factor. In both
gauges, taking ﬁrst the time integral in (1) by parts and then
integrating over the spatial coordinates we obtain
⎛
s2 ⎞
P (p, t ) = i ⎜ I p + ⎟ ψ0 (s).
2⎠
⎝
(A.1)
Here s = p (velocity gauge) or s = vp (t ) (length gauge). For
an SR potential with the radius rc the bound state wave
function at r > rc is
ψκ, l, m (r) ≈ 2Cκl κ 3 2(κr )ν − 1e−κr Ylm (n),
(A.2)
where κ = 2I p , ν = Z κ is the effective principal quantum
number, Z is the charge of the atomic residual (Z = 0, ν = 0
for negative ions, Z = 1 for neutral atoms, etc) and Ylm is the
spherical function with n = r r . A dimensionless asymptotic
coefﬁcient Cκl can be taken from tables (see, e.g. table 1 in
[4]) or calculated from the Hartree approximate formula [194]
(Cκl = 1 for hydrogen and 1/ 2 for the ZRP)
• The development of a fully gauge-invariant version of
the Keldysh ionization model. This would make the
theory applicable to the description of the ionization of
molecules, nanoparticles and bulk dielectrics to the same
extent as is currently achieved for atoms.
• Incorporation of bound states. Currently this is only
performed for a trivial case when the initial state is a
superposition of two bound atomic states. This extension
would allow a consistent theoretical description of
intermediate resonances, of ionization from Rydberg
states and of recently discovered frustrated tunneling
ionization [190].
• The development of a many-body version of the theory
allowing the inclusion of correlation effects on an equal
footing with the effect of a strong electromagnetic wave.
The very ﬁrst steps towards this goal were made by the
theory of collective tunneling [191–193] and by the
ARM method [134].
Cκ2l ≈
22ν− 2
, x != Γ (x + 1).
ν(ν + l) ! (ν − l − 1)!
(A.3)
The Fourier transform of (A.2) reads [13]
ψκlm (s) =
Cκl 2ν + 3/2κ 2ν + 1 2ν!
Ylm ( n s ),
π s2 + κ 2 ν+1
(
)
(A.4)
where n s is the unit vector along s .
Second, the integral is taken by the SPM
+∞
∫−∞
None of these tasks is currently being explored and there
is no clear understanding of how one could do this, though a
solution to any of them will give new life to the theory.
P (p, t )e−iS0(p, t ) dt ≡
∑ ( p, tsα )e−iS ( p,t
0
sα )
.
(A.5)
α
For ν = 0 the prefactor (A.1) is regular at the saddle point
v 2p (ts ) = − κ 2 and application of the SPM gives immediately
(54). When ν ≠ 0 , the prefactor is singular at the saddle point.
A correct saddle-point calculation requires expanding the
photoelectron velocity in the denominator of (A.4)
Acknowledgements
The author acknowledges numerous discussions with D
Bauer, C F Faria, A M Fedotov, M Ivanov, B M Karnakov, N
L Manakov, V D Mur, V S Popov, H R Reiss and B A Zon.
He is particularly grateful to W Becker, S P Goreslavsky, M
v2p (t ) ≈ −κ 2 − 2vp ( ts ) E ( ts )( t − ts )
= − κ 2 − 2S0′′ ( ts )( t − ts ) .
31
J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204001
Review Article
This leads to the integral
Next, expanding the Coulomb-free action
+∞
Iν =
∫−∞
e−iS0(p, t )
dt .
( t − t s )ν
Using the formula
(
)
and taking into account that
1
1
=
xν
Γ (ν )
∫0
+∞
∂S 0
=
∂p
dyy ν − 1e−xy ,
− 2π i
2iS0″( ts)
S0″ ( ts )
Γ (ν 2)
2 Γ (ν )
(
ν 2 −iS ( p, t )
0
s
)
e
∫t
td
(p + A (t ) ) dt = r0 ( td ) − r0 ( ts ),
s
∂S 0
( p, t = ts ) = 0
∂t s
one obtains for the integral (A.6) in the saddle-point
approximation:
Iν ≈
⎞
∂S 0
∂S 0 ⎛
p′ − p +
⎜ts′ − ts ⎟
∂p
∂t s ⎝
⎠
( )
S0 p′, td ≈ S0 ( p, td ) +
(A.6)
(B.4)
we obtain
.
(A.7)
( )
S0 p′, td ≈ S0 ( p, td ) − Δp [ r0 ( td ) − r0 ( ts ) ] .
(B.5)
Instead of (53) we obtain for the amplitude of ionization [16]
Collecting all the terms and taking into account that
v1(td ) = Δp we obtain:
MK (p) ≈ Cκl i1/2 − 3ν /223ν /2 + 1 κ 2ν + 1Γ (ν 2 + 1)
× ∑S0′′( tsα)−1 2 − ν 2 Ylm ( n α)e−iS0( p, ts ) ,
S ( p, td ) ≈ S0 ( p, td ) + r0 ( ts ) [ Δp − v1( ts ) ] .
(A.8)
(B.6)
α
The correction v1 diverges at t → ts , therefore the limit
has to be taken. Substituting the expansion (90) into the
equation of motion (82) we obtain
with n α = ∓ ivp (tsα ) κ being a complex ‘unit’ vector along
the saddle-point velocity vp (ts ) [16]. It is easy to check that
for ν = 0 this formula recovers the SR case (54).
v1(t ) ≈
Appendix B. Perturbative calculation of Coulomb
corrections with Hamiltonian and Lagrangian forms
of action
r0 (p, t ) v1(p, t )
∫t
td
s
( )
where r0 (p, t ) is the Coulomb-free trajectory (64), v0 = r˙0
and v1 are to be found from (81). Integrating by parts one
obtains
∫t
s
td
−
ts
∫t
td
s
td
r0 v˙1dt = r0 v1
+
ts
∫t
td
s
= ± iZ
2 I p ≡ ± iν ,
(B.8)
(B.9)
This result means that the momentum distribution
remains unaffected, while the total rate changes by the ﬁeldindependent factor exp ( ± 2ν ) ∼ 1. This result contradicts
the whole bulk of our knowledge about Coulomb effects in
strong ﬁeld ionization, including the Coulomb-corrected
ionization rates proven by comparisons with static-ﬁeld
results.
Consider now the same calculation made for the
Lagrangian form of the action (71), (72). To shorten equations
we introduce W˜ = W + F . Substituting there the same Coulomb-corrected trajectories we obtain
(B.1)
⎧1
⎫
Z
⎨ v(p, t )2 −
+ Ip⎬ dt ≈ S0 p′, td
⎩2
⎭
r (p , t )
td ⎧
Z⎫
⎨ v0 v1 − ⎬ dt
+
(B.2)
ts ⎩
r0 ⎭
v0 v1dt = r0 v1
t → ts
S ( p , t d ) ≈ S 0 ( p , t d ) ± iν .
∫
td
(B.7)
)
where ν is given by (87). Finally the Coulomb-corrected
action (in the linear approximation with respect to Z) is
Then
S ( p, td ) =
(
Then
Here we calculate Coulomb corrections to the laser-induced
action presented in the Hamiltonian (45) and the Lagrangian
(71) forms, assuming that the Coulomb perturbation is small,
so that the ﬁrst order of the perturbation theory with respect to
the atomic core charge Z is sufﬁcient:
Δp = p − p′ ∼ Z , v1 ∼ Z , Δts = ts − ts′ ∼ Z .
Z p + A ( ts )
, t → t s.
t − t s − 2I 3 2
p
( )
W˜ ( p, td ) ≈ W˜ 0 p′, td +
Z
dt . (B.3)
r0
− rpv1 ttds
32
∫t
− r1vp
td
s
td
ts
.
⎧
Z⎫
⎨ v0 v1 + E (t ) r1 + ⎬ dt
r0 ⎭
⎩
(B.10)
J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204001
Review Article
Integration by parts gives
∫t
td
td
v0 v1dt = v0 r1
s
−
ts
−
∫t
∫t
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td
td
v˙ 0 r1dt = v0 r1
s
ts
td
E (t ) r1dt .
(B.11)
s
Therefore, in the linear approximation
( )
W˜ ( p, td ) ≈ W˜ 0 p′, td +
∫t
td
s
Z
d t ± iν .
r (p , t )
(B.12)
This result looks physically meaningful. The ﬁrst term is
the Coulomb-free action calculated along the new trajectory
while the second is the Coulomb action calculated along the
Coulomb-free trajectory. The third term is identical to that of
(B.9) and gives a constant (ﬁeld- and momentum-independent) factor slightly correcting the total ionization probability.
The two principally different results (B.9) and (B.12)
obtained by calculating corrections to the same Coulomb-free
classical action demonstrate the fact that the perturbation
theory in the action is essentially sensitive to the choice of
independent variables which are different for the actions (45)
and (71).
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