Cent. Eur. J. Phys. • 11(9) • 2013 • 1099-1106
DOI: 10.2478/s11534-013-0180-x
Central European Journal of Physics
Two-photon double ionization of helium: investigating
the importance of correlation in the final state
Research Article
Aleksander S. Simonsen ∗ , Sigurd Askeland, Morten Førre †
Department of Physics and Technology, University of Bergen,
N-5007 Bergen, Norway
Received 21 December 2012; accepted 22 January 2013
Abstract:
In this paper, we present theoretical results for the process of non-sequential two-photon double ionization
of helium at the photon energy 42 eV. Our approach is based on solving the time-dependent Schrödinger
equation in a B-spline based numerical framework. Information about the process is obtained by extracting
the double-ionized component by means of uncorrelated final states. The total (generalized) cross section
for the process is extracted, as well as differential cross sections resolved in electron energies and ejection
angles. We focus on the impact the final-state correlation has on the accuracy of the cross sections.
PACS (2008): 32.80.Rm, 32.80.Fb, 42.50.Hz
Keywords:
double ionization • ionization of helium • two-photon double ionization
© Versita sp. z o.o.
1.
Introduction
The pioneering work of Byron and Joachain in 1967 [1]
provided the first theoretical insight into the breakup process of helium involving the absorption of a single photon,
i.e., one-photon double ionization (OPDI). In such a process, one electron absorbs a photon and then knocks out
the other one on its way out, making electron-electron energy transfer (correlation) strictly necessary for this process to occur. With the development of high-order harmonics [2, 3] and free-electron laser (FEL) [4, 5] light
sources, experimental investigation of the role of electron correlation in few-photon processes became possi∗
†
E-mail: [email protected] (Corresponding Author)
E-mail: [email protected]
ble. The process of OPDI is today considered to be well
understood as full agreement has been obtained between
theory and experiments [6]. The corresponding process involving two photons are far more difficult to pin down, especially experimentally, due to the weak two-photon ionization yield. When two photons are involved, the sequential process differs from the non-sequential (direct) one as
electron energy transfer is a prerequisite only for the latter. Non-sequential two-photon double ionization (TPDI)
of helium has been studied extensively during the last
decade, both theoretically [7–26] and experimentally [27–
32]. The sizable interest in this particular problem is presumably due to the fact that helium is the simplest system
where such processes occur, and that insight into TPDI
of helium could pave the way for further understanding
of few-photon multi ionization processes in more complex
systems. In addition, there are large discrepancies in the
reported results, and as such, the process is not consid1099
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Two-photon double ionization of helium: investigating the importance of correlation in the final state
ered to be fully understood.
In this paper, we revisit the problem of non-sequential
TPDI of helium using a theoretical approach. We solve
the problem from first-principles, i.e., the time-dependent
Schrödinger equation (TDSE) is solved in full dimensionality. Our numerical framework has been applied to several two-electron problems in the past, including stabilization of helium [33], two-photon double ionization of
helium [21, 34], the hydride ion [35], and molecular hydrogen [36]. With this numerical scheme, we are able to
solve the TDSE for two-electron systems for relatively
long periods of time (in the order of 10 fs) in boxes extending to several hundred atomic units in the radial direction. One of the interesting challenges that arises in
our approach is how to separate double from single ionization. Due to the electron-electron interaction, the independent particle picture cannot describe such systems and
the single-ionized component is hard to tell apart from the
double-ionized component in terms of total energy. In order to isolate these channels, we apply a similar method
as that employed by [17] and others. In short one neglects the Coulomb interactions between the electrons in
final states and thereby obtains a way to separate individual electron energies. This approximation is expected
to be more accurate when the electrons are far away from
each other as the electron-electron interaction becomes
less important at large distances. The distance between
the electrons can be controlled by letting the electrons
drift further apart after the end of the laser pulse (postpropagation). In this work we pay special attention to
how the amount of post-propagation influences the obtained results. We calculate the total (generalized) as
well as differential cross sections for the process of nonsequential TPDI of helium. In order to trigger the TPDI
events, we expose the helium ground state to a linearly
polarized laser pulse with a mean photon energy of 42 eV,
which is below the sequential energy threshold, i.e., only
non-sequential TPDI will occur. Our findings are in good
agreement with some of the previous findings both for the
total cross section and the triple-differential cross section
(TDCS). In addition to comparing the magnitudes of the
obtained total cross section, extracted at different times
after the conclusion of the laser pulse, we also compare
the shapes of the differential cross sections.
Atomic units, where ~, me , e and a0 are scaled to unity,
are applied throughout this paper unless stated otherwise.
2.
Theory and methods
The problem of non-sequential two-photon double ionization (TPDI) is approached by solving the time-dependent
Schrödinger equation (TDSE) numerically. In the case of
helium, assuming an infinitely heavy nucleus, the problem
takes six spatial dimensions, three for each electron. The
laser field is modelled in the dipole approximation and
the system Hamiltonian takes the following form in the
velocity gauge,
2
2
1
b = p1 + p2 − 1 − 1 +
+ A(t) · (p1 + p2 ), (1)
H
2
2
r1 r2 |r1 − r2 |
where A(t) is the time-dependent classical vector potential
describing the laser pulse. The linearly polarized laser
pulse is modelled with a sine-squared carrier-envelope,
A(t) = A0 sin2
πt
T
cos(ωt)û,
(2)
where û is the unit vector pointing in the direction of polarization, A0 = E0 /ω, E0 being the electric field amplitude at
peak intensity, ω is the angular frequency, and T is the total duration of the laser pulse. In our approach, spherical
coordinates are employed and the time-dependent wave
function is expanded in a basis of B-splines [37] and coupled spherical harmonics,
Ψ(r1 , r2 , Ω1 , Ω2 , t) =
X
i,j,k
ci,j,k (t)
Bi (r1 ) Bj (r2 ) L,M
Yl1 ,l2 (Ω1 , Ω2 ),
r1
r2
(3)
B(r) being a B-spline function and YlL,M
(Ω
,
Ω
)
being
a
1
2
,l
1 2
coupled spherical harmonic. k is a combined index for
the angular basis functions, running over all the allowed
combinations of l1 , l2 , L and M. The coupled spherical
harmonics are weighted sums over spherical harmonics,
YlL,M
=
1 ,l2
X
m
(Ω2 ), (4)
hl1 , l2 , m, M − m|L, MiYlm1 (Ω1 )YlM−m
2
where the weights are the Clebsch-Gordan coefficients.
By choosing the polarization direction to be along the
z−axis, the system is azimuthally symmetric at all times.
This means we only need to include coupled spherical
harmonics with M = 0. Furthermore, the angular part of
the basis is truncated at lmax = 7 and Lmax = 5. It has
previously been shown that this number of partial waves
is sufficient to obtain converged results [17, 21]. In order
to resolve the electron-continuum accurately in terms of
energy, 248 B-splines of order 6 in each radial dimension are required. When these splines are distributed in
our radial box, extending to 250 a.u., we obtain reliable
continuum-states up to 60 eV, which is more than sufficient
to represent the energy domain of the outgoing electrons.
By using the expansion in Eq. (3), the time-dependent
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Aleksander S. Simonsen, Sigurd Askeland, Morten Førre
Schrödinger equation becomes a system of ordinary differential equations,
iS
∂
c(t) = H(t)c(t),
∂t
(5)
where H(t) is the Hamiltonian matrix and c(t) is the vector of the coefficients defined by Eq. (3). S is the basis
overlap matrix, made necessary by the non-orthogonal Bspline functions. Equation (5) is solved numerically with
the Cayley-Hamilton propagation scheme, which takes the
form
S+
i∆t
i∆t
H(t) c(t + ∆t) = S −
H(t) c(t).
2
2
(6)
The initial condition c(t = 0) is the set of coefficients
that corresponds to the helium ground state wave function.
These are obtained by partial diagonalization of the fieldfree Hamiltonian matrix. With the matrix dimensions at
hand, ordinary linear system solvers will not suffice and
our numerical schemes are based on a Krylov subspace
expansion of the matrix on the left-hand side of Eq. (6).
The main principles of our numerical scheme are described
in [21].
Extraction of physical information regarding the process of
TPDI is a non-trivial matter as the single-continuum part
of the wave packet overlaps with the double-continuum
part in terms of the total energy. We approximate the
double-continuum by products of one-electron Coulomb
He+
(r), and project them onto the
waves with Z = 2, φE,l
different angular channels of the final wave function,
Ψl1 ,l2 ,L (r1 , r2 , t = τ),
Pl1 ,l2 ,L (E1 , E2 )
l ,l2 ,L
= |hφEHe1 ,l1 (r1 )|hφEHe2 ,l2 (r2 )|Ψf1
+
+
(r1 , r2 , t = τ)i|2 ,
(7)
where Pl1 ,l2 ,L (E1 , E2 ) is the probability contribution at time
τ from the angular channel (l1 , l2 , L) that electrons one
and two have the energies E1 and E2 , respectively. When
considering non-sequential two-photon processes, a generalized cross section is introduced. Any further use of the
term cross section refers to this generalized one, defined
as
2
ω
Pion
σ=
,
(8)
I0
TEff
where I0 is the peak intensity and TEff is the effective pulse
35
duration, given as TEff = 128
T for a sine-squared pulse
[17]. Pion is the total probability for two-photon double
ionization,
Pion =
X XZ Z
L∈2Z∗ l1 ,l2
Pl1 ,l1 ,L (E1 , E2 )dE1 dE2 ,
(9)
where only the positive energies are integrated over. Due
to the L = 0 symmetry in the helium ground state and
the selection rules in the dipole approximation, only even
L’s contribute in a two-photon process. The probability
is further resolved in the electron ejection angles in this
way [17, 38, 39],
P(E1 , E2 , Ω1 , Ω2 ) =
2
X − iπ (l +l )+i(σ +σ ) L,M=0
l1
l2
2 1 2
Y
(Ω
,
Ω
)P
(E
,
E
)
e
1
2
l1 ,l2 ,L
1
2 ,
l1 ,l2
l1 ,l2 ,L
(10)
√
where σl = arg Γ(1 + l − iZ / 2E) is the Coulomb phase.
From Eq. (10), the triple differential cross sections (TDCS)
are obtained by scaling the probability to units of cross
section and integrating out the energy of electron two,
2
Z
ω
1
P(E1 , E2 , Ω2 , Ω2 )dE2
I0
TEff
(11)
While the electron-electron interaction is included
throughout the entire propagation of the wave function,
Eq. (7) does not take into account any final state correlation. As such, the electrons should be far away from
each other in order to increase the overlap between uncorrelated Coulomb waves and the propagated wave function. If the wave function is propagated in a field-free environment (post-propagation), any ionized electrons will
drift away from each other and hence reduce the magnitude of the Coulombic repulsion between them. As such,
post-propagating the solution wave function for as long as
possible will yield more accurate results. The maximum
propagation time is in practice limited by the extent of
the radial box, as the double-ionized wave packet must
be completely contained at all times.
dσ
=
dE1 dΩ1 dΩ2
3.
Results
In this section we present the results of our simulation
of non-sequential two-photon double ionization (TPDI) of
helium at 42 eV photon energy. The laser pulse applied
in our calculations has a 1012 W/cm2 peak intensity, and
is shaped by a sine-squared envelope function. The total duration of the pulse is 4.03 fs which corresponds to
41 optical cycles for our photon energy. The initial condition of the time-dependent Schrödinger equation is the
helium ground state, obtained by partial diagonalization
of the field-free Hamiltonian of helium, i.e., Eq.(1) without
the last term. In our expansion basis with lmax = 7, we
obtain the ground state energy E = −2.90357 a.u., which
is in satisfactory agreement with the benchmark value reported in [40], E = −2.90367 a.u., obtained with the same
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Two-photon double ionization of helium: investigating the importance of correlation in the final state
number of partial waves. The time-dependent Schrödinger
equation is solved for the complete duration of the laser
pulse, and for an additional 3.94 fs (40 optical cycles) of
field-free propagation. At the edges of the radial box, extending to 250 a.u., an absorber dampens the high energy
portions of the single-ionized wave-packet. This minimizes reflections that may contaminate the results. We
extract the double-ionized wave-packet at several instants
of time after the conclusion of the pulse. Figure 1 depicts
the probability density of the two-electron wave function
as a function of the electrons distances to the nucleus,
r1 and r2 respectively. The two upper panels show the total wave functions while the isolated double-ionized wave
packets are displayed in the two lower. The First and
third panels from the top correspond to the wave packets
extracted just after the conclusion of the laser pulse while
the second and forth panels from the top correspond to the
wave packets after 40 optical cycles of post propagation.
From the isolated double-ionized wave-packets, the total
cross section is obtained. It is plotted as a function of
post-propagation time in Figure 2. All our values are in
the range from σ = 0.454 × 10−52 cm4 s to σ = 0.464 ×
10−52 cm4 s which are in close agreement with the value
reported in [17]. The largest deviation due to different
choices of post-propagation time is 2.12 percent, which is
also similar to the value reported in [17]. Interestingly,
the total cross section seems to increase monotonically
towards some asymptotic value as the electrons are moved
further apart from each other. And, as such, it seems like
the method of extraction of the double-ionized component
by means of projections onto uncorrelated final states is
accurate to within a few percent.
We now turn our attention to the single-differential cross
section (SDCS), i.e., the likelyhood of double ionization as
a function of how the electrons share the available excess
energy (energy sharing). The SDCS extracted at different
moments of post-propagation time is depicted in Figure 3.
Naturally, the error in the total cross section, evident in
Figure 2, is inherited by the differential quantities, but as
a matter of fact, the shape of the SDCS seems to be almost
independent on post-propagation time, c.f., lower panel in
Figure 3. In the lower panel the differential cross sections are scaled to fit to the same area under the curves.
It is interesting to note that there is almost perfect agreement between the shapes of the SDCSs extracted at a
large variety of times after the termination of the laser
pulse. As such, it seems like the shape of the relative
energy sharing is already converged and that the change
in the SDCS is attributed only to a scaling factor associated with the (tiny) increase in the total cross section, c.f.,
Figure 2. As for typical energy distributions associated
with non-sequential TPDI processes, the energy resolved
Figure 1.
Probability density plots for the propagated two-electron
wave function. The probability is presented as a function
of r1 and r2 , i.e., the distances from the nucleus to electrons one and two, respectively. In descending order the
panels depict, 1) the final propagated wave function just
after the conclusion of the laser pulse, 2) The final propagated wave function 40 optical cycles later, 3) the isolated
double-ionized wave packet just after the conclusion of the
pulse, and 4) the isolated double-ionized wave packet 40
optical cycles later.
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Aleksander S. Simonsen, Sigurd Askeland, Morten Førre
0.466
0.460
Total cross section
0.462
SDCS (10 cm s)
Cross section (10 cm s)
0.464
0.460
0.458
0.456
0.454
0
0.445
0.440
0.430
30
35
40
5
15
10
20
25
Time of post-propagation (Number of optical cycles)
The total (generalized) cross section for non-sequential
two-photon double ionization of helium at the photon energy of 42 eV as a function of time after the conclusion of
the laser pulse. The duration is given in number of optical
cycles.
cross section shows that equal energy-sharing is the least
probable outcome, resulting in the U-shaped curves.
Now, turning to the triple-differential cross section
(TDCS), we have focused on the angular distribution in
the co-planar geometry, i.e., φ1 = φ2 = 0. Our TDCSs
are visualized as angular distributions in the case where
electron one has the energy E1 = 2.5 eV. Similar to others,
we have focused on the conditional angular distributions
in the cases where electron one has the fixed angles 0, 30,
60 and 90 degrees relative to the direction of polarization.
Our calculations are in good agreement with previous findings presented in [17, 23], both in shape and in magnitude,
c.f., Fig 4. One can observe a strong forward-backward
asymmetry for all the angular distributions, indicating that
the second electron will favour the opposite direction of
electron one. A forward-backward scattering parallel to
the polarization direction of the laser field seems most
favourable. The angular distributions for electron two is
also shown in Figure 5, where the cross sections are plotted for different periods of post propagation. The general
trend seems to be that the conditional TDCS increases in
magnitude with post-propagation time, in accordance with
our observations in the total cross section and the SDCS.
However, as opposed to the total cross section and the
SDCS, the conditional TDCS does not seem to always
follow this trend. This is evident for the case 60◦ , where
the leftmost lobe is decreasing in magnitude with longer
periods of post-propagation.
In order to investigate this further, the angular distributions have also been scaled by applying the condition that
the integral over the second electron angle is fixed. As opposed to the scaled SDCSs shown in Figure 3, the scaled
angular distributions are not in quantitative agreement.
One of the more interesting cases is the scaled angular
0.2
0.460
0.4
0.6
E / (E + E )
T+
T+
T+
T+
T+
T+
T+
T+
0.455
scaled SDCS (10 cm s)
Figure 2.
0.450
0 optcial cycles
6 optcial cycles
12 optcial cycles
18 optcial cycles
24 optcial cycles
30 optcial cycles
36 optcial cycles
40 optcial cycles
0.435
0.452
0.450
T+
T+
T+
T+
T+
T+
T+
T+
0.455
0.450
0.445
0.8
0 optical cycles
6 optical cycles
12 optical cycles
18 optical cycles
24 optical cycles
30 optical cycles
36 optical cycles
40 optical cycles
0.440
0.435
0.430
0.2
0.4
0.6
E / (E + E )
0.8
Figure 3.
Single-differential cross section (SDCS) for nonsequential two-photon double ionization of helium at
the photon energy of 42 eV. The SDCSs are resolved
in relative energy between the electrons. The different
curves show the SDCS extracted at different times (given
in number of optical cycles) after the conclusion of the
laser pulse. The SDCSs are depicted in the upper panel
whereas the lower panel shows the same SDCSs scaled
to the same total cross section.
distribution when the ejection angle of electron one is kept
fixed at θ1 = 90◦ . This is drawn in Figure 6 where the upper and lower panels depict the sectors θ2 = [155◦ , 230◦ ]
and θ2 = [310◦ , 25◦ ], respectively. First of all, it is interesting that the lobes get longer with increased postpropagation time, even if the angular distributions are
scaled to the same total value. Moreover, the black
curve, corresponding to the longest post-propagation, is
the right-most in the upper panel and the left-most in
the lower panel while the golden curve (shortest postpropagation) is its opposite. This means that for longer
periods of post-propagation, the lobes are slighly shifted
towards θ2 = 270◦ , i.e., back-to-back scattering. This
is due to the long-range nature of the electron-electron
interaction, the repulsive force bending the electron trajectories away from each other. This is what we observe
in Figure 6 where electron one is ejected with the angle
θ1 = 90◦ and the lobes in the probability distributions
shift towards θ2 = 270◦ .
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Two-photon double ionization of helium: investigating the importance of correlation in the final state
Present Results
[17]
[23]
15
10
5
0
0
50
100
10
5
0
Present Results
[17]
[23]
12
300
TDCS (10 cm s sr eV )
TDCS (10 cm s sr eV )
12
15
350
150 200 250
Angle (degrees)
10
8
6
4
0
50
100
100
150 200 250
Angle (degrees)
300
350
T + 0 optical cycles
T + 10 optical cycles
T + 30 optical cycles
T + 40 optical cycles
8
6
4
3.0
350
0
Present Results
[17]
[23]
3.5
150 200 250
Angle (degrees)
300
2.5
2.0
1.5
1.0
0
350
Present Results
[17]
[23]
0.8
1.0
TDCS (10 cm s sr eV )
0.8
300
0.6
0.4
0.2
350
T + 0 optical cycles
T + 10 optical cycles
T + 30 optical cycles
T + 40 optical cycles
1.0
0.5
150 200 250
Angle (degrees)
300
1.5
0.0
100
150 200 250
Angle (degrees)
2.0
0.5
50
100
2.5
0.0
0
50
3.0
TDCS (10 cm s sr eV )
TDCS (10 cm s sr eV )
50
2
3.5
TDCS (10 cm s sr eV )
0
10
2
0
T + 0 optical cycles
T + 10 optical cycles
T + 30 optical cycles
T + 40 optical cycles
20
TDCS (10 cm s sr eV )
TDCS (10 cm s sr eV )
20
0
50
100
150 200 250
Angle (degrees)
300
350
T + 0 optical cycles
T + 10 optical cycles
T + 30 optical cycles
T + 40 optical cycles
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Figure 4.
0
50
100
150 200 250
Angle (degrees)
300
350
Conditional angular distributions for electron two, Eq. (11),
in a non-sequential two-photon double ionization process
at the photon energy of 42 eV. The angular distributions
are extracted in the case of co-planar geometry, i.e., φ1 =
φ2 = 0 and with electron one having the energy E1 =
2.5 eV. The results are presented together with the calculations reported in [17] and [23]. The panels depict, in descending order, the angular distributions with electron one
having the fixed angles 0, 30, 60 and 90 degrees, respectively. The vertical dashed lines show the fixed angles for
electron one. The left-right arrows in the polar plots indicate the polarization axis.
0.0
Figure 5.
0
50
100
150 200 250
Angle (degrees)
300
350
Angular distributions extracted in the case of co-planar
geometry, i.e., φ1 = φ2 = 0 and with E1 = 2.5 eV being the
energy of electron one. The different curves correspond to
varying choices of the post-propagation time. The panels
depict, in descending order, the angular distributions with
electron one having the fixed angles 0, 30 ,60 and 90 degrees, respectively. The vertical dashed lines show the
fixed angles for electron one. The left-right arrows in the
polar plots indicate the polarization axis.
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Scaled TDCS (10 cm s sr ev )
0.9
T + 0 optical cycles
T + 10 optical cycles
T + 30 optical cycles
T + 40 optical cycles
0.8
0.7
0.6
0.5
0.4
0.3
Acknowledgments
0.2
0.1
0.0
160
170
180 190 200
Angle (degrees)
0.9
Scaled TDCS (10 cm s sr eV )
magnitude, but their shapes seem to be well-converged
even right after the conclusion of the laser pulse. For the
TDCS, we observe small changes in the shapes of the angular distributions, which are attributed to the long-range
nature of the electron-electron interaction.
210
220
230
T + 0 optical cycles
T + 10 optical cycles
T + 30 optical cycles
T + 40 optical cycles
0.8
0.7
This work was supported by the Bergen Research Foundation and the Norwegian metacenter for computational
science (Notur). All calculations were performed on the
Cray XE6 (Hexagon) supercomputer installation at Parallab, University of Bergen (Norway).
0.6
0.5
References
0.4
0.3
0.2
0.1
0.0
310
Figure 6.
320
330
340 350 360
Angle (degrees)
370
380
Scaled angular distributions for electron two, Eq. (11) in
a non-sequential two-photon double ionization process in
helium at the photon energy of 42 eV. The scaling factor is
determined such that the integral over the angular distribution is constant. The angular distributions are extracted
in the case of co-planar geometry, i.e., φ1 = φ2 = 0 and
with the energy E1 = 2.5 eV for electron one. Electron
one is also fixed at the angle θ1 = 90◦ and the different
curves represent the cross section as a function of θ2 for
various amounts of post-propagation. The upper panel
shows the sector θ2 = [155◦ , 230◦ ] while the lower panel
depicts θ2 = [310◦ , 25◦ ]. The dotted lines in the polar plots
enclose the depicted sectors. The left-right arrows in the
polar plots indicate the polarization axis.
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
4.
Concluding remarks
The process of non-sequential two-photon double ionization (TPDI) of helium has been investigated at the photon
energy 42 eV. Total (generalized) and differential cross
sections were extracted from a numerical solution to the
time-dependent Schrödinger equation. Our findings are in
excellent agreement with the results presented in [17, 23]
for the total cross section as well as the triple-differential
cross section. The impact of the post-propagation time has
been discussed, and in accordance with earlier findings,
the magnitude of the total cross section seems to increase
monotonically, converging slowly to some finite asymptotic
value. From this we anticipate that our extracted value for
the cross section is correct within an error of a few percent. Interestingly, the differential cross sections, both
energy-resolved and angular-resolved, seem to change in
[12]
[13]
[14]
[15]
[16]
[17]
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