Time-Dependent B-Spline R-Matrix Approach to Double Ionization of Atoms by XUV Laser Pulses

4.8 MB·English

by  Xiaoxu Guan

Time-Dependent B-Spline R-Matrix Approach to Double
9 Ionization of Atoms by XUV Laser Pulses
0
0
2
XiaoxuGuan†,1,O Zatsarinny†,2, CJ Noble†,§,3,K Bartschat†,4, and
n
a BI Schneider‡,5
J
6 †DepartmentofPhysicsandAstronomy,DrakeUniversity,DesMoines,IA50311,USA
1 §ComputationalScienceandEngineeringDepartment,DaresburyLaboratory,Warrington

WA44AD,UK
]
h ‡PhysicsDivision,NationalScienceFoundation,Arlington,Virgina22230,USA
p
- E-mail:[email protected],[email protected],[email protected],
m [email protected],[email protected]
o
t
a
.
s Submittedto: J.Phys. B:At. Mol. Opt. Phys.
c
i
s
y Abstract. We present an ab initio and non-perturbative time-dependent approach to the
h problem of double ionization of a general atom driven by intense XUV laser pulses. After
p
using a highly flexible B-Spline R-matrix method to generate field-free Hamiltonian and
[
electric dipole matrices, the initial state is propagated in time using an efficient Arnoldi-

1
Lanczosscheme. Test calculations for double ionization of He by a single laser pulse yield
v
3 good agreement with benchmark results obtained with other methods. The method is then
7 appliedtotwo-colorpump-probeprocesses,forwhichmomentumandenergydistributionsof
3
thetwooutgoingelectronsarepresented.
2
.
1
0
9 PACSnumbers:42.50.Hz,32.60.+i,32.80.Rm,32.80.Fb
0
:
v
i 1. Introduction
X
r
a The two-photon double ionization (DI) of the helium atom induced by intense short XUV
laserpulseshasreceivedconsiderableattentionfromboththeoristsandexperimentalistsalike.
Insteadoflistingalargenumberofreferenceshere,wenotethatmuchoftherecentworkwas
quoted in recent papers [1, 2]. Even within the past few months, however, several additional
papers appeared.
Given the intensities and lengths of the laser pulses involved, the numerical approaches
used to tackle this problem are all essentially attempts to solve the time-dependent
Schro¨dinger equation (TDSE), beginning with a well-defined initial state before the laser
strikes and then propagating this state in the presence of the laser field by one of a number
of numerical approaches. Once the laser is switched off, various probabilities and, in some
cases, generalized cross sectionscan beextracted.
DoubleIonizationof AtomsbyXUVLaser Pulses 2
Over the past two years, our group has been working on the development of a general
ab initio theoretical approach, which is applicable to complex targets beyond (quasi) two-
electron systems. In two recent papers [3, 4] we outlined how field-free Hamiltonian and
electricdipolematricesgenerated withthehighlyflexibleB-SplineR-matrix(BSR) [5]suite
of codes may be combined with an efficient Arnoldi-Lanczos time propagation scheme to
describe the interaction of short intense laser pulses with a complex atom, leading to multi-
photon excitation and single ionization. The key points of our method are the following:
1) We employ the BSR code, which allows for the use of non-orthogonal orbital sets, to
generate field-free Hamiltonian and dipole matrices. 2) We then set up an efficient Arnoldi-
Lanczos scheme to propagate the initial state in time. 3) Finally, we extract the information
by standardprojectionschemes.
In the present paper, we report on the extension of this approach and the corresponding
computer code [6] to allow for two electrons in the continuum and hence the possibility
to describe double ionization processes. After outlining the general method, we present
a test application to the He problem, for which many benchmark results are available for
comparison. Finally, we use the method to investigate pump-probe processes involving two
XUV laser pulseswhosecharacteristics, includingatimedelay, are assumed to becontrolled
separately inthecorrespondingexperiment.
Unlessspecified otherwise,atomicunits(a.u.) are usedthroughoutthismanuscript.
2. THEORY
2.1. The TDBSRApproach
The ingredients of an appropriate theoretical and computational formulation require an
accurateandefficientgenerationoftheHamiltonianandtheelectron fieldinteractionmatrix
−
elements, as well as an optimal approach to propagate the TDSE in real time. As mentioned
above, there have been numerous calculations for two-electron systems such as He and also
H . While these investigations emphasize the important role of two-electron systems in
2
studyingelectron electroncorrelationinthepresenceofastronglaserfield,initspresumably
−
purest form, experiments with He atoms are difficult and other noble gases, such as Ne and
Ar, areoften favored by theexperimentalcommunity.
Fully ab initio theoretical approaches, which are applicable to complex targets beyond
(quasi) two-electron systems, are still rare. For (infinitely) long interaction times, the R-
matrix Floquet ansatz [7] has been highly successful. A critical ingredient of this method is
the general atomic R-matrix method developed over many years by Burke and collaborators
in Belfast. A modification of the method, allowing for relatively long though finite-length
pulseswas describedby Plummerand Noble[8]. Recently, atime-dependentformulation[9]
of this method was applied to short-pulse laser interactions with Ar [10] and a pump-probe
XUV scenario forNe[11].
Following our recent work on excitation and single-ionization of Ne [3] and Ar [4], we
now describe what needs to be done to extend our approach to double ionization. We start
DoubleIonizationof AtomsbyXUVLaser Pulses 3
withthetime-dependentSchro¨dingerequation
∂
i Ψ(r ,...,r ;t) = [H (r ,...,r )+V(r ,...,r ;t)]Ψ(r ,...,r ;t) (1)
1 N 0 1 N 1 N 1 N
∂t
for the N-electron wavefunction Ψ(r ,...,r ;t), where H (r ,...,r ) is the field-free
1 N 0 1 N
Hamiltonian containing the kinetic energy of the N electrons, their potential energy in the
field ofthenucleus,and theirmutualCoulombrepulsion,while
N
V(r ,...,r ;t) = E(t) r (2)
1 N i
·
Xi=1
represents theinteractionoftheelectrons withthelaserfield E(t) in thedipolelengthform.
Thetasks to be carried outin order to computationallysolvethis equationand to extract
thephysicalinformationofinterest are:
1. Generate arepresentationofthefield-free Hamiltoniananditseigenstates;theseinclude
the initial bound state, other bound states, autoionizing states, as well as single-
continuumanddouble-continuumstatestorepresentelectronscatteringfromtheresidual
ion.
2. Generate thedipolematrices torepresent thecouplingto thelaserfield.
3. Propagatetheinitialboundstateuntilsometimeafter thelaserfield is turnedoff.
4. Extract thephysicallyrelevantinformationfrom thefinal state.
Of particular interest in the experiments mentioned above are processes, in which
one, two, or even more electrons undergo significant changes in their quantum state in the
presenceofanatomiccore. Theseincludeexcitation,singleanddoubleionization,ionization
plus simultaneous excitation, or inner-shell ionization with subsequent rearrangement in the
hollow ion. The latter processes, in particular, can only be investigated in systems beyond
the frequently studied two-electron helium atom or H molecule. This generalization to two
2
electronsoutsideamulti-electroncoreisfarfromtrivial,buttheflexibilityoftheBSRmethod
ishighlyadvantageousforthetasks1and2. Incontrasttomanyotherapproaches[12,13,14],
ourmethodisformulatedinasufficientlygeneralwaytobeapplicabletocomplexatoms,such
as inert gases other than helium and even open-shell systems with non-vanishing spin and
orbitalangular momenta. In reality,ofcourse, thesizeoftheproblemsthat can behandled is
alsodetermined bytheavailablecomputationalresources.
The solution of the TDSE requires an accurate and efficient generation of the
Hamiltonian and electron field interaction matrix elements. In order to achieve this goal,
−
weapproximatethetime-dependentwavefunctionas
Ψ(r ,...,r ;t) C (t)Φ (r ,...,r ). (3)
1 N q q 1 N
≈
Xq
TheΦ (r ,...,r )areasetoftime-independentN-electronstatesformedfromappropriately
q 1 N
symmetrizedproductsofatomicorbitals. Theyare expandedas
Φ (r ,...,r ) = a Θ (x ,...,x ;rˆ σ ;rˆ σ )R (r )R (r ). (4)
q 1 N ijcq c 1 N−2 N−1 N−1 N N i N−1 j N
A
Xc,i,j
DoubleIonizationof AtomsbyXUVLaser Pulses 4
Here is the anti-symmetrization operator, the Θ (x ,...,x ;rˆ σ ;rˆ σ ) are
c 1 N−2 N−1 N−1 N N
A
channel functions involving the space and spin coordinates (x ) of N 2 core electrons
i
−
coupled to the angular (rˆ) and spin (σ) coordinates of the two outer electrons, R (r) is a
i
radial basis function, and the a are expansion coefficients. Although resembling a close-
ijcq
coupling ansatz with two continuum electrons, the expansion (4) contains bound states and
singly ionized states as well. In general, the atomic oritals, R (r), are not orthogonal to one
i
another or to the orbitals used to describe the atomic core. If orthogonality constraints are
imposed on these functions, additional terms would need to be added to the expansion to
relax theconstraints. Thispossibilitystillexistsas an optioninourcomputercode.
In addition to simplifying the expansion, a significant advantage of not forcing such
orthogonality conditions is the flexibility gained by being able to tailor the optimization
procedures to the individual neutral, ionic, and continuum orbitals. In the BSR code, the
outer orbitals (i.e., the R functions above) are expanded in B-splines. Factors that depend
on angular and spin momenta are separated from the radial degrees of freedom through
the construction of the channel functions. Since many Hamiltonian matrix elements share
common features, this enables the production of a “formula tape”, resulting in an efficient
procedureto generatetherequiredmatrixelements.
When theexpansion(3)isinserted intotheSchro¨dingerequation,weobtain
∂
iS C(t) = [H +E(t)D] C(t), (5)
0
∂t ·
whereS istheoverlapmatrixofthebasisfunctions,H and D are matrixrepresentationsof
0
the field-free Hamiltonian and the dipole coupling matrices, and C(t) is the time-dependent
coefficient vectorin Eq.(3).
The price to pay for the flexibility in the BSR approach, at least initially, is the
representationofthefield-freeHamiltonianandthedipolematricesinanon-orthogonalbasis.
In our previous work [3], we described two methods of combining the BSR method with a
highly efficient Arnoldi-Lanczos propagation scheme. In the first one, we diagonalized the
overlap matrix S and transformed the problem back to an orthogonal basis before applying
the standard propagation scheme [15, 16]. Alternatively, the propagation can be done more
directly in the non-orthogonal basis. This only requires a Cholesky decomposition of S, but
someadditionaloperationsatevery timestep.
A third approach involves a transformation into the eigenbasis by solving the field-free
generalized eigenvalue problem first. Details can be found in Guan et al [4]. This method,
also used byLaulan and Bachau [17], has themajoradvantagethat thetransformationmakes
it possibleto cut unphysically high eigenvalues and the corresponding eigenvectors from the
time propagation scheme, thereby making the scheme more stable. At the same time, it
simplifies the definition of the initial state and the extraction of the physically interesting
information, and it once again allows for the use of the standard Arnoldi-Lanczos time
propagation scheme. It was applied in the present work as well. We checked that the results
werestableagainstavariationofboththecut-offenergy(5.5a.u. fortheresultsshownbelow)
and thetime-step(100 stepsperopticalcycle).
We first tested our method for the He problem, i.e., two electrons in the presence of a
DoubleIonizationof AtomsbyXUVLaser Pulses 5
bareHe2+ core, andwillshowsomeexampleresultsbelow. Withaboxsizeof60a.u. and81
B-splines distributed over this range, with a spacing of 1.0 a.u. near the edge of the box and
smaller steps near the origin, the ranks of the field-free Hamiltonian blocks ranged between
10,000and20,000. Weexpecttheserankstoincreaseto50,000 100,000foracomplextarget
−
likeNe. Fortunately,it has becomerather straightforwardto diagonalizematrices ofthat size
and perform the transformationof thedipolematrices to the eigenbasis on massivelyparallel
computingplatforms.
2.2. Extractionof thecrosssectionsintwo-photondoubleionizationprocess
Detailsregardingtheextractionofthefullydifferential(energiesandanglesresolved)aswell
asthetotalcrosssectionweregivenbyGuanetal[2]andthereforewillnotberepeated here.
Tomakethispaperself-consistent,however,wesummarizethemostimportantformulasused
for single-colorprocesses and then indicate what has to be donein thecase of morethan one
laser.
In a time-dependent formulation, the dependence of the N-photon cross section on the
number of photons absorbed not only occurs through the factor (ω/I )N, where ω is the
0
angular frequency and I is the peak intensity of the laser, but also through the effective
0
interaction time. The result for the latter depends explicitly on the number of photons being
absorbed. Fortwo photonsitis givenby[18]
τ
T(N) f2N(t)dt (6)
eff ≡ Z
0
withtheresult
T(2) = 35τ/128 (7)
eff
forasine-squared pulseshapeofdurationofτ.
The energy sharing between the two escaping electrons can be uniquely determined
through the hyperangle α tan−1(k /k ). We set E = E cos2α and E = E sin2α,
2 1 1 exc 2 exc
≡
where E = 2ω I2+ is the excess energy for two-photon double ionization. For a given
exc
−
energy sharingα, thetriple-differentialcross section(TDCS) can bewrittenas
d3σ ω 2 1 k′
= dk′dk′k′2k′2δ(α tan−1( 2)) Ψ(−) Ψ(t) 2, (8)
dαdkˆ1dkˆ2 (cid:16)I0(cid:17) Te(ff2) Z 1 2 1 2 − k1′ |h k1′,k2′| i|
where Ψ(t) is the time-propagated wavefunction. Also, the singlet two-electron continuum
| i
wavefunctionsatisfyingtheincoming-waveboundarycondition( ) isgivenby
−
1
Ψ(−) (r ,r ) = Φ(−)(r )Φ(−)(r )+ Φ(−)(r )Φ(−)(r ) . (9)
k1,k2 1 2 √2h k1 1 k2 2 k2 1 k1 2 i
Withtheone-electron Coulombfunctiongivenby
1
Φ(−)(r) = ile−iσl(k)ϕ(c)(r)Y (rˆ)Y∗ (kˆ), (10)
k k kl lm lm
Xlm
and theasymptoticbehavior
2 Z lπ
ϕ(c)(r) sin(kr + ln2kr +σ ), (11)
kl r−→ rπ k − 2 l
→∞
DoubleIonizationof AtomsbyXUVLaser Pulses 6
with σ as the Coulomb phase, the two-electron Coulomb function (9) is normalized in
l
momentumspaceaccording to
Ψ(−) Ψ(−) = δ(k k′)δ(k k′)+δ(k k′)δ(k k′). (12)
h k1,k2| k1′,k2′i 1 − 1 2 − 2 1 − 2 2 − 1
In the present work, we first generated the one-electron Coulomb functions ϕ(c)(r) by
kl
employingtheroutineCOULFGofBarnett[19],onthesamegridusedfortheB-splines. This
choice guarantees a self-consistent grid representation of the time-evolved wavepacket and
the final continuum states. We then expressed the function in terms of our B-spline basis in
orderto usethestandardintegrationschemes inthat basis.
It is important to remember that there are two indistinguishable electrons in the final
channel, and hence the energy sharings described by α and π/2 α represent the same
−
observable event. Therefore, we either need to consider 0 α π/4 or π/4 α π/2 to
≤ ≤ ≤ ≤
avoiddoublecounting. TheTDCSwithrespect totheenergy ofoneelectronis thengivenby
d3σ 1 ω 2 1
= dk′dk′k′δ(k′ k′ tanα)
dE1dkˆ1dkˆ2 k1k2cos2α(cid:16)I0(cid:17) Te(ff2) Z 1 2 1 2 − 1
2
χ (k′,k′) LM(kˆ ,kˆ ) L (k′,k′) . (13)
×(cid:12)(cid:12)L=X0,2,l1l2l1l2 1 2 Yl1l2 1 2 Fl1l2 1 2 (cid:12)(cid:12)
(cid:12) (cid:12)
Herewe havedefined thetwo-electronprojection
L (k ,k ) ΨL(−) Ψ(t) , (14)
Fl1l2 1 2 ≡ h k1l1,k2l2| i
withthephasefactor
χ (k ,k ) = ( i)l1+l2ei(σl1(k1)+σl2(k2)), (15)
l1l2 1 2 −
and
ΨL(−) ϕ(c) (r )ϕ(c) (r ) L (kˆ ,kˆ ) (16)
k1l1,k2l2 ≡ A k1l1 1 k2l2 2 Yl1l2 1 2
n o
as the properly antisymmetrized product of two one-electron radial Coulomb functions and
coupled spherical harmonics L for the angular part. [Since M = 0 for linearly polarized
Yl1l2
radiation,weomitittosimplifythenotation.]
By collecting all ionization events, the total cross section for two-photon double
ionizationis obtainedas
d3σ ω 2 1 d2 (k ,k )
σ = dα dkˆ dkˆ = dk dk P 1 2 , (17)
NS Z Z 1 2dαdkˆ1dkˆ2 (cid:16)I0(cid:17) Te(ff2) Z 1 2 dk1dk2
wherewehaveintroducedthemomentumdistribution
d2 (k ,k )
P 1 2 = L (k ,k ) 2. (18)
dk dk |Fl1l2 1 2 |
1 2 LX=0,2Xl1l2
Similarly,themomentumdistributionisgivenby
d2 (E ,E ) 1 d2 (k ,k )
E 1 2 = P 1 2 . (19)
dE dE k k dk dk
1 2 1 2 1 2
Below we will also show results obtained for two linearly polarized XUX laser pulses,
forwhichtheelectricfield is givenby
E(t) = [E f (t)sin(ω t)+E f (t T )sin(ω (t T ))]e . (20)
1 1 1 2 2 d 2 d z
− −
DoubleIonizationof AtomsbyXUVLaser Pulses 7
Here ω and ω are the two central frequencies while T is the time delay between the two
1 2 d
pulses. Thetotaltimedurationcan bewrittenas τ = max τ ,τ +T .
1 2 d
{ }
Note that this time duration depends on the durations of pulse 1, pulse 2, and the time
delay between them. Hence, the definition of an “effective interaction time” is by no means
straightforwardandevena“generalizedcrosssection”(whichshouldbeindependentofthese
parameters) cannot be defined in the standard sense. Momentum distributions, however, can
be obtained by calculating and then squaring the magnitude of the function (14). Energy
distributionsfortheHecasewere recently reportedby Foumouoet al [20].
3. RESULTS AND DISCUSSION
As a first test of our newly developed approach, we applied it to the single-color two-photon
doubleionizationofhelium,forwhichavarietyofbenchmarkdataexistforcomparison(see,
forexample,refs.[1,2]andreferencestherein). Inthisparticulartest,weconfinedthesystem
to a spatial box of r = 60a.u. We used 81 B-splines of order 8 to span this configuration
max
space, with the knot sequence chosen in such a way that the spatial variation of the wave-
function close to the nucleus (determined by the nuclear charge Z) and far away from the
nucleus(determinedbythehighestenergyofafreeelectron)couldberepresentedaccurately.
Havingobtainedtheinitialstateofthesystembydiagonalizingthefield-freeHamiltonian
ofthe1Se symmetryandtakingtheeigenvectortothelowesteigenvalue( 2.90330a.u.inour
−
particular case), the time propagation in the laser field is accomplished through the Arnoldi-
Lanczos algorithm. Compared to other time-propagation approaches, such as leapfrog or
a split-operator approach [21], the present scheme allows us to take relatively large steps
in time. Specifically, using only 100 steps per optical cycle (o.c.) is sufficient to achieve
converged solutionsoftheTDSE forthecases presentedin thispaper.
Theresultspresentedbelowwereobtainedwithasine-squaredpulsewithapeakintensity
of5 1014W/m2 and atimedurationof10 opticalcycles. Forthiscase, wefirst checked the
×
total cross sections for photon energies of 42 and 50eV, respectively. These values were
chosen to ensure that the box size and the length of the laser pulse were suitable to properly
define a generalized cross section for two-photon double ionization. The present results of
4.02 10−53cm4sand2.20 10−52cm4sagreeinasatisfactorywaywiththecorresponding
× ×
values of 3.76 10−53cm4s and 1.89 10−52cm4s obtained in an entirely different finite-
× ×
elementdiscrete-variablerepresentation(FE-DVR) [2].
3.1. Triple-differentialcrosssectionsfor one-colortwo-photondoubleionization
Figure 1 displays the triple-differential cross section for two-photon double ionization of
helium in a 10-cycle sine-squared laser pulse of central photon energy 42eV and peak
intensity 5 1014W/cm2 for equal energy sharing (E = E = 2.5eV) of the two outgoing
1 2
×
electrons. These results are for the coplanar geometry, where the electric field vector of the
linearly polarized laser field and the momentum vectors of the two escaping electrons all lie
inthesameplane.
DoubleIonizationof AtomsbyXUVLaser Pulses 8
12 8
eV) 10 θ1=0◦ Present eV) 7 θ1=30◦
2/sr 8 FE-DVR 2/sr 6
s s 5
4m 4m
c 6 c 4
5510− 4 5510− 3
S( S( 2
DC 2 DC 1
T T
0 0
0 60 120 180 240 300 360 0 60 120 180 240 300 360
θ2(degree) θ2(degree)
0.5 8
eV) 0.4 θ1=90◦ eV) 7 θ1=150◦
2/sr 2/sr 6
4ms 0.3 4ms 5
c c 4
550− 0.2 550− 3
1 1
CS( 0.1 CS( 2
D D 1
T T
0.0 0
0 60 120 180 240 300 360 0 60 120 180 240 300 360
θ2(degree) θ2(degree)
Figure 1. The coplanar triple-differentialcross section for two-photondouble ionization of
heliumina10-cyclesine-squaredlaserpulseofcentralphotonenergy42eVandpeakintensity
5 1014W/cm2forequalenergysharing(E1 =E2 =2.5eV)ofthetwooutgoingelectrons.
×
Theanglelistedinthefigureistheanglebetweenthelaserpolarizationvectorandoneofthe
two escapingelectrons, while the emission angle of the second electron varies. The present
TDBSRresultsarecomparedwiththosefromourpreviousFE-DVRcalculation[2].
Given the very different implementations of the current TDBSR method and the FE-
DVRapproach,weseesatisfactoryagreementbetween thepresentresultsandthoseobtained
earlier. Note that these results were obtained with (L ,l ,l ) = (3,3,3). We
max 1,max 2,max
are aware of potential convergence problems with these parameters, especially for kinematic
situations where the cross section is small (e.g., at θ = 90◦). However, since the principal
1
motivation for performing these particular calculations was to test the method and the
accompanying computer code, we consider the test successful and hence move on to the
two-color-caseinthenextsubsection.
3.2. Momentumand energydistributionsfortwo-colortwo-photondoubleionization
We now consider the process of double photoionization by absorption of two photons at
different central photon energies. In other words, the target helium atom is exposed to the
irradiation by two laser pulses, of potentially different frequencies and with a controllable
timedelay. Thisallowsustostudythemechanismofthebreakupproblembyapplyingpulses
of various durations for each pulse and also modifying the time delay between them. The
following two-color laser parameters were considered in this work: pulse 1 has a central
photon energy of 35.3eV, a timeduration of 12o.c., and a peak intensity of 1 1014W/cm2,
×
while the corresponding parameters for pulse 2 are 57.1eV, 14o.c., and 1 1013W/cm2,
×
DoubleIonizationof AtomsbyXUVLaser Pulses 9
0.06 0.06 0.06
0.04 0.04 0.04
u.) u.) u.)
a. 0.02 a. 0.02 a. 0.02
( ( (
d d d
el 0.00 el 0.00 el 0.00
fi fi fi
c c c
ctri-0.02 ctri-0.02 ctri-0.02
e e e
El El El
-0.04 -0.04 -0.04
-0.06 -0.06 -0.06
0 10 20 30 40 50 60 0 10 20 30 40 50 0 10 20 30 40 50 60
Time(a.u.) Time(a.u.) Time(a.u.)
Figure2. Electricfieldofthetwo-colorlaserpulse. Thelaserparametersare: ω1 =35.3eV,
τ1 =12o.c.atapeakintensityof1014W/cm2(solidcurve),andω2 =57.1eV,τ2 =14o.c.at
apeakintensityof1013W/cm2 (dashedcurve). Thetime delaysbetweenthemaximaofthe
twopulsesare,fromlefttoright, 16.5,0.0,and16.5a.u.
−
respectively. Asimilartwo-colorlaserpulsewasstudiedintherecentworkbyFoumouoetal
[20].
In thepresentwork,weareinterestedinthemechanismfortheejectionoftwoelectrons
when the time delay between the two pulses is varied. Figure 2 displays the time-dependent
electric fields of the two-color pulses for time delays of 16.5a.u. (i.e., pulse 2 comes first),
−
no delay (both pulses come simultaneously, and a delay of +16.5a.u. (pulse 1 comes first).
Notethat+16.5a.ucorrespondto approximately400attoseconds.
Figure 3 depicts the momentum and energy distributions of the two escaping electrons
for the above two-color laser pulses. For the negative time delay with pulse 2 arriving first,
we observe a strong peak around E = E 6.7eV, as well as two weaker peaks around
1 2
≃
(E ,E ) (2.7,32)eVand(E ,E ) (32,2.7)eV,respectively. Thissuggestthatthetarget
1 2 1 2
≃ ≃
electrons predominantly absorb the two photons in the highly correlated way similar to the
nonsequential double ionization mechanism, even though the photon energies are different.
Recent calculations by Feist et al [22], using a single-color laser, also demonstrate that the
conventional scenario of “direct” vs “sequential” two-photon double ionization breaks down
forattosecondXUV pulses,evenwhenthesequentialprocess isenergetically allowed.
In the above two-color case, the two ejected electrons share the excess energy of
E = ω + ω I2+ 13.5eV almost equally. However, there is apparently another
exc 1 2
− ≃
ionization channel open for the present laser parameters. Since the target electrons interact
with pulse 2 of energy 57.1eV first, the first electron is ejected with a kinetic energy of
about 32 ( 57.1 24.6)eV before the second electron is ionized with an energy of
≃ −
2.7 ( 57.1 54.4)eV from the ground state of the residual He+(1s) ion. This channel
≃ −
is characteristic of sequential ionization by two photons of the same high frequency and very
different from the direct channel. In fact, in this case there is no electron left for the second
pulseto interactwith.
To improvethevisibilityof theremainingfigures, we willshow theenergy distributions
only in the energy region below 20eV for the cases of zero and positive time delays. When
there is no time delay between the two maxima of the electric fields, the principal structure
observed is stillaround E = E 6.7eV in the energy distribution. The mechanism in this
1 2
≃
DoubleIonizationof AtomsbyXUVLaser Pulses 10
2.0 1.2 40 2.5
35
2.0
1.5 30
0.8
u.) V)25 1.5
(a. 1.0 (e20
k2 E2 1.0
15
0.4
0.5 10
0.5
5
0.0 0.0 0 0.0
0.0 0.5 1.0 1.5 2.0 0 5 10 15 20 25 30 35 40
k1(a.u.) E1 (eV)
2.0 1.5 20 3.0
2.5
1.5 15
1.0 2.0
u.) V)
(a. 1.0 (e10 1.5
k2 E2
0.5 1.0
0.5 5
0.5
0.0 0.0 0 0.0
0.0 0.5 1.0 1.5 2.0 0 5 10 15 20
k1(a.u.) E1 (eV)
2.0 2.0 20 5.0
4.0
1.5 1.5 15
u.) V) 3.0
(a. 1.0 1.0 (e10
k2 E2 2.0
0.5 0.5 5
1.0
0.0 0.0 0 0.0
0.0 0.5 1.0 1.5 2.0 0 5 10 15 20
k1(a.u.) E1 (eV)
Figure 3. Momentum (left column) and energy (right column) distributions of the two
escapingelectronsinatwo-colorlaserpulse. Thelaserparametersarethesameasinfigure2.
Thecolorbarsforthefirstrow(negativetimedelay)correspondtounitsof10−5a.u.,whilethe
othertworowsaregiveninunitsof10−4a.u. Thetimedelaysbetweenthetwolaserpulses,
fromtoptobottom,are 16.5,0.0,and+16.5a.u.,respectively.
−
caseis thussimilarto whatweobservedfor thenegativetimedelay.
Ontheotherhand,whenthetimedelayisincreasedfurther,forexampleupto+16.5a.u.,
the two electrons are ejected a very different way. In this case, the main structure revealed
from figure 3 exhibits two peaks at (E ,E ) (10.0,2.5)eV and (2.5,10.0)eV. This is due
1 2
≃
tothefactthatpulse1comesfirstandhence,afterabsorptionofonephotonoffrequency(ω )
1
one electron with the kinetic energy of 10.7 (= 35.3 24.6)eV is ejected. The time delay
−
of +16.5a.u. is sufficiently long for the residual He+ ion to relax into its ground state after