Table Of ContentElectron correlation in two-photon double ionization of helium from
attosecond to FEL pulses
J. Feist1, R. Pazourek1, S. Nagele1, E. Persson1, B. I. Schneider2,3, L. A. Collins4, and
J. Burgd¨orfer1
1 InstituteforTheoreticalPhysics,ViennaUniversityofTechnology,1040Vienna,Austria,EU
2 PhysicsDivision,NationalScienceFoundation,Arlington,Virginia22230,USA
3 ElectronandAtomicPhysicsDivision,NationalInstituteofStandardsandTechnology,Gaithersburg,Maryland20899,USA
4 TheoreticalDivision,T-4,LosAlamosNationalLaboratory,LosAlamos,NewMexico87545,USA
E-mail: [email protected]
9
Abstract. We investigate the role of electron correlation in the two-photon double
0
ionization of helium for ultrashort XUV pulses with durations ranging from a
0
2 hundred attoseconds to a few femtoseconds. We perform time-dependent ab initio
calculations for pulses with mean frequencies in the so-called “sequential” regime
n
a ((cid:126)ω >54.4eV). Electron correlation induced by the time correlation between
J emission events manifests itself in the angular distribution of the ejected electrons,
6 which strongly depends on the energy sharing between them. We show that for
2 ultrashort pulses two-photon double ionization probabilities scale non-uniformly with
pulse duration depending on the energy sharing between the electrons. Most
]
h interestingly we find evidence for an interference between direct (“nonsequential”)
p and indirect (“sequential”) double photo-ionization with intermediate shake-up
- states, the strength of which is controlled by the pulse duration. This observation
m
may provide a route toward measuring the pulse duration of FEL pulses.
o
t
a PACSnumbers: 32.80.Rm,32.80.Fb,42.50.Hz
.
s
c
1. Introduction priori necessary [11, 35–40].
i
s
y In a previous paper [11], we investigated the role
h The role of electron correlation is of central interest of energy and angular correlations in the shortest
p in our understanding of atoms, molecules and solids. pulses available today, where the distinction between
[ The recent progress in the development of ultrashort “sequential” and “nonsequential” becomes obsolete. In
1 and intense light sources [1–10] provides unprecedented this contribution, we explore the dependence of two-
v opportunities to study the effects of correlation not photon double ionization (TPDI) on the pulse duration
2 only in stationary states, but also in transient states T ranging from 100 attoseconds (the duration of the
p
7 (i.e.,resonances),andeventoactivelyinducedynamical shortest pulses p∼roduced by high-harmonic generation
0
4 correlations [11]. [3]) to a few femtoseconds (the expected duration of
1. The helium atom is the simplest atomic system FEL pulse “bursts”). Tp can be used as a control knob
whereelectron-electroninteractionscanbestudied,with to change from a “direct” to an “indirect” process. In
0
9 its double ionization being the prototype reaction for a section 3, we discuss the behavior of the one-electron
0 three-body Coulomb breakup. While computationally ionization rate PDI(E)/T , which displays non-uniform
p
: challenging, the full dynamics of the helium atom can scaling with T . In section 4, we investigate the angular
v p
i still be accurately simulated in ab initio calculations correlations, with a focus on longer pulses, which reveal
X
[12]. With the advent of intense XUV pulses, the focus thedetaileddynamicsoftheTPDIprocess. Insection5,
r has shifted from single-photon double ionization [13– weshowthatforenergiesabovethethresholdassociated
a
17] and intense-IR laser ionization by rescattering ([18– with shake-up ionization of the He atom, interferences
20] and references therein) to multiphoton ionization. between sequential and nonsequential contributions can
Two-photon double ionization (TPDI) has recently be observed, the strength of which can be varied by
received considerable attention, both in the so-called changing the pulse duration. One consequence is that
“nonsequential” or “direct” regime (39.5eV < (cid:126)ω < from the size and shape of these Fano-like resonances,
54.4eV), where the electrons necessarily have to share thepulsedurationofXUVpulsesmightbededuced. All
energyviaelectron-electroninteractiontoachievedouble this information is encoded in the final joint momentum
ionization[21–34], andinthe“sequential”regime((cid:126)ω > distribution PDI(k ,k ) PDI(E ,E ,Ω ,Ω ), which
1 2 1 2 1 2
≡
54.4eV), where electron-electron interaction is not a is experimentally accessible in kinematically complete
2
COLTRIMSmeasurements[41]. Inthiscontribution,we r = 800a.u. for the longest pulses. All presented
max
focus on integrated quantities, which are more readily quantities were tested for numerical convergence and
accessible because of better statistics. Unless otherwise gauge independence.
stated, atomic units are used.
3. Pulse length dependence of TPDI
2. Method
The nature of the two-photon double ionization (TPDI)
Our theoretical approach (described in more detail in process depends strongly on the photon energy. In
[23]) is based on a direct solution of the time-dependent order to doubly ionize the helium atom (ground state
Schr¨odinger equation (TDSE) by the time-dependent energy E0 79eV), each photon must have an
close-coupling (TDCC) scheme [22, 29, 38, 42]. The energy of at≈le−ast (cid:126)ω = E0/2 39.5eV. For
TDSE is solved in its full dimensionality including all 39.5eV < (cid:126)ω < 54.4eV, a−single≈photon does not
inter-particle interactions. The laser field is linearly provide sufficient energy to ionize the He+ ion. Thus,
polarized and treated in dipole approximation. The TPDI can only occur if the two electrons exchange
interaction operator is implemented in both length and energy during the ionization process. In a temporal
velocity gauge, such that gauge independence can be picture, this implies that the “first”, already ejected,
explicitly checked. In the TDCC scheme the angular electron still has to be close to the nucleus when the
part of the wave function is expanded in coupled secondphotonisabsorbed,i.e.,bothphotonshavetobe
spherical harmonics. For the discretization of the absorbedquasi-simultaneously(ornonsequentially). For
radial functions, we employ a finite element discrete photon energies larger than the ground state energy of
variable representation (FEDVR) [43–46]. A local DVR the He+ ion ((cid:126)ω > 54.4eV), an independent-particle
basis within each finite element leads to a diagonal picture is applicable for long pulses: each electron
representation of all potential energy matrices. The absorbs one photon and electron-electron interaction is
sparse structure of the kinetic energy matrices enables a priori not required for double ionization to occur. The
efficient parallelization, giving us the possibility to firstelectronisreleasedfromtheHeatomwithanenergy
employ pulses with comparably long durations (up to a of E1 = (cid:126)ω I1, while the second electron is released
few femtoseconds) in our simulations. For the temporal from the He−+ ion with an energy of E2 = (cid:126)ω I2.
−
propagation of the wave function, we employ the short Here, I1 24.6eV (I2 54.4eV) is the first (second)
≈ ≈
iterative Lanczos (SIL) method [47–49] with adaptive ionization potential of helium. For shake-up satellites
time-step control. thepartitioningofionizationpotentialsisdifferent(I(cid:48) =
2
Dynamical information is obtained by projecting I2/n2),andsoarethepeakpositionsE1(cid:48),2,buttheoverall
the wave packet onto products of Coulomb continuum picture of sequential and independent photoionization
states. As these independent-particle Coulomb wave events remains unchanged.
functions are not solutions of the full Hamiltonian, For ultrashort pulses of a few hundred attoseconds,
projection errors are, in principle, inevitable. However, the notion of sequentiality loses its meaning. The
since we are able to propagate the wavepacket for long breakdown of the independent-particle picture and
times after the conclusion of the pulse, errors in the strongcouplingbetweentheoutgoingelectronsisinthat
asymptotic momentum distribution can be reduced to case not imposed by the necessity of energy-sharing but
the one-percent level by delaying the time of projection is enforced by the ultrashort time correlation between
untilthetwoelectronsaresufficientlyfarapartfromeach the two photoemission events occurring within Tp.
other [23]. Electron-electron interaction therefore plays a decisive
Most of the results presented were obtained at role in the correlated final momentum distribution. In
a mean photon energy of (cid:126)ω = 70eV, which would particular,theelectronsarepreferablyemittedinaback-
correspond to the sequential regime for long pulses. We to-back configuration at approximately equal energy
choose the vector potential to be of the form A(t) = sharing, corresponding to a Wannier ridge configuration
ˆzA sin2(πt/(2T ))sin(ωt) for 0 < t < 2T . The [11].
0 p p
duration T corresponds to the FWHM of the sine- A key indicator for sequential TPDI is that for
p
squared envelope function. The peak intensity was sufficiently low intensities (when ground state depletion
cdheopsleetnioansaIn0d=th1r0e1e2-oWr/mcomre2ptohoetnosnuereffetchtastagreronuengdligsitbaltee. (cid:82)is∞ne(cid:82)g∞ligIib(tl)eI),(t(cid:48))tdhte(cid:48)dttotalT2y,iewldhesrcealTes iswitthhe dPusDerqaItio∝n
−∞ t ∝ p p
Inordertoreachconvergenceoftheangulardistribution, of the laser pulse [23, 37]. This is an immediate
singleelectronangularmomentauptovaluesofl = consequence of two independent subsequent emission
1,max
l = 10 were used. The highest total angular processes, the probability for each of which increases
2,max
momentumincludedinthetimepropagationwasL = linearly with T , such that PDI (PI)2 T2.
max p seq ∼ ∼ p
3. Forextractingthefinalprobabilitydistributions,only Equivalently,foreachofthetwoprocessesawell-defined
the two-photon channels L = 0 and L = 2 were taken transitionrateW =lim PI/T exists. Thisimplies
Tp→∞ p
into account. The radial grid was composed of FEDVR thatthetotalratePDI/T ofthetwo-stepprocessgrows
seq p
elementsof4a.u.withorder11, withanextensionupto linearly with T in the limit of long pulses. By contrast,
p
3
(a) 10−1 ¯hω−I1−E2 ¯hω−I2 ¯hω−I1 ¯hω−I2+E2 (b) 60 10
75as 8
300as V] 40 6
750as e
[
u.] 10−2 1500as E2 20 4
b. 4500as 2
r
[a 0 0
/Tp 10−3 0 20 40 60
) (c) 60 10
1
E
( 8
DIP 10−4 eV] 40 6
[
2 4
E 20
2
10−50 10 20 30 40 50 60 00 20 40 60 0
E1 [eV] E1 [eV]
Figure 1. (a)Doubleionization(DI)ratePDI(E)/Tp (i.e.,DIprobabilitydividedbythepulseduration)forTPDIbyanXUVpulse
at (cid:126)ω = 70eV with different pulse durations Tp. For sufficient pulse duration, the DI rate converges to a stable value except near
the peaks of the sequential process. (b) and (c) show the two-electron energy spectrum PDI(E1,E2) for (b) Tp = 300as and (c)
Tp=750as.
thenonsequentialordirectdoubleionizationprobability the first electron is already far from the nucleus, the
PDI scaleslinearlywithT andaconvergedtransition electrons cannot exchange a sufficient amount of energy.
nonseq p
rate exists in the limit W =lim PDI /T . For each final state, there is a maximum delay t(ii)
Tp→∞ nonseq p c
For ultrashort pulses, the scaling of the ionization between ionization events that can lead to that specific
yield with T varies between T and T2 highlighting the energy sharing. This implies that the pulse has to be
p p p
non-uniform convergence over different regions of the considerably longer than this maximal delay in order to
electron emission spectrum and the breakdown of the resolve all contributions to a specific final state.
distinctionbetweendirectandindirectprocesses. Fig.1a In order to estimate the size of effect (ii), we
illustrates the dependence of the energy differential employ a simple classical model: the first electron
electron emission probability (projection of the joint is emitted with energy ESI = (cid:126)ω I1. In order
−
energy distribution Fig. 1b,c, onto the E1 (or E2) axis) to reach a specific final state with energies (E1,E2),
for different pulse durations, divided by T , dW/dE = the liberated electron has to gain or lose the energy
p
PDI(E)/Tp. This quantity converges to a duration- ∆E = min(|ESI−E1|,|ESI−E2|) by interacting with
independent cross section value (apart from constant the second electron. Therefore, the first electron can be
factors) except in the regions near E = (cid:126)ω I1 and atmostadistancerSI(t(cii))=1/∆E fromthecoreatthe
E = (cid:126)ω I , i.e., those values of the energy w−here the moment of the second photon absorption. This leads to
2
−
sequential process is allowed [40]. The peak areas grow a critical time
linearlywithTp indicativeofanoverallquadraticscaling 2(cid:112)α(α+1) ln(cid:16)2α+2(cid:112)α(α+1)+1(cid:17)
characteristic for the sequential process (cf. Fig. 2a). If t(ii) = − , (1)
one divides the yield contained in the peak areas by Tp2, c (2ESI)3/2
theresultisjustproportionaltotheproductofthesingle
with α=E /∆E.
SI
ionization cross sections for one-photon absorption from
Likewise, the spectral width of the pulse gives a
theHegroundstateandone-photonabsorptionfromthe (i)
corresponding time t = 1/∆E. Linear scaling should
He+ ground state. c
be observed for pulse durations T much longer than
c
The region within which the linear scaling prevails (i,ii) (i,ii) (i,ii)
t . Setting T 10t leads to good agreement
c c c
is determined by the pulse duration for two different ≈
with the full numerical simulation (Fig. 2b). Moreover,
reasons:
both criteria give similar results thereby precluding a
(i) Due to Fourier broadening, the photon energy is clear distinction between them. Fig. 2b displays the
not well defined for a finite pulse, limiting the energy (i,ii)
estimates T and the fraction of double ionization
c
resolution. Thus, if the broadened sequential peak
probability that scales linear with T as a function of
p
overlaps with the final energy of interest, the long-pulse
emission energy and pulse duration,
limit PDI(E) T can not be observed.
p
∝ PDI(E,T ) T
(ii) There is an intrinsic maximum time delay PDI(E,T )= p max , (2)
rel p PDI(E,T ) T
between ionization events that can lead to a specific max p
combination of final energies of the ejected electrons. where T = 4.5fs is the longest pulse we used.
max
When the second electron is ionized at a time when PDI takes on the value one when the double ionization
rel
4
(a) 10 7 (b) 60 2
−
totalTPDIyieldPDI
PDI(E=Eeq) 50
10−8 PDI(E=41eV)
1.5
40
u.] 10−9
arb. eV] 30 1
[ [
d E
el 10 10
yi −
20
0.5
10 11
− 10
10 12 0 0
−
75 125 250 500 1000 2000 4000 75 125 250 500 1000 2000 4000
Tp [as] Tp [as]
Figure 2. (a)Scalingoftwo-photondoubleionizationyieldswithpulsedurationTp (FWHMofthesin2 XUVpulses)at(cid:126)ω=70eV.
ThegreenpointsarethetotalionizationyieldPDI,theredsquaresgivethedifferentialyieldatequalenergysharingPDI(E=Eeq),
with Eeq = (2(cid:126)ω+E0)/2, and the blue diamonds give the differential yield at E = 41eV. The dashed lines show fits to quadratic
andlinearscalingwithTp forthetotalandsinglydifferentialyield. (b)ContourplotofPrDelI(E,Tp). Avalueof1forPrDelI (whitein
thecolorscaleusedhere)markstheregionwherelinearscalingofthesinglydifferentialyieldwithpulsedurationTp isobserved. The
orange lines indicate the positions of the peaks from the sequential process. The violet and green lines indicate the pulse durations
(i) (ii)
Tc andTc afterwhichlinearscalingoftheyieldwithTp isexpectedduetoFourierbroadeningofthesequentialpeakandbecause
ofthemaximumtimedelaybetweenthephotonabsorptions(seetext).
probability at energy E shows linear scaling with pulse (θ =0◦) and calculating the probability for the second
1
duration. We note that the estimate of effect (ii) could electron to be emitted into the forward half-space θ
2
≤
be validated in a time-independent perturbation theory π/2 or backward half-space θ >π/2. The probabilities
2
calculation. ThelatterdoesnotshowFourierbroadening thus defined are
butintroducesaneffectivecutofffortheinteractiontime
t(cii) because of the limited box size. P±(E1,E2)=
For long enough pulses, there is an additional (cid:90)
interestingfeatureatenergiesE =(cid:126)ω I andE = 4π2 P(E1,E2,θ1=0◦,θ2)sinθ2dθ2, (3)
(cid:126)ω I + , with = 2 2/1n2 the−exc1i−taEt2ion ene2rgy θθ22><ππ//22
2 2 n
to t−he n-tEh excitedEstate −in He+. At these energies, wherethefactor4π2 stemsfromintegrationoverφ and
1
sequential ionization via the excited ionic (shake-up) φ . The forward-backward asymmetry is then given by
2
state nl is allowed. We discuss this is in more detail in
| (cid:105) P+(E ,E ) P−(E ,E )
section 5. (E ,E )= 1 2 − 1 2 , (4)
A 1 2 P+(E ,E )+P−(E ,E )
The non-uniform scaling with T described here 1 2 1 2
p
shouldoccurforanyphotonenergywherethesequential which varies in the range [ 1,1]. Values close to zero
process is allowed. This is confirmed by calculations at indicate vanishing correlat−ion between the electrons,
(cid:126)ω = 91eV, shown in Fig. 6. At these higher photon while large absolute values identify strong angular
energies, the ionized electrons obtain higher momenta, correlations. Positive values ( > 0) indicate a
such that larger box sizes are required in the simulation preference for ejection of both eAlectrons in the same
for the same pulse duration. Therefore, the maximum direction while negative values ( <0) indicate ejection
durationofthelaserpulsewasrestrictedtoTp =1.5fs. in opposite directions. Note Athat (E1,E2) is not
A
symmetricunderexchangeofE andE ,astheemission
1 2
4. Angular correlations direction of the electron with energy E1 is fixed in the
laser polarization direction. Analogously, the reduced
Additional information on the dynamics of the two one-electron asymmetry (E ) can be determined by
1
A
ionized electrons can be extracted from the angular integrating P±(E ,E ) over the energy of the second
1 2
correlations in the TPDI process. To that end, we electron,i.e.,P±(E )=(cid:82) P±(E ,E )dE ,and (E )=
1 1 2 2 1
A
introducetheforward-backwardasymmetrydistribution (P+(E ) P−(E ))/(P+(E )+P−(E )).
1 1 1 1
(E ,E ), obtained by fixing the ejection direction of Fig. −3 shows the asymmetry of TPDI at (cid:126)ω =
1 2
A
one electron in the direction of the laser polarization 70eV photon energy for pulses of different duration
5
1 ¯hω−I1−E2 ¯hω−I2 ¯hω−I1 ¯hω−I2+E2
75as
300as
750as
0.5 1500as
4500as
)
1
E 0
(
A
0.5
−
1
− 0 10 20 30 40 50 60
E1 [eV]
Figure 3. Forward-backwardasymmetry (E1)forTPDIbyanXUVpulseat(cid:126)ω=70eV,fordifferentpulsedurationsTp. Thegray
A
linesshowtheexpectedpositionsofthepeaksforthesequentialprocess(withandwithoutshake-up).
T , from T = 75as up to T = 4500as. For the onlyelectronsemittedinoppositedirectionareobserved
p p p
shortest pulses, the electrons are dominantly ejected in bothin“sequential”((cid:126)ω >54.4eV)and“nonsequential”
opposite directions independent of energy, as observed (39.5eV < (cid:126)ω < 54.4eV) TPDI [23]. The main
previously [11]. As the duration is increased, a differenceisthatinnonsequentialTPDI,onlythatregion
stable pattern emerges: at the “sequential” peaks, is energetically accessible, such that no other angular
the electrons are essentially uncorrelated, leading to configurations are observed.
vanishing asymmetry. As most electrons are ejected in
this channel, the total (energy-integrated) asymmetry
5. Shake-up interferences
is very small for long pulses. However, for energies in
between the two main peaks at E =(cid:126)ω I and E =
(cid:126)ω−I2, theelectronsareemitted1inoppo−site1directio2ns. W(Ee re(cid:126)tωurnIn2o+w2t)oatnhdeloawdedrit(iEonal(cid:126)sωtrucIt1ures2)atenheirgghieesr
This is precisely because these final state energies are ≈ − E ≈ − −E
visible in Figs. 1 and 3. They correspond to shake-
reachedonlywhenthetwoelectronsareejectedinsucha up satellites in He+ which can serve as intermediate
configuration. This back-to-back Wannier-like emission
states in sequential TPDI. In the shake-up process,
near equal energy sharing remains pronounced even for the He+ ion is left in an excited state, while the free
long pulses. electron obtains an energy of E(cid:48) = (cid:126)ω I
1 − 1 − En
For energies outside the energy interval delimited (with the excitation energy to the n-th shell of
n
by the sequential peaks, the asymmetry is equally He+). EIn the long-pulse limit, this simply leads to the
strong, but now positive pointing to the same emission appearanceofshake-upsatellitelinesatenergiesE(cid:48) and
1
direction for both electrons. When the second electron E(cid:48) =(cid:126)ω I + in the one-electron energy spectrum.
2 − 2 En
is emitted in the same direction as the first one, the For ultrashort pulses, however, the nonsequential (or
well-known post-collision interaction [50–53] tends to direct) double ionization channel becomes available as
increase the asymmetric sharing of the available energy well and can lead to the same final states. Post-
[11]. Thedividinglinebetweenthetwodifferentregimes collision interactions (PCI) lead to a broad distribution
of ejection in opposite or in the same direction is quite of electron energies (see section 4), so that the electrons
sharp and lies directly at the position of the sequential can obtain the same final energies of EPCI = E(cid:48) and
1 2
peaks. A more complete representation of the two- EPCI = E(cid:48) as the electrons emitted via He+(nl) in
2 1
electron energy and angular correlations is presented in the sequential process. Both indistinguishable pathways
Fig. 4 for a pulse duration of T = 450as. While the lead to the same final state and thus to an interference
p
height gives the joint probability PDI(E ,E ), the color pattern in the double ionization yield, as observed
1 2
represents the asymmetry distribution (E ,E ). The in Fig. 1 and Fig. 3. This interference bears some
1 2
A
borderline between positive and negative (i.e., 0, resemblance to the well-known exchange interference
A A≈
white) is precisely near the peaks associated with the between e.g. photo-electrons and Auger electrons [54–
sequential process. In the central region in between 57]. There is, however, a fundamental difference: while
the “sequential” peaks the emission is preferentially on the exchange interference is intrinsically controlled by
opposite sides while emission into the same hemisphere atomic parameters, namely the energy and lifetime
prevails outside the main peaks. For completeness we (width) of the Auger electron, the novel interference
note that in the region between the two main peaks, observedhereistrulyadynamicaleffectpresentonlyfor
6
E
2
E
1
Figure4. CombineddoubleionizationprobabilityPDI(E1,E2)andforward-backwardasymmetry (E1,E2)afterTPDIbyanXUV
pulseat(cid:126)ω=70eVwithadurationof450as. Thez-axisgivesPDI(E1,E2)(inarbitraryunits),whileAthecolorencodestheasymmetry,
withcyantobluesignifyingnegativevalues(ejectioninoppositedirection)andyellowtoredsignifyingpositivevalues(ejectioninthe
samedirection). Vanishing correspondstowhite. ForenergieswherePDI(E1,E2)isnegligible,thecolorissettogray.
A
shortpulsesandcanbecontrolledbythepulseduration background contribution c is added to account for the
bg
T . different angular distributions of the different channels,
p
As the dependence of the yield on pulse duration which prevent complete interference. This gives
is different for the different channels (proportional to PDI(E) (qΓ/2+E E )2
Tp for the nonsequential channel, proportional to Tp2 in PDI (E) ≈cbg+cF(E E )2+−(ΓF/2)2 , (5)
the sequential channel), the observed spectrum strongly nonres − F
changes with pulse duration. For short pulses (T < The simple fitting procedure used here only works well
p
500as, cf. Fig. 1), the yield is completely dominated by for pulse durations T 1.5fs, as for shorter pulses,
p
≥
the nonsequential channel without any trace of a shake- the employed approximation for the “nonresonant”
up interference. As the pulse duration is increased, the backgroundbreaksdown,andtheshake-uppeakitselfis
sequential channel with shake-up becomes increasingly lessstrongandconsiderablybroadened. Fig.5illustrates
important. As expected from the interference of a the dependence of the obtained parameters on the pulse
relatively sharp peak with a smooth background, the duration, confirming the expected behavior: for long
peak resembles a Fano lineshape [58]. Thus, the pulses, the peaks converge to the satellite lines, i.e.,
position of the maximum is shifted from the position Lorentzians of vanishing width, such that E (cid:126)ω
F
expected in the limit of infinitely long pulses. Even I + (E (cid:126)ω I ), Γ 0, q→ −1.
2 n F 1 n
E → − − E → | | (cid:29)
for relatively long pulses (T = 4.5fs), similar to The overall strength I of the shake-up peak relative
p F
those produced in free electron lasers, the position of to the nonresonant background is obtained from the
the shake-up peak in the one-electron energy spectrum integral over the Fano lineshape, I c (q2 1)Γ.
F F
∝ −
PDI(E) is shifted by a considerable fraction of an eV. This behaves approximately linear with T , confirming
p
The structural similarity to a Fano resonance (a quasi- the scaling of the sequential shake-up channel with T2
p
discrete resonance due to the shake-up intermediate versus the scaling of the nonresonant background with
state embedded in a smooth continuum due to the T (Fig. 5c). Also shown in Fig. 5a is the position E
p max
direct double ionization) suggests to characterize the of the maximum of the spectrum PDI(E) without any
interference in terms of Fano resonance parameters further processing.
for the position E (T ), width Γ(T ), and asymmetry Such effects could possibly be observed in FEL
F p p
parameter q(T ), as well as its strength I (T ) (Fig. 5). pulses, which reach focused intensities of up to
p F p
To apply Fano’s parametrization [58], the calculated 1016W/cm2. To confirm that the results shown
energy spectrum PDI(E) is divided by the nonresonant here (calculated for 1012W/cm2) also apply for these
spectrum PDI (E), taken to be proportional to the high intensities, we performed an additional calculation
nonres
singly differential cross section as predicted from the at a peak intensity of I = 5 1015W/cm2 with
0
·
model by Horner et al [40, Eq. (8)]. Away from the a pulse duration of T = 4.5fs. The shape of
p
peaks, this fits the form of the spectrum very well. A the differential yield PDI(E) (not shown) is almost
7
58.5 2.6 6
E (a) (b) q (c)
F 2.4
5
58 E I
max 2.2 F
q
V] 57.5 V] 2 u.], 4
e e 1.8 b. 3
[ [ r
E 57 Γ a
1.6 [
2
F
56.5 ¯hω I2+ 2 1.4 I
− E 1
1.2
56
1 0
1.5 2 2.5 3 3.5 4 4.5 1.5 2 2.5 3 3.5 4 4.5 1.5 2 2.5 3 3.5 4 4.5
T [fs] T [fs] T [fs]
p p p
Figure 5. Parameters of the shake-up interference peaks around 57eV for TPDI by an XUV pulse at (cid:126)ω = 70eV obtained from
fittingtoaFanolineshape. (a)FanoresonanceenergyEF andpositionEmax ofthemaximuminthespectrum,(b)widthΓ,(c)Fano
parameterq andintegratedyieldIF fromtheshake-uppathway. Seetextfordetails.
unchanged compared to the result at 1012W/cm2 of this contribution, we became aware of work by
peak intensity, even though the ground state survival Palacios et al [59] who also observe the interference
probability is only 32%. The total double ionization between these different channels.
probability is PDI = 36%, i.e., more than a third of
the helium atoms in the laser focus are doubly ionized.
6. Summary
Even though the yield in the shake-up peak is only
0.6% of the total yield for that duration, this could be
We have presented a detailed study of the dynamics of
seen in experiment as only the integrated one-electron
the two-photon double ionization process in helium in
energy spectrum has to be observed. Moreover, from
theso-called“sequential”energyregimeforawiderange
theposition,strengthandasymmetryoftheinterference
of ultrashort pulse durations (75as to 4.5fs). We have
peaks, information on the poorly known pulse duration
shown how electron interaction and thereby correlation
of FEL pulse “bursts” could possibly be deduced.
enforced by the short pulse duration influences the
The results shown up to now were obtained at a observed energy spectra and angular distributions.
photon energy of (cid:126)ω = 70eV, where only the n = 2 The one-electron ionization rate PDI(E)/T con-
p
shake-up channel plays a role. While the qualitative verges to a stable value with increasing pulse duration
behaviour of each shake-up peak is expected to be forenergiesawayfromthesequentialpeaks(E =(cid:126)ω I
1
independent of (cid:126)ω, new intermediate ionic states nl and E = (cid:126)ω I ), giving rise to a well-defined (dir−ect)
become accessible at (cid:126)ω > I1 + n, converging| to(cid:105) differential d−oub2le ionization cross section. However,
(cid:126)ω > E0 for n (where E0E 79eV is the near the peaks where the sequential process is allowed,
− → ∞ ≈ −
ground state energy of helium). This is demonstrated PDI(E)/T grows with T . We have thus observed a
p p
in Fig. 6 at a photon energy of (cid:126)ω = 91eV. As the non-uniform scaling of the double ionization probability
shake-up probability strongly decreases with increasing with T . Even though in this spectral range the sequen-
p
n, only the peaks associated with n = 2 and n = 3 can tial process is allowed, both the direct and sequential
clearly be identified at the pulse lengths used here (up co-exist, giving rise to interferences which are induced
to Tp = 1.5fs). For longer pulses, more highly excited by the short time correlation between the two emission
stateswouldstarttoplayaroleaswell. Inthatcase,one events. Thenonsequentialchannelwithoutshake-upand
wouldneedtotakeintoaccountthatthepeaksforhigher the sequential shake-up channel, where the intermediate
noverlapwitheachotheraswellaswiththenonresonant state after one-photon absorption is an excited state of
background. theHe+ion,caninterfere. Inattosecondpulses,onlythe
It should be noted that in order to observe these nonsequential channel contributes, while in long pulses
interference effects, the asymptotic vectorial momenta (longer than the 4.5fs used here), the sequential shake-
k ,k (i.e., not only the asymptotic energies (E ,E )) upchannelwoulddominate. Forpulsedurationsofafew
1 2 1 2
of the two pathways have to coincide. The shake- femtoseconds,asobtainedinfreeelectronlasers,thetwo
up channel has an angular distribution considerably channels are similarly important, such that interference
different from that of nonsequential channel, such that can be clearly observed. This interferences may open
only partial interference between the final states is up the possibility to measure the duration of ultrashort
expected. This leads to a rich structure in the observed XUV pulses in the femtosecond regime.
angular distributions (not shown), a more detailed We have also found that the angular distributions
analysis of which is in progress. During the preparation in the final states populated by nonsequential processes
REFERENCES 8
10−2 ¯hω−I1−E2 ¯hω−I2 ¯hω−I1 ¯hω−I2+E2
450as
n=4 n=3 n=3 n=4
750as
1500as
u.] 10−3
b.
r
a
[
/Tp 10−4
)
1
E
(
DI
P 10−5
10−60 10 20 30 40 50 60 70 80 90 100
E1 [eV]
Figure 6. Doubleionization(DI)ratePDI(E)/Tp (i.e.,DIprobabilitydividedbythepulseduration)forTPDIbyanXUVpulseat
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