Impact of two-electron dynamics and correlations on high-order harmonic generation in He

0.69 MB·

by  Anton N. Artemyev

Impact of two-electron dynamics and correlations
on high-order harmonic generation in He
Anton N. Artemyev,1 Lorenz S. Cederbaum,2 and Philipp V. Demekhin1,3,∗
1Institut fu¨r Physik und CINSaT, Universita¨t Kassel, Heinrich-Plett-Str. 40, 34132 Kassel, Germany
2Theoretische Chemie, Physikalisch-Chemisches Institut,
Universita¨t Heidelberg, Im Neuenheimer Feld 229, 69120 Heidelberg, Germany
3Research Institute of Physics, Southern Federal University, Stachkiav.194, 344090 Rostov-on-Don, Russia
(Dated: February 14, 2017)
7
1 The interaction of a helium atom with intense short 800nm laser pulse is studied theoretically
0 beyondthesingle-active-electron approximation. Forthispurpose,thetime-dependentSchr¨odinger
2 equation for the two-electron wave packet driven by a linearly-polarized infrared pulse is solved
bythetime-dependentrestricted-active-spaceconfiguration-interactionmethod(TD-RASCI)inthe
b
dipolevelocitygauge. Bysystematicallyextendingthespaceofactiveconfigurations,weinvestigate
e
F theroleofthecollectivetwo-electrondynamicsinthestrongfieldionizationandhigh-orderharmonic
generation(HHG)processes. Ournumericalresultsdemonstratethatallowing bothelectronsinHe
3 tobe dynamically active results in a considerable extension of thecomputed HHGspectrum.
1
PACSnumbers: 32.80.Rm,33.20.Xx,42.65.Ky
]
s
u
l I. INTRODUCTION active-electron (SAE) approximation (for recent results
c
see, e.g., Refs. [12–20] and references therein).
-
m Anelectronreleasedfromanatomormoleculeandfur- The role of inactive electrons, which are kept frozen
t ther on steered in the field of the parent ion by intense within the SAE approximation, is not negligible for the
a laser fields gives rise to many fundamental phenomena strong-fieldprocesses. However,exactnumericalsolution
.
s [1]. For instance, the field driven electron-ion recombi- of the time-dependent Schr¨odinger equation (TDSE) for
c
i nation results in the emission of coherent radiation with many-electron systems in laser fields is very formidable
s frequencieswhichareoddintegermultiplesofthecarrier task. At present, it has been realized only for He atom
y
frequency of exciting pulse, a phenomenon knownas the [21–27] to calculate single- and double-electron multi-
h
p high-order harmonic generation (HHG) process. Over photon ionization rates. The method of full dimensional
[ the last two decades, HHG has been intensively studied numerical integration of TDSE for two electrons, devel-
experimentallyandtheoretically(see,e.g.,reviewarticles opedintheseworks,utilizesabasissetofcoupledspheri-
2
[2–4] and references therein), owing to its powerful ap- calharmonics[21],anditisparticularlyefficienttostudy
v
0 plication to the generationof coherent XUV laser pulses strong-fieldproblemswhichcanbedescribedbyimplying
1 down to the attosecond regime [5–8]. very small radial grids of about 100 Bohr.
3 Inprinciple,highharmonicscanbegeneratedbytran- There are several theoretical studies of HHG process
3 sitions among excited bound electronic states as well performedbeyondthe SAEapproximationinatoms[28–
0 as by the involvement of unbound continuum electronic 31] and molecules [32–36]. Ref. [28], for instance, pro-
.
1 states. Detailedcalculationsonanarrayofquantumdots poses generalization of the three-step model for many-
0 with only bound discrete electronic states have demon- electronsystems by introducing perturbative corrections
7 stratedanefficientHHG[9]. However,asthecurrentex- due to exchange and electron correlations. A more con-
1 periments on atoms and molecules are performed in the sistent tracking of electron correlations and dynamics
:
v strong-field ionization regime, below we concentrate on is provided by the multi-configuration time-dependent
i this regime. Here, the essential mechanism behind the Hartree-Fock (MCTDHF) method [37–42]. Its straight-
X
HHG process is explained within the simplified three- forwardimplementationtothesolutionofthestrong-field
ar step model [1, 10, 11], in which an electron: (i) es- problems is, however, a challenging computational task
capes from the potential formed by the superposition [43,44]. Inorderto be ableto studyHHG byMCTDHF
of the ionic core and linearly-polarized laser field po- method in realistic systems, one can either lower dimen-
tentials; (ii) is accelerated back to the parent ion dur- sionality of the problem [35] (i.e., consider only one or
ing the next half-cycle of the driving pulse; and (iii) re- twodimensionsforeachelectron),or/andneglecttheex-
combineswiththeionemittingtherebyhigh-energypho- change interaction[29] (i.e., utilize the MCTDH method
tons. The validity of this illustrative model has been [45]).
confirmed by numerous theoretical studies of HHG pro- Alternatively, relaxing the full configuration interac-
cessinatomsandmoleculesperformedwithinthesingle- tion character of MCTDHF by cleverly selecting and in-
corporating only the important electron correlation and
dynamicaleffectsintotheansatzforthewavepacket,one
can fully catch the essential physics of the process at
∗Electronicaddress: [email protected] hand, and at the same time reduce considerably the nu-
2
merical effort and thus make the computation tractable. photons (the ionizationpotentialis 24.587eV [55]). The
This can be realized by limiting the space of active elec- correspondingKeldysh[56]parameterγ =0.64indicates
tron configurations utilizing, e.g., the time-dependent thationizationtakesplaceintheintermediateregimebe-
restricted-active-space configuration-interaction method tweenthestrong-fieldtunnel-ionization(γ 1)andmul-
≪
(TD-RASCI, [46, 47]), the time-dependent generalized- tiphotonionization(γ 1)extremes. Atthechosenfield
≫
active-space configuration-interaction (TD-GASCI, [36, strength, the rates for double ionization of He are very
48]), or the time-dependent restricted-active-space self- small compared to its single ionization rates [23, 26, 57].
consistent-field theory (TD-RASSCF, [30]). The latter We may, therefore, neglect in the present study of the
approach has already been applied to calculate HHG HHG process the double ionization and permit only one
spectra of the 1D beryllium atom [30]. of the electrons of He to be ionized by the pulse. Nev-
Recently[47],wehaveappliedthe TD-RASCI method ertheless, we allow the bound electron to interact with
developed in Ref. [46] to investigate the photoioniza- the laserfield as wellas with the photoelectron,i.e., it is
tion of He by intense high-frequency laser pulses. In fully active but kept bound.
the present work, we utilize this method to investigate To accomplish the above description, we utilized the
the influence of the correlative electron dynamics on the following symmetrized ansatz for the spatial part of the
HHG spectra of this simplest system with two electrons, totaltwo-electronwavefunctionΨ(~r ,~r ,t)inthesinglet
1 2
treatedeachinthree dimensions(i.e., wetreatherea six spin state:
dimensionalproblem). Thepaperisorganizedasfollows.
Sec.II outlinesourtheoreticalapproachandjustifies the Ψ(~r ,~r ,t)= a (t)φ (~r )φ (~r )
1 2 α α 1 α 2
present choice of the active space of electron configura- α
X
tions. In the numerical calculations, we systematically 1
increased the space of active configurations and proceed + bαα′(t)√2[φα(~r1)φα′(~r2)+φα′(~r1)φα(~r2)]
asfaraswecould. Thesenumericalresultsarepresented αX>α′
1
and discussed in Sec. III. We conclude with a brief sum-
+ [φ (~r )ψ (~r ,t)+ψ (~r ,t)φ (~r )]. (2)
α 1 β 2 β 1 α 2
mary. √2
αβ
X
As justified above, the wave function (2) is constructed
II. THEORY by using two different mutually-orthogonal one-electron
spatial basis sets φ (~r) and ψ (~r,t) . The for-
α β
{ } { }
The present theoretical approach is fully described in mer describes dynamics of the electron which remains
our previous work [47]. More details on its numerical bound to the nucleus, and it includes selected discrete
implementationcanbefoundinRefs.[49–53]. Therefore, orbitals φα(~r) φnℓm(~r) . The latter is formed by
{ ≡ }
only essential relevant points are discussed below. the time-dependent wave packets of a photoelectron
Wedescribethelight-matterinteractioninthevelocity {ψβ(~r,t)≡ψℓαm(~r,t)}, which are built to be orthogonal
gauge, which is the most suitable gauge for strong field to all incorporated discrete orbitals, i.e., φα ψβ(t) =0.
h | i
problems [54], since it ensures rapid convergence of the In order to describe these one-electron basis sets,
numerical solution over angular momentum of released we applied the finite-element discrete-variable repre-
photoelectrons[17]. Intheelectricdipoleapproximation, sentation (FEDVR) scheme and introduced the three-
the total Hamiltonian governing dynamics of two elec- dimensional basis element ξλ(~r) as:
tronsofHeexposedtointensecoherentlinearly-polarized
χ (r)
ik
laser pulse reads (atomic units are used throughout) ξλ(~r) ξik,ℓm(~r)= Yℓm(θ,ϕ). (3)
≡ r
1 1 2 2 1 Here,theradialcoordinateisrepresentedbythebasisset
Hˆ(t)= ~2 ~2 +
−2∇1− 2∇2− r − r ~r ~r of the normalized Lagrange polynomials χik(r) [49–53]
1 2 1 2
| − | constructedoveraGauss-Lobattogrid r (indexiruns
ik
−i(∇z1 +∇z2)A0g(t)sin(ωt). (1) over the finite intervals [ri,ri+1] and i{ndex}k counts the
basisfunctions ineachinterval). UsingthisFEDVR,the
Here, g(t) is the time-envelope of the pulse, ω is its car-
normalized stationary orbitals and the time-dependent
rier frequency, is the peak amplitude of the vector
0
A wave packets can be expanded as follows (note that λ
potential (the vector potential and the electric field vec- ≡
tor are related via E = ∂tA), and the peak intensity {ik,ℓm} is four-dimensional index):
of the pulse is I0 = 8ωπ2α−A20 (α ≃ 1/137.036 is the fine φα(~r)= dλαξλ(~r), (4a)
structure constant, and 1 a.u. of intensity is equal to
λ
6.43641 1015 W/cm2). X
×
The present calculations were performed for laser
ψ (~r,t)= cβ(t)ξ (~r). (4b)
pulses with carrier frequency of ω = 0.0569 a.u. (cor- β λ λ
responding to a wavelength of λ = 800 nm) and peak Xλ
intensity of 5 1014 W/cm2. For this photon energy, Intheusedbasissetofthe three-dimensionalelements
×
ionization of He requires the absorption of at least 16 Eq. (3), all matrix elements of the Hamiltonian (1) can
3
beevaluatedanalytically. Thecorrespondingexplicitex- there in the dipole length gauge. In the velocity gauge
pressions can be found in our previous work [47], apart used here, the dipole transition matrix element reads
fromthelight-matterinteractiontermwhichwastreated
χik ∂ sinθ ∂ χi′k′ (2ℓ+3)(ℓ m+1)(ℓ+m+1) δi,i′δk,k′
ξλ z ξλ′ = Yℓm cosθ Yℓ′m′ = − δℓ′,ℓ+1
h |∇ | i (cid:28) r (cid:12) ∂r − r ∂θ(cid:12) r (cid:29) "r 2ℓ+1 rik
(cid:12) (cid:12)
(cid:12) (cid:12) ∞
(ℓ m+1)(ℓ+m(cid:12)+1) (ℓ(cid:12) m)(ℓ+m) d ′ δi,i′δk,k′
+ − δℓ′,ℓ+1+ − δℓ′,ℓ−1 χik χi′k′dr (ℓ +1) δm,m′,
s (2ℓ+3)(2ℓ+1) s(2ℓ+1)(2ℓ−1) !Z0 dr − rik 
  (5)
where the first derivative dχ /dr can be evaluated ana- we performed a few key calculations for a sine-squared
ik
lytically via Eqs. (10) from Ref. [47]. pulse of the same length g(t) = sin2(π t ). The size of
Tf
Evolutionofthetotalwavefunction(2)intimeisgiven the radial box was chosen to be R = 3500 a.u. The
max
by the vector of the time-dependent expansion coeffi- interval [0,R ] was represented by 1750 equidistant
max
cients A~(t) = aα(t);bαα′(t);cλβ(t) , which was propa- finite elements of the 2 a.u. size, each covered by 10
Gauss-Lobatto points. The photoelectron wave packets
gated accordingnto the Hamiltoniano(1):
were described by the partial harmonics with ℓ 50.
≤
Finally, in order to avoid reflection of the wave packets
A~(t+∆t)=exp iPHˆ(t)P∆t A~(t). (6)
− fromtheboundaryatRmax,theyweremultipliedateach
n o time-step by the following mask-function [13, 59]
Theone-particleprojectorP =1 φ φ inEq.(6)
− α| αih α|
actsonthe cβ(t) subspaceandensurestheorthogonal- 1, r <R
λ P 0
g(r)= 1 (10)
ritiyedcoountdibtyionnthheφsαh|ooψrβt(-itt)eira=ti0v.eLTahneczporsopmaegtahtoiodn[5w8a].s Tcahre- ( cos π2Rmra−xR−0R0 8 , R0 <r <Rmax,
initial groundstate A~(t=0) wasobtained by the propa- (cid:16) h i(cid:17)
The mask-function (10) was set-in at R = 3400 a.u.
0
gation in imaginary time (by relaxation) in the absence
Simultaneously,wepaidattentionthatthe totalelectron
of the field. The total three-dimensional photoemission
densityinthe intervalof[0,R ]doesnotdeviatefromits
0
probability was computed as the Fourier transformation initial value of 2 by more than 10−8.
ofthefinalelectronwavepacketsattheendoflaserpulse:
2
W(~k)= (2π1)3/2 ψβ(~r)e−i~k·~rd3~r . (7) III. RESULTS AND DISCUSSION
β (cid:12)Z (cid:12)
X(cid:12) (cid:12)
(cid:12) (cid:12) The present calculations were performed on differ-
Finally, the HHG spectrum(cid:12) I(ω) was comput(cid:12)ed as the
ent levels of approximationwhich are systematically im-
squaredmodulus ofthe Fouriertransformedacceleration
proved by extending in the total wave function (2) the
of the total electric dipole moment:
set of electronic configurations describing the dynamics
1 d2D(t) 2 of the electronwhich remains bound. The photoelectron
I(ω)= e−iωtdt , (8) is described fully. For this purpose, in each improve-
(2π)1/2 dt2
(cid:12)Z (cid:12) ment step we expanded the basis set φα(~r) describ-
(cid:12) (cid:12) { }
with D(t) given by (cid:12) (cid:12) ing the bound electron by one additional hydrogen-like
(cid:12) (cid:12) nℓ function of the He+ ion with large quantum num-
+
D(t)= Ψ(~r ,~r ,t) ~r +~r Ψ(~r ,~r ,t) . (9) bers n and ℓ. The smallest basis set used includes only
1 2 1 2 1 2
h | | i one 1s orbital of He+. Hence, in this simplest level of
+
The present study was conducted for two different approximationthe bound electron is frozen and exhibits
pulse envelopes g(t). The main set of calculations was no dynamics. This approximation is an analogue of the
performedforatrapezoidalpulsewithalinearly-growing SAE approximation. One should stress, however,that it
front-edge, a constant plateau with unit height, and a differs from the usual one-electron SAE approximation
linearly-fallingback-edge,eachsupportingfiveopticalcy- in hydrogen, since the Hamiltonian (1) includes direct
cles. The propagation was thus performed in the time and exchange Coulomb interactions between bound and
interval of [0,T ] with T = 152π 40 fs. In addition, continuum electrons. The largest basis set of active or-
f f ω ≃
4

Radial coordinate (a.u.)
1.5 n = 1
0 500 1000 1500 2000 2500 3000
n = 2
101 u.) 1.0 n = 3
y (a.u.) 1100--31 ment (a. 0.5 n = 4
sit -5 n = 4 mo
en 10 e 0.0
n d 10-7 n = 3 pol
Electro 10-1-91 n = 2 Total di-0.5
10
-1.0
-13
10 n = 1

10-15 -1.5
u.) 107 n = 4 0 200 400 600 800 1000 1200 1400 1600
m (a. 105 Time (a.u.)
ectru 103 n = 3 FdiIpGo.le2:mo(mCeonlotrEoqn.lin(9e)), Tinhdeucteodtalbytimthee-delipneenadrleyn-tpoellaercitzreidc
p
s 800nm trapezoidal pulse described in the text. The calcu-
on 101 n = 2 lations are performed in a systematic series of improving ap-
Electr 10-1 proximations as indicated in thecaption of Fig. 1.
-3 n = 1
10
see from this figure that enabling the bound electron in
-5
10 He to be dynamically active causes considerable changes
0 10 20 30 40 50 60 in the computed radial density. Indeed, systematically
Electron energy (eV) extending the nℓ basis set by the layers of orbitals
+
{ }
FIG. 1: (Color online) The final photoelectron radial densi- with n=2, 3, and 4 results in a significant lowering of
ties(upperpanel)andthefinalphotoionizationspectra(lower thecomputedelectrondensityaroundthenucleusandto
panel) after the linearly-polarized 800nm trapezoidal pulse its slight enhancement in the outer region. This already
has expired. The time-shape, duration and intensity of the indicates the loss of low-kinetic-energy and gain of high-
pulse are indicated in the text. Note the logarithmic scales kinetic-energy electrons in the spectrum.
ontheverticalaxes. Thecalculations areperformed inasys-
The final photoelectron spectra obtained as a Fourier
tematic series of improving approximations by sequentially
transformation from the final wave packets via Eq. (7)
extendingthebasissetofdiscreteorbitals{nℓ+},usedtode-
arecomparedinthelowerpanelofFig.1. Becauseofthe
scribedynamicsoftheboundelectron. Ineachstep,thisbasis
wasextendedbythelayerofallnℓ+ orbitalswithlargerprin- trapezoidalpulseenvelope,thecomputedspectraconsist
cipal quantumnumbern, as indicated at the right-hand side of a comb of sharp peaks. These peaks with exponen-
ofeachspectrum. Theresultslabeledbyn=1correspondto tiallyfallingintensityareseparatedbythephotonenergy
the SAE approximation. To enable for a better comparison, ω (note also the logarithmic scale on the vertical axis).
the different spectra in each panel are vertically shifted up- Eachpeak represents photoelectrons with kinetic energy
wards by multiplying successively with 103 starting with the of ε = E E jω released by the above-threshold
nj n 0
spectrum of theSAE(n=1 layer) approximation. multiphoton−ioniza−tionofHe(herej isthenumberofab-
sorbedphotons,E isthegroundstateenergyofHe,and
0
E stands for the energy of the nℓ ionic state of He+).
n +
bitals used in the calculations consists of all the discrete By comparing slopes of the electron spectra obtained in
one-electron functions nℓ+ with n 4 and ℓ 3. differentapproximation(seelowerpartofFig.1),onecan
{ } ≤ ≤
Westartthepresentdiscussionwithanalysisofresults now directly observe the already announced diminution
obtained for the trapezoidal pulse. The upper panel of of the low-energy and enhancement of the high-energy
Fig. 1 depicts the final radial photoelectron wave packet parts of the computed spectra, caused by the dynamics
densities at the end of the laser pulse, computed in the of the bound electron.
different approximations explained above. The electron Figure 2 compares the total time-dependent electric
density obtained in the SAE approximation (labeled as dipole moments (9) computed in different approxima-
n=1)fallsexponentiallyasafunctionofdistancetothe tions. As is evident from this figure, the impact of the
nucleus (note the logarithmic scale on the vertical axis). dynamics of the bound electron on this quantity is dra-
This density is modulated by weak sharp features which matic: The maximal value of D(t), computed for the
representbunchesoffastelectronsreleasedbythestrong- largest basis set used here (labeled as n = 4), drops by
field multiphoton above-threshold ionization. One can almosta factorof7 as comparedto the SAE approxima-
5

tion (labeled as n=1).
31
10
We now turn to Fig. 3 which collects all HHG spec-
tra computed for the trapezoidal pulse by sequentially 29
10
extending the basis set of discrete orbitals nℓ by one
+
{ }
additional function. For a better eye view, the spectra 27
10
obtained in each step are shifted vertically: The lower-
most spectrum corresponds to the SAE approximation, 1025
whereas the uppermost one corresponds to the largest
23
basissetusedhere. TheHHGspectrumcomputedinthe 10
SAEapproximation(thelowermostspectruminFig.3la-
21 4f+
beled as 1s ) exhibits a set of sharp harmonics kω with 10
+
odd numbers k 100, which, as expected [2–4], build a
typical plateau w≤ith a cutoff. 1019
This ‘classical’ picture changes if the dynamics of the 17 4d+
10
boundelectroninHeisallowed. As onecanseefromthe
second from the bottom spectrum, already adding the 15
10
2s statetothe basissetofactiveorbitalsforthe bound
+
electronextendsthenumberofgeneratedharmonics. Al- 13 4p+
10
lowingtheboundelectrontooccupythe2s and2p or-
+ +
bitals, further extends the number of generatedharmon- 1011
ics. Onecanspeakoftheformationofasecondrelatively nits) 9 4s+
weak plateau in the HHG spectrum, which starts at the u10
cutoff of the main plateau and exhibits its own cutoff at b.
ar 7
much higher kω. Including sequentially the additional m (10
43fs++,o3rbp+it,al3sdi+n,thaendnfℓu+rthbearsoisnsethtere4sus+lt,s4inp+a,sy4sdt+em, aantid- ectru105 3d+
{ } p
calbroadeningofthesecondplateaushiftingitscutoffto G s103
harmonics kω of higher and higher order k (see Fig. 3). H
The HHG spectra computed by sequentially adding a H 1 3p+
10
layerof nℓ states with fixed principal quantum num-
+
bern={1,2,}3,and4aresummarizedagainintheupper -1
10
panel of Fig. 4. One can see that the extension of the
first plateau of the HHG spectrum does not change as 10-3 3s+
the basis set is enlarged. The extension of the second
-5
plateau, which is due to the presence of a second active 10
electron, appears once the basis set contains quantum
numbers n larger than 1, and grows fast with additional 10-7 2p+
basis functions n = 2 and n = 3. The extension of this
-9
plateau computed for the largest basis set with n = 4 10
does, however, not differ much from that obtained with -11 2s+
10
n = 3 indicating a noticeable trend in the convergence
of the present computational results with respect to the -13
10
basis set of discrete orbitals. We stress that the calcula-
tionsperformedhereforthelargestbasiswithn=4were -15 1s+
10
already at the limit of our computational capabilities.
-17
As a final point of our study, we ensure that the effect 10
observedhere is independent of the time-envelope of the
0 20 40 60 80 100 120 140 160 180 200
laserpulse employed. Forthis purpose,we performedan
Harmonic order
analogous set of calculations using a sine-squared pulse
FIG.3: (Color online) TheHHGspectraof Hecomputedfor
of the same length T and intensity I (for details see
f 0 thetrapezoidallaserpulse. Thecalculationsareperformedin
the lastparagraphof the precedingsection). The results
a systematic series of improving approximations by sequen-
of these calculations are collected in the lower panel of tially extendingthe basis set of discrete orbitals {nℓ+}, used
Fig. 4. The HHG spectrum computed in the SAE ap- to describe dynamics of the bound electron. In each step,
proximation (labeled as n = 1) exhibits a main plateau this basis was extended by one orbital, which is indicated at
and cutoff which are very similar to those obtained for the right-hand side of each spectrum. The results labeled by
the trapezoidal pulse in the same approximation (com- 1s+ correspond to the SAE approximation. To enable for a
pare with the lowermost spectrum in the upper panel of bettercomparison,thespectraforeachnℓ-statearevertically
this figure). From the lower panel of Fig. 4 one can also shiftedupwardsbymultiplyingsuccessivelywith104 starting
with the1s+ spectrum.
6

nucleus,neglectingtherebythedoubleionizationprocess
7
10
which is very weak for the pulse applied. For this pur-
5
10 pose,thepresentactivespacewasrestrictedtoconfigura-
3 tions which permit only one of the electrons to populate
10
1 continuum states. This photoelectron was described in
10
the time-dependent wavepacketswith angularmomenta
-1
10 ℓ 50. The dynamics of the bound electron was de-
-3 n = 4 scr≤ibedbya setofselecteddiscreteorbitals nℓ ofthe
10 +
-5 He+ ion. Inthenumericalcalculations,this{discre}teone-
10
electron basis was systematically increased by including
-7 n = 3
10 stateswithlargerquantumnumbersn andℓupto n 4
nits) 10-9 and ℓ≤3. ≤
b. u 10-11 n = 2 The pulse-drivencollectivecorrelateddynamicsoftwo
m (ar 10-13 enluemctbroenrsofingeHneerraetseudlthianrmaocnoincssidinertahbelecoinmcrpeuatseedoHfHthGe
u -15 n = 1
ctr 10 spectrum. In particular, compared to the SAE approx-
pe -17 imation, we observe the formation of a second plateau
s 10
G with weaker harmonics of higher order, which starts at
0
HH 10 thecutoffofthemainplateauandendswithitsowncut-
-2
10 off. Increasingsequentiallythebasissetofactiveorbitals
-4 describing the bound electron results in a systematic ex-
10
tension of the second cutoff to the high-energy side. For
-6
10 n = 3 the presently usedpulse parameters,the computedmain
-8
10 plateauextendsinthespectrumofharmonicskω uptok
-10 ofabout100. Forthelargestbasissetofdiscreteorbitals
10 n = 2 used here, the second plateau, which is about three or-
-12
10 ders of magnitude weaker than the main one, consists of
-14
10 additional harmonics with 100<k<180.
n = 1
-16 Takingintoaccountthatourcalculationsareprobably
10
0 20 40 60 80 100 120 140 160 180 200 still not fully converged over the basis set of discrete or-
Harmonic order bitals nℓ+ , it is rather difficult to exactly predict the
{ }
finalfateofthesecondplateaufoundathigherorderhar-
FIG. 4: (Color online) Summary of the HHG spectra of He,
computed for thetrapezoidal (upper panel) and sine-squared monics. Nevertheless, the main theoretical conclusion of
(lowerpanel)laserpulsesindifferentapproximations(seecap- the present work – that going beyond the SAE approxi-
tion of Fig. 1 and text for details). To enable for a better mation and allowing more electrons to be active and to
comparison,thespectraforeachn-layerareverticallyshifted interact is important and leads to the generationof con-
upwards by multiplying successively with 104 starting with siderably more harmonics – will remain unchanged. A
thespectrum of the SAE(n=1 layer) approximation. full understanding of how two or more active electrons
impact the HHG is a rather involved subject and goes
much beyond the present work.
seethatallowingthe boundelectroninHe to occupythe
Nevertheless,wecansayalreadynowthathavingmore
next layer of orbitals with n = 2 results in the forma-
discrete orbitals implies many more bound states of the
tionofasecondplateauinthecomputedHHGspectrum.
twoandinothersystemspossiblymoreelectronspartici-
As demonstrated by the calculations which also include
pating in the process. These bound states alonecanalso
the next layer of orbitals with n = 3, the second cutoff
give rise to some HHG [9]. In Ref. [9] a realistic model
further moves toward higher photon energies.
for an array of quantum dots with six active correlated
boundelectronshasbeensolvednumericallyexactlyand
shown to unambiguously give rise to a second plateau in
IV. CONCLUSION the HHG spectrum. Although the situation in Ref. [9]
differsfromours,thisresultsupportsourfindingthatal-
Generation of high-order harmonics in the He atom lowing more electrons to be active generates higher har-
exposed to intense linearly-polarized 800nm laser pulse monics and is likely to give rise to a second plateau. In
is studied beyond the single-active-electron approxi- our present example of He, the ionization of an electron
mation by the time-dependent restricted-active-space plays a crucial role, but the existence of the additional
configuration-interaction method. During the propaga- boundstates certainly leadsto additionaldifferentpath-
tion of the two-electron wave packets in strong laser ways of the important ionization channel and this will
fields, we allowed only one of the electrons to be ion- alsoinfluencetheHHG.Finally,afteroneelectronision-
ized and kept the other electron always bound to the ized, the He+ ion can stay in severalexcited states. The
7
recombinationstepisthenalsodifferentfromthatinthe by the F¨orderprogramm zur weiteren Profilbildung in
SAE approximationas there aremany new pathwaysfor der Universit¨at Kassel (Fo¨rderlinie Große Bru¨cke), by
it now. the Deutsche Forschungsgemeinschaft the within DFG
project No. DE 2366/1-1, and by the U.S. ARL and
the U.S. ARO under Grant No. W911NF-14-1-0383.
Acknowledgments Ph.V.D. acknowledges Research Institute of Physics,
Southern Federal University for the hospitality and sup-
This work was partly supported by the State Hes- port during his researchstay there.
sen initiative LOEWE within the focus-project ELCH,
[1] P.B. Corkum, Phys. Rev.Lett. 71, 1994 (1993). [24] J.S. Parker, L.R. Moore, K.J. Meharg, D. Dundas, and
[2] T. Brabec and F. Krausz, Rev. Mod. Phys. 72, 545 K.T. Taylor, J. Phys.B 34, L69 (2001).
(2000). [25] J.S.Parker,B.J.S.Doherty,K.J.Meharg,andK.T.Tay-
[3] C.Winterfeldt,C.Spielmann,andG.Gerber,Rev.Mod. lor, J. Phys.B 36, L393 (2003).
Phys.80, 117 (2008). [26] J.S. Parker, B.J.S. Doherty, K.T. Taylor, K.D. Schultz,
[4] F. Krausz and M. Ivanov, Rev. Mod. Phys. 81, 163 C.I. Blaga, and L.F. DiMauro, Phys. Rev. Lett. 96,
(2009). 133001 (2006).
[5] P.AgostiniandL.F.DiMauro,Rep.Prog.Phys.67,813 [27] J.S.Parker,K.J.Meharg,G.A.McKenna,andK.T.Tay-
(2004). lor, J. Phys.B 40, 1729 (2007).
[6] G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. [28] R. Santra and A. Gordon, Phys. Rev. Lett. 96, 073906
Avaldi,R.Flammini, L.Poletto, P.Villoresi, C.Altucci, (2006).
R.Velotta, S.Stagira, S. DeSilvestri, M. Nisoli, Science [29] S.Sukiasyan,C.McDonald,C.Destefani,M.Yu.Ivanov,
314, 443 (2006). and Th. Brabec, Phys.Rev.Lett. 102, 223002 (2009).
[7] J.J. Carrera, X.M. Tong, and S.-I. Chu, Phys. Rev. A [30] H. Miyagi and L.B. Madsen, Phys. Rev. A 87, 062511
74, 023404 (2006). (2013).
[8] E. Goulielmakis, V.S. Yakovlev, A.L. Cavalieri, M. Uib- [31] HuShi-LinandShiTing-Yun,Chin.Phys.B22,013101
eracker, V. Pervak, A. Apolonski, R. Kienberger, U. (2013).
Kleineberg, and F. Krausz, Science 317, 769 (2007). [32] S. Patchkovskii, Z. Zhao, Th. Brabec, and D.M. Vil-
[9] I. Bˆaldea, A.K. Gupta, L.S. Cederbaum, and N. Moi- leneuve, Phys.Rev. Lett.97, 123003 (2006).
seyev,Phys. Rev.B 69, 245311 (2004). [33] O.Smirnova,Y.Mairesse, S.Patchkovskii,N.Dudovich,
[10] M.Lewenstein, Ph.Balcou, M.Yu.Ivanov,A.L’Huillier, D. Villeneuve, P. Corkum, and M.Yu. Ivanov, Nature
and P.B. Corkum, Phys.Rev.A 49, 2117 (1994). 460, 972 (2009).
[11] M.Yu.Ivanov,T. Brabec, andN.Burnett,Phys.Rev.A [34] O.Smirnova,S.Patchkovskii,Y.Mairesse, N.Dudovich,
54, 742 (1996). D.Villeneuve,P.Corkum,andM.Yu.Ivanov,Phys.Rev.
[12] G. Lagmago Kamta and A.D. Bandrauk, Phys. Rev. A Lett. 102, 063601 (2009).
71, 053407 (2005). [35] S.Sukiasyan,S.Patchkovskii,O.Smirnova,Th.Brabec,
[13] A.D. Bandrauk, S. Chelkowski, D.J. Diestler, J. Manz, and M.Yu.Ivanov,Phys. Rev.A 82, 043414 (2010).
and K.-J. Yuan,Phys.Rev. A 79, 023403 (2009). [36] S. Chattopadhyay, S. Bauch, and L.B. Madsen, Phys.
[14] I.A. Ivanov and A.S. Kheifets, Phys. Rev. A 79, 053827 Rev. A 92, 063423 (2015).
(2009). [37] J. Zanghellini, M. Kitzler, C. Fabian, T. Brabec, and A.
[15] A.-T. Le, R.R. Lucchese, S. Tonzani, T. Morishita and Scrinzi, Laser Phys. 13, 1064 (2003).
C.D. Lin, Phys. Rev.A 80, 013401 (2009). [38] T. Kato and H. Kono, Chem. Phys. Lett. 392, 533
[16] M.V. Frolov, N.L. Manakov, T.S. Sarantseva, M.Yu. (2004).
Emelin, M.Yu. Ryabikin, and A.F. Starace, Phys. Rev. [39] M.Nest,T.Klamroth,andP.Saalfrank,J.Chem.Phys.
Lett.102, 243901 (2009). 122, 24102 (2005).
[17] Y.-C. Han and L.B. Madsen, Phys. Rev. A 81, 063430 [40] O.E.Alon,A.I.StreltsovandL.S.Cederbaum,J.Chem.
(2010). Phys. 127, 154103 (2007).
[18] S.Chelkowski,T.Bredtmann,andA.D.Bandrauk,Phys. [41] D.J. Haxton, K.V. Lawler, and C.W. McCurdy, Phys.
Rev.A 85, 033404 (2012). Rev. A 83, 063416 (2011).
[19] C. O´ Broin and L.A.A. Nikolopoulos, Phys. Rev. A 92, [42] D.HochstuhlandM.Bonitz,J.Chem.Phys.134,084106
063428 (2015). (2011).
[20] Y.Fu,J. Zeng, and J. Yuan,Computer Phys.Commun. [43] J.Zanghellini,M.Kitzler,Th.Brabec,andA.Scrinzi,J.
210, 181 (2017). Phys. B 37, 763 (2004).
[21] E.S. Smyth, J.S. Parker, and K.T. Taylor, Computer [44] G. Jordan and A. Scrinzi, New J. Phys. 10, 025035
Phys.Commun. 114, 1 (1998). (2008).
[22] J.S.Parker,E.S.Smyth,andK.T.Taylor,J.Phys.B31, [45] M.H. Beck A. J¨ackle, G.A. Worth, and H.-D. Meyer,
L571 (1998). Phys. Rep.324, 1 (2000).
[23] J.S.Parker,L.R.Moore,D.Dundas,andK.T.Taylor,J. [46] D. Hochstuhl and M. Bonitz, Phys. Rev. A 86, 053424
Phys.B 33, L691 (2000). (2012).
[47] A.N. Artemyev, A.D. Mu¨ller, D. Hochstuhl, L.S. Ceder-
8
baum, and Ph.V. Demekhin, Phys. Rev. A 93, 043418 (1996).
(2016). [55] A. Kramida, Y. Ralchenko, and J. Reader, NIST
[48] S. Bauch, L.K. Sørensen, and L.B. Madsen, Phys. Rev. Atomic Spectra Database (National Institute of Stan-
A 90, 062508 (2014). dards and Technology, Gaithersburg, MD, 2012),
[49] D.E. Manolopoulos and R.E. Wyatt, Chem. Phys. Lett. http://physics.nist.gov/PhysRefData/ASD/index.html.
152, 23 (1988). [56] L.V. Keldysh, Sov.Phys. JETP 20 1037 (1965).
[50] T.N. Rescigno and C.W. McCurdy, Phys. Rev. A 62, [57] U. Mohideen, M.H. Sher, H.W.K. Tom, G.D. Aumiller,
032706 (2000). O.R.WoodII,R.R.Freeman,J.Bokor,P.H.Bucksbaum,
[51] C.W. McCurdy, M. Baertschy and T.N. Rescigno, J. Phys. Rev.Lett. 71, 509 (1993).
Phys.B 37, R137 (2004). [58] T.J. Park and J.C. Light, J. Chem. Phys. 85, 5870
[52] Ph.V. Demekhin, D. Hochstuhl, and L.S. Cederbaum, (1986).
Phys.Rev.A 88, 023422 (2013). [59] J.L.Krause,K.J.Schafer,andK.C.Kulander,Phys.Rev.
[53] A.N. Artemyev, A.D. Mu¨ller, D. Hochstuhl, and Ph.V. A 45, 4998 (1992).
Demekhin,J. Chem. Phys. 142, 244105 (2015).
[54] E. Cormier and P. Lambropoulos, J. Phys. B 29, 1667