Laser-induced nonsequential double ionization at and above the recollision-excitation-tunneling threshold

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by  T. Shaaran

Laser-induced nonsequential double ionization at and above the
recollision-excitation-tunneling threshold
T. Shaaran1, M. T. Nygren1,2 and C. Figueira de Morisson Faria1
1Department of Physics and Astronomy, University College London,
Gower Street, London WC1E 6BT, United Kingdom
2Theoretical Physics, Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom
(Dated: January 28, 2010)
Weperformadetailedanalysisoftherecollision-excitation-tunneling(RESI)mechanisminlaser-
0 inducednonsequentialdoubleionization (NSDI),inwhich thefirstelectron, uponreturn,promotes
1 asecond electron toan excited state, from which it subsequentlytunnels,based on thestrong-field
0 approximation. Weshowthattheshapesoftheelectronmomentumdistributionscarryinformation
2 about the bound-state with which the first electron collides, the bound state to which the second
electron is excited, and the type of electron-electron interaction. Furthermore, one may define a
n
driving-field intensity threshold for the RESI physical mechanism. At the threshold, the kinetic
a
J energyofthefirstelectron,uponreturn,isjustsufficienttoexcitethesecondelectron. Wecompute
thedistributionsforheliumandargoninthethresholdandabove-thresholdintensityregime. Inthe
8
latter case, we relate our findings to existing experiments. The electron-momentum distributions
2
encountered are symmetric with respect to all quadrants of the plane spanned by the momentum

] componentsparallel tothelaser-field polarization, instead of concentrating on only thesecond and
h fourth quadrants.
p
-
m
I. INTRODUCTION p =p = 2 U , where U is the ponderomotive en-
1k 2k p p
±
o ergy, and the recently reported V-shaped structure [11],
t which is a signpature of the long-range character of the
a Electron-electron correlation in strong laser fields has
s. raisedconsiderableinterestfor overa decade,inparticu- electron-electroninteraction [7, 8, 10, 12, 13]. Moreover,
c lar in the context of laser-induced nonsequential double fromthetheoreticalviewpoint,thismechanismisconsid-
i erablyeasiertomodel,ascomparedtootherrescattering
s and multiple ionization [1]. Concrete examples are the
y early measurements of a “knee” in the double ionization processes. Thisisparticularlytrueinthecontextofsemi-
h analyticalmodels,suchasthestrong-fieldapproximation
yieldasafunctionofthelaser-fieldintensity,whichdevi-
p [5–7, 9, 13, 18].
atesfromthepredictionsofsequentialmodelsinordersof
[
magnitude [2], and the peaks in the electron momentum Recent experimental results, however, reveal other

1 distributions in nonsequential double ionization (NSDI), rescatteringmechanismsapartfromelectron-impaction-
v as functions of the electron components p (n = 1,2) ization. For instance, if the driving-field intensity is be-
nk
5
parallel to the laser-field polarization [3]. Such peaks low the threshold intensity, the kinetic energy of the
2
occur at nonvanishing parallel momenta and cannot be recolliding electron is no longer sufficient to make the
2
5 explained by a sequential mechanism. second electron overcome the ionization potential of the
. Presently, it is an established fact that NSDI occurs singly ionized core and reach the continuum [14, 15]. In
1
duetotheinelasticrecollisionofanelectronwithitspar- this intensity range, the energy transferred to the core
0
0 ention[4]. Inthisrecollision,thefirstelectrongivespart is only enough to raise the second electron to an excited
1 ofthe kinetic energyitacquiredfromthe drivingfield to bound state, from which it subsequently tunnels. This
: a second electron, which is then freed. process is known as recollision-excitation-tunneling ion-
v
i The simplest type of recollision which can lead to this ization(RESI).Thismechanismisalsoimportantforspe-
X phenomenon is electron-impact ionization. Thereby, the cific targets, such as argon [17], even if intensities above
r first electron, upon return, provides the second electron thethresholdaretaken,providedthedrivingpulseislong
a with enough energy so that it is able to overcome the enough.
second ionization potential of the target in question and For this specific mechanism, the first electron leaves
reach the continuum. Both electrons leave simultane- near a crossing of the driving field, whilst the second
ously and lead to distributions peaked at nonvanishing electron is freed near a field maximum. The time delay
momenta, occupying the first and third quadrant of the betweenbothelectronssuggeststhatthey leavewith op-
plane p p spanned by the parallelmomentum compo- posite momenta. This implies that the second and the
1k 2k
nents. fourth quadrant of the p p plane are expected to be
1k 2k
Electron-impactionization is possibly the most exten- populated[16]. Apopulationinsuchregionsoftheparal-
sively investigated NSDI mechanism [5–13]. This is due lel momentum plane has also been observed for NSDI of
to the fact that, until the past few years, it was suf- diatomic molecules, especially if the molecules in ques-
ficient to describe key features observed experimentally tion are aligned perpendicular to the laser-field polar-
in the electron momentum distributions. Concrete ex- ization[19–22]. Furthermore,ithas been recently shown
amples are the peaks near the non-vanishing momenta thatelectron-impactionization,alone,wouldnotbevery
2
usefulforretrievingthemolecularstructureintheexperi- II. TRANSITION AMPLITUDE
mentallyrelevantintensityrange[23]. Alltheseexamples
suggest that the RESI mechanism is becoming increas- A. General expressions
ingly important, in view of the driving field intensities
and targets employed.
The transition amplitude describing the recollision-
This mechanism, however, is considerably less under-
excitation-tunneling ionization (RESI) mechanism,
stood than electron-impact ionization. In fact, apart
within the SFA, can be written as (for details on the
fromearlyresults in whichonly the RESI yieldhas been
derivation see [26]).
calculatedwithin a semi-analyticalframework[18], most
existing results are the outcome of computations, either ∞ t t′
classical [16, 19, 20, 24, 25] or quantum mechanical [22], M(p1,p2) = dt dt′ dt′′ d3k
for which the different rescattering mechanisms cannot Z−∞ Z−∞ Z−∞ Z
be easily disentangled. Vp2eVp1e,kgVkgeiS(p1,p2,k,t,t′,t′′), (1)
Inpreviouswork[26], we havestudied the RESI phys-
with the action
ical mechanism, within the strong-field approximation.
Ianmthuilstifprlaemienwteogrrkalt,hewtitrhanasittiiomnea-dmeppleintuddeentisawctriiotntenanads S(p1,p2,k,t,t′,t′′)= ∞[p2+A(τ)]2dτ
− 2
slowly-varying prefactors. This approach has the partic- Zt
∞[p +A(τ)]2
ular advantage of providing a clear space-time picture of 1
dτ
the process in question, in terms of electron trajectories, −Zt′ 2
while retaining featuressuchasquantum-interferenceef- ′
t [k+A(τ)]2
fects [27]. dτ
− ′′ 2
We have shown that the RESI mechanism may be un- Zt
′′ ′ ′
derstood as the combination of two processes. For the +E1gt +E2gt +E2e(t t )(2)
−
first electron, a behavior similar to rescattered above-
threshold ionization is present. The main difference lies and the prefactors
onthefactthatpartofthekineticenergyofthefirstelec-
1
tron is given to the parent ion. This leads to a maximal Vkg = k˜(t′′) V ψg(1) = (2π)3/2
kinetic energy slightly smaller than the high-order ATI
D (cid:12) (cid:12) E
cutoff 10Up. The second electron, which is tunnel ion- d3(cid:12)(cid:12)r V(cid:12)(cid:12)(r )exp[ ik˜(t′′) r ]ψ(1)(r ), (3)
ized at a subsequent time, behaves in the same way as × 1 1 − · 1 g 1
Z
in direct above-thresholdionization. Hence, its maximal
kineticenergyisgivenbythe directATIcutoff, i.e.,2U .
p 1
This allowed one to determine kinematic constraints for Vp1e,kg = p˜1(t′),ψe(2) V12 k˜(t′),ψg(2) = (2π)3
the RESI mechanism, which lead to distributions which
D (cid:12) (cid:12) E
equally occupy the four quadrants of the p1kp2k plane. d3r2d(cid:12)(cid:12)3r1ex(cid:12)(cid:12)p[ i(p1 k) r1]
For an early derivation of somewhat different kinematic × − − ·
Z Z
constraints for this mechanism see, e.g., Ref. [28]. V (r r )[ψ(2)(r )]∗ψ(2)(r ) (4)
× 12 1, 2 e 2 g 2
In the above-stated investigations, however, we as-
sumed the prefactors in the strong-field approximation and
amplitude to be constant. Nonetheless, in a more re-
1
alistic scenario, one should take into account that such Vp2e = hp˜2(t)|Vion ψe(2) = (2π)3/2
prefactors cause a bias in momentum space. This may
(cid:12) E
leadtoelectronmomentumdistributionsconcentratedon d3r V (cid:12)(cid:12)(r )exp[ ip˜ (t) r ]ψ(2)(r ).(5)
the second and fourth quadrants of the p p plane. × 2 ion 2 − 2 · 2 g 2
1k 2k Z
This work is organized as follows: In Sec. II, we will
Eq. (1) describes the physical process in which an
briefly recall the SFA transition amplitude for the RESI
electron, initially in a bound state ψ(1) >, is released
mechanism,whichhasbeenderivedin[26]. Wewillstart g
by tunneling ionization at a time t′′|into a Volkov state
from the general expressions (Sec. IIA), which will be
k˜(t′) >. Subsequently, this electron propagates in the
solved by saddle-point methods. The pertaining saddle-
|continuum from t′′ to a later time t′. At this time, it
point equations are derived in Sec. IIB. Subsequently,
is driven back by the field and rescatters with its parent
in Sec. IIC we will provide the specific prefactors em-
ion. Inthiscollision,itexcitesthesecondelectron,which
ployed for the hydrogenic systems to be investigated in
this work. In Sec. III, we employ this approach to com- is bound at ψg(2) >, to the state ψe(2) >, through the
| |
pute electron momentum distributions for Helium and interaction V12. Finally, the second electron, which is in
Argon. For the latter species, an explicit comparison a bound excited state ψ(2) >, is released by tunneling
e
|
withthe resultsinRef.[14]isperformed. Finally,inSec. ionization at a time t into a Volkov state p˜ (t)>. The
2
|
IV we state the main conclusions of this paper. final electron momenta are described by p (n = 1,2).
n
3
In the above-stated equations, E (n = 1,2) give the Eq. (8) gives the conservation of energy at the time
ng
ionization potentials of the ground state, E (n = 1,2) t′′,which,physically,correspondstotunnelingofthefirst
ne
give the ionization potentials of the excited state and electron. Since tunneling has no classical counterpart,
V(r ) and V (r ) correspond to the atomic binding this equation possesses no real solution. Eq. (9) con-
1 ion 2
potential of the system as seen by the first and second strains the intermediate momentum k of the first elec-
electron, respectively. tron,sothatitcanreturntotheparention. Eq. (10)ex-
The form factors (3) and (5) contain all the informa- pressesthefactthatthefirstelectronreturnstoitsparent
tion about the binding potential of the first and second ionatatimet′ andrescattersinelasticallywithit,giving
electron, respectively. The form factor (4) contains all partofitskineticenergyE (t′)=[k+A(t′)]2/2tothe
ret
the information about the interaction of the first elec- core. Underthisinteractionthesecondelectronisexcited
tron with the singly ionized atom, which is an inelastic froma state with energy E to a state with energyE .
2g 2e
interaction. If the electron interactionis only dependent Thefirstelectronleavesimmediately andreachesthede-
on the difference between both electroncoordinates,i.e., tector with momentum p . Finally, Eq. (11) describes
1
if V12(r1,r2)=V12(r1 r2), Eq. (4) may be rewritten as the fact that the second electron tunnels at a later time
−
t from an excited state of energy E .
V (p k) 2e
12 1
Vp1e,kg = (2π)3−/2
1. Momentum constraints
d3r e−i(p1−k)·r2[ψ(2)(r )]∗ψ(2)(r ),(6)
× 2 e 2 g 2
Z
The saddle-point equations (10) and (11) provide use-
with
ful information on the momentum-space regions popu-
1
V (p k)= d3rexp[ i(p k)r]V (r) (7) lated by the RESI mechanism, and on the shapes of the
12 1− (2π)3/2 − 1− · 12 electron-momentumdistributions. Forinstance,Eq. (11)
Z
isidenticalto thatdescribingtunnelionizationfordirect
and r=r r .
1 2
− above-threshold ionization. This implies that the maxi-
Clearly, V andV aregauge dependent. In fact, in
kg p2e malkineticenergyofthesecondelectronatthedetector,
the length gauge p˜ (τ) = p +A(τ) and k˜(τ) = k+
n n if the field can be approximated by a monochromatic
A(τ)(τ =t′,t′′), while in the velocity gauge p˜ (τ)=p
n n wave, is roughly given by 2U .
and k˜(τ) = k. This is a direct consequence of the fact p
Furthermore, the electron is leaving the excited state
that the gauge transformation χ = exp[ iA(τ) r]
l→v with largest probability when the electric field E(t) =
− ·
fromthelengthtothevelocitygaugecausesatranslation
dA(t)/dt is maximum. If the time dependence of the
p p A(τ) in momentum space. Due to the factthat −
laser field is such that A(t) vanishes when E(t) is at its
→ −
these shifts cancel out in Eq. (6), V remains the
p1e,kg peak (for instance, monochromatic fields), then
same in the length and velocity gauges.
2 U p 2 U . (12)
p 2 p
− ≤ ≤
B. Saddle-point analysis If, to first approximpation, we nepglect the momentum
componentsperpendicularto the laser-fieldpolarization,
onecanseethatthemomentumofthesecondelectron,in
The multiple integral in Eq. (1) will be solved us-
the parallelmomentumplane,is expectedto be centered
ing saddle-point methods (for details see Ref. [29]).
around vanishing momentum p and be limited by the
For that purpose, we must find the coordinates 2k
(t ,t′,t′′,k ) for which S(p ,p ,k,t,t′,t′′) is stationary, bounds p2k = 2 Up.
s s s s 1 2 ±
i.e., for which the conditions ∂ S(p ,p ,k,t,t′,t′′) = Oneshouldnotethat,duetothefactthatEq. (11)de-
t 1 2 p
∂∂t′SS((pp1,,pp2,,kk,,tt,,tt′′,,tt′′′′))==0∂ta′′rSe(spa1ti,spfi2e,dk.,Tt,hti′s,tl′e′)ad=st0oatnhde srecgriiboensfaortutnhneesleincgonpdroecleescst,rothne.rAeisnonno-cvlaansissihcainllgypaellropweend-
k 1 2
dicularmomentum,effectively,willleadtoanincreasein
equations
the potential barrier and a suppression in the yield. In
fact, Eq. (11) can also be written as
[k+A(t′′)]2 = 2E , (8)
− 1g [p +A(t)]2 = 2E˜ , (13)
2k 2e
−
1 t′ whereE˜ =E p2 isaneffectiveionizationpotential.
k= dτA(τ) (9) 2e 2e− 2⊥
− t′−t′′ Zt′′ The saddle-point equation (10), on the other hand,
yields information about the momentum of the second
[p +A(t′)]2 =[k+A(t′)]2 2(E E ). (10) electron. According to this equation,
1 2g 2e
− −
A(t) 2E p A(t)+ 2E , (14)
diff 1k diff
− − ≤ ≤−
and
where E =pE (t′,t′′) (E Ep) p2 /2 and
[p +A(t)]2 = 2E (11) E (t′,t′d′)iffdenoteksinthekine−ticen2egrg−yof2tehe−firs1t⊥electron
2 2e kin
−
4
upon return. For a monochromatic field, the electron To overcome the singularity in V , one needs to em-
p2e
returns most probably near a crossing of the laser field, bed this prefactor into the action, which now reads
one may use the approximation A(t) 2 U in the
≃ p S˜(p ,p ,k,t,t′,t′′)=S(p ,p ,k,t,t′,t′′) ilnV .
above-stated equation. In this case, we also know that 1 2 1 2 − p2e
thekineticenergyE (t′,t′′) 3.17U .Hencpe,E(max) (16)
3.17Up−(E2g −E2ek)in−p21⊥/2≤and p diff ≤ Ttiohnis∂wtSi˜ll(ple1a,dp2t,okm,to,dt′i,fitc′′a)t,iownhsicihnitshneoswadgdilvee-npobiynt equa-
−2 Up− 2Ed(miffax) ≤p1k ≤−2 Up+ 2Ed(miffax). (15) [p2+A(t)]2 =−2E2e+i∂tlnVp2e. (17)
q q
Eq. p(15) allows one to delimit apregion in momentum The main consequence of such a modification is that the
space for p centered around 2 U and bounded by drift velocity of the second electron is no longer pure
1k p
2E(max). In contrast to the p−revious case, there may imaginary. This will lead to a splitting in the ionization
diff p time t for each orbit, as compared to the non-modified
be a classically allowed region for the momentum of the
case. Depending on the velocity in question, the barrier
first electron if the parameters inside the square root
the electron must tunnel through in order to reach the
are positive, i.e., if 3.17U (E E ) + p2 /2.
p ≥ 2g − 2e 1⊥ continuumwilleitherwidenornarrow. Thismeansthat,
For increasing perpendicular momentum and/or bound-
with regard to the non-modified action, Im[t] will either
stateenergydifference,this regionwillbecomemoreand
increase or decrease.
more localized around 2 U until it collapses. There-
p
−
fore, it is also possible to distinguish a threshold and
p
anabove-thresholdbehaviorinthecontextofrecollision-
C. Prefactors
excitation-tunneling. One should note, however,that in-
tensitiesbelowtherecollision-excitationthreshold(E
2g
− In this work, we are particularly interested in expo-
E ) = 3.17U do not make physically sense, as the en-
2e p
nentiallydecaying,hydrogenicboundstates. Thismeans
ergy of the returning electron would not be sufficient
that, in general, the bound-state wavefunction reads
to promote the bound electron to an excited state. If
(E2g E2e) 3.17Up the well-known cutoff of 10Up for ψ(α)(r )=R (r )Ym(θ ,ϕ ), (18)
− ≪ α nl n l α α
rescattered above-threshold ionization is recovered[31].
In view of the above-mentioned constraints, the ex- where n,l and m denote the principal, orbital and mag-
pected maxima of the electron momentum distribution neticquantumnumbers,theindexαreferstotheelectron
are located at the most probable momenta (p1||,p2||) = in question, and the angular coordinates are given by θα
( 2 Up,0),and,aftersymmetrizingwithrespecttothe and ϕα. In this case, the binding potentials V(r1) and
± V (r ) will be given by
exchangep p ,at(p ,p )=(0, 2 U ). Thisim- ion 2
p 1 ↔ 2 1|| 2|| ± p
plies that, if the field canbe approximatelydescribed by Z
amonochromaticwave,theoutcomeofouprmodelshould Vα(rα)= eff, (19)
− r
α
be distributions in the p p plane, which are symmet-
1k 2k
ric upon pn pn, n = 1,2 and upon p1 p2, and where Vα yields either V or Vion, and Zeff corresponds
which equally→oc−cupy the four quadrants of t↔he parallel to the effective electronic charge. The general expres-
momentum plane. sions for the prefactors in this work are provided in the
appendix.
Below, we state the specific prefactors to be employed
2. Bound-state singularity in Sec. III, for Helium and Argon. In the former case,
upon collision, the second electron may be excited from
the 1s state to either the 2s or the 2p state, while in the
Finally, due to the saddle-point equations (8) and
latter species it may undergo a transition from the 3p
(11), for exponentially decaying bound states the pref-
state to the 4s orthe 4p state. One should note that the
actors (3) and (5) exhibit singularities in the length-
prefactor V is gauge dependent. In the length gauge,
gauge formulation of the strong-field approximation. p2e
p˜ (t) = p + A(t), and p˜ (t) = p + A(t) while, in
This is due to the fact that these prefactors will be 2 2 2 k 2k
n the velocity gauge, p˜ (t) = p and p˜ (t) = p . The
inversely proportional to [k+A(t′′)]2+2E and 2 2 2 k 2k
1g prefactor V , on the other hand, is gauge invariant.
[p +A(t)]2+2E m,wh(cid:16)eren,mareintegers.(cid:17)Forthe p1e,kg
2 2e
problem addressed in this specific work, however, only
t(cid:0)he prefactor V w(cid:1)ill influence the shape of the elec- 1. Excitation 1s→2s
p2e
tron momentum distributions. The prefactor V will
kg
affect the electron momentum distributions only quanti- Let us first consider the simplest case, in which the
tatively. Hence, to first approximation,one can consider secondelectronisexcitedto2s.Thisgivesthe prefactors
Eq. (3) as constant. A similar problem for the electron-
impactionizationmechanisminNDSIhasbeendiscussed V(2s) [p˜2(t)]2−2E2e (20)
in detail in [7]. p2e ∼ [[p˜ (t)]2+2E ]2
2 2e
5
and and
η (κ2,E ,E )
Vp(11es,→kg2s) ∼V12(p1−k)[κ21+ζ2(E2g,E2e)]3, (21) Vp(11es,→kg2p) ∼V12(p1−k)η2(κ2,E2g,E2e)β(κ), (31)
2g 2e
where the angular dependency is given by
with
η (κ2,E ,E ) = κ2( 2E +2 2E )+(2E )3/2 β(q)=(sinθ cosϕ +sinθ sinϕ +cosθ ). (32)
1 2g 2e 2g 2e 2g q q q q q
−2(p2E2e)3/2−p6E2e 2E2g. (22) Thereby,weemployedtheusualrelationsbetweenspher-
and p ical polar coordinates and the spherical harmonics.
One may write the above-stated expressions in terms
ζ(E ,E )= 2E + 2E . (23)
2g 2e 2e 2g of the electron momentum components parallel and per-
The above-stated equatpions canpalso be written in pendicular to the laser-field polarization. In this case,
terms of the momentum components parallel and per- Eq. (30) reads
pendicularto thelaserfieldpolarization,denotedbyp
α||
and pα⊥, (α=1,2), respectively. In this case, V(2p) p˜2(t)k 2+p22⊥ β(p˜ (t)). (33)
p2e ∼ 2Eq+(cid:2) p˜ (t(cid:3)) 2+p2 2 2
p˜ (t) 2+p2 2E 2e 2 k 2⊥
Vp(22es) ∼ [(cid:2)p˜22(t)kk(cid:3)2+p222⊥⊥+−2E22ee]2 (24) In β(p˜2(t)),(cid:16)the angle(cid:2)s θp˜2 a(cid:3)nd ϕp˜2 a(cid:17)re given by
(cid:2) (cid:3)
V(1s→2s) V (p k) η1 k−p1k 2+p21⊥,E2g,E2e , θp˜2 =arccos p˜2(t)k/ p˜2(t)k 2+p22⊥ (34)
p1e,kg ∼ 12 1− [ k h(cid:0)p 2+(cid:1)p2 +ζ2(E ,E i)]3 (cid:20) q(cid:2) (cid:3) (cid:21)
− 1k 1⊥ 2g (22e5) and ϕp˜2 = arccos[p˜2(t)x/p˜2(t)⊥], respectively. In Eq.
(cid:0) (cid:1) (31), κ2 = (k p )2 +p2 and the angles θ and ϕ
− 1k 1⊥ κ κ
read
2. Excitation 1s→2p
θ =arccos k p / (k p )2+p2 (35)
κ − 1k − 1k 1⊥
If, on the other hand, the second electron is excited (cid:20) q (cid:21)
(cid:0) (cid:1)
to 2p, one must consider three degenerate states, corre-
and ϕ = arccos[p /p ], respectively. This angular
sponding to the magnetic quantum numbers m = 1,0. κ 1x 1⊥
± dependence will be washed out when the transverse mo-
This yields
mentum components are integrated over (see Sec. III).
[p˜ (t)]2
V(2p) 2 [Ym(θ ,ϕ )]∗ (26)
p2e ∼ 2Eq+[p˜ (t)]2 2 1 p˜2 p˜2 3. Excitation 3p→4s and 3p→4p
2e 2
(cid:16) (cid:17)
and Finally, we will assume that the second electron, ini-
V(1s→2p) V (p k)η (κ2,E ,E )[Ym(θ ,ϕ )]∗, tially in 3p, will be excited either to the 4s or to the 4p
p1e,kg ∼ 12 1− 2 2g 2e 1 κ κ state. Similarly to the procedure adopted in the previ-
(27)
ous section, we will consider a coherent superposition of
with
the 3p , 3p and 3p states for the initial state of the
x y z
ζ(E ,E )√κ2 electron, i.e.,
η (κ2,E ,E )= 2g 2e . (28)
2 2g 2e (ζ2(E ,E )+κ2)3
2g 2e 1
ψ(2) = ψ(2) + ψ(2) + ψ(2) . (36)
Sincetheelectronmaybeexcitedtoanyofthe2pstates, 3p √3 3px 3py 3pz
(cid:12) E (cid:16)(cid:12) E (cid:12) E (cid:12) E(cid:17)
we will consider the coherent superposition (cid:12) (cid:12) (cid:12) (cid:12)
If the(cid:12)electron is ex(cid:12)cited to t(cid:12)he 4s sta(cid:12)te, the excitation
ψ(2) = 1 ψ(2) + ψ(2) + ψ(2) , (29) prefactor V(3p→4s) will exhibit an angular dependence
2p √3 2px 2py 2pz p1e,kg
(cid:12)(cid:12) E (cid:16)(cid:12)(cid:12) E (cid:12)(cid:12) E (cid:12)(cid:12) E(cid:17) givenby β(κ), andthe tunneling prefactorVp(24es) willnot
where(cid:12) r ψ(2) =(cid:12)ψ(2)(r ),(cid:12)with j =(cid:12) x,y,z. This im- depend on the angular variables. Both prefactors also
h 2 2pj 2pj 2 have a radial dependence on p˜ (t) or κ. If, however, the
plies that (cid:12) E 2
(cid:12) electronisexcitedtothe4pstate,onemusttakethefinal
(cid:12)
state as
[p˜ (t)]2
2
Vp(22ep) ∼ 2E2qe+[p˜2(t)]2 2β(p˜2(t)) (30) ψ4(2p) = √13 ψ4(2p)x + ψ4(2p)y + ψ4(2p)z , (37)
(cid:16) (cid:17) (cid:12)(cid:12) E (cid:16)(cid:12)(cid:12) E (cid:12)(cid:12) E (cid:12)(cid:12) E(cid:17)
(cid:12) (cid:12) (cid:12) (cid:12)
6
i.e., as a coherent superposition of 4p , 4p and 4p . In angulardependence ofthese prefactorscanbe neglected.
x y z
this case,the angulardependence ofV will be embed- If, however, states of higher orbital quantum numbers
p2e
dedinβ(p˜ (t)).TheangulardependenceofV(3p→4p) will are involved, or if the initial and excited bound states of
be more co2mplex and will involve the sum ofp1the,ekgorbital the second electron are p states, this dependence will be
angular momenta of the two electronic bound states in- more complex.
volved. Due to the higher quantum numbers involved, The simplest scenario is if all prefactors are nonsin-
theprefactorsaremessierthanthoseintheprevioussec- gular, such as in the velocity-gauge formulation of the
tions and will not be written down explicitly. They can, SFA. In this case, they will contribute to the electron
however,be obtainedfromthe generalexpressionsinthe momentumdistributions as |Vp2e|2 and|Vp1e,kg|2. In the
appendix. length-gauge SFA, however, Vp2e exhibits a singularity,
andthereforemustbe incorporatedinthe actionaccord-
ing to Eq. (17). This singularity, however, is present
III. ELECTRON MOMENTUM only in the radial part of this prefactor. Therefore, the
DISTRIBUTIONS angularpartsarestillslowlyvarying,andmaybetreated
asabove. Theradialpartoftheprefactor,however,must
In this section, we will compute electron momentum be incorporated in the action.
distributions,asfunctionsofthemomentumcomponents
(p ,p ) parallel to the laser-field polarization. We ap-
1k 2k
proximate the external laser field by a monochromatic A. Intensity dependence
wave, i.e.,
We will commence by having a closer look at how
E(t)=ε sinωte . (38)
0 x the momentum-space constraints affect the electron mo-
This is a reasonable approximation for pulses whose du- mentum distributions for different driving-field intensi-
ration is of the order of ten cycles or longer (see, e.g. [9] ties. For that purpose, we will assume that the prefac-
for a more detailed discussion). In this case, the elec- tors Vp1e,kg and Vp2e are constant, and vary the laser-
tron momentum distributions, when integrated over the field intensity. For the lowest intensity, the kinetic en-
transverse momentum components, read ergy of the returning electron is just enough to pro-
mote the second electron to an excited state, i.e., we
F(p ,p ) = d2p d2p M (p ,p ) (39) are considering the recollision excitation (RESI) thresh-
1k 2k 1⊥ 2⊥| R 1 2 old E E 3.17U . This intensity, however, is be-
ZZ 2g − 2e ≃ p
+ ML(p1,p2)+p1 p2 2, low the electron-impact ionization threshold, i.e., E2g >
↔ | 3.17U . The intermediate intensity has been chosensuch
p
where M (p ,p ) is given by Eq. (1) and d2p = that E E < 3.17U , i.e., above the threshold for
R 1 2 n⊥ 2g 2e p
−
p dp dϕ . If A(t T/2)= A(t), where T =2π/ω recollision-excitation. Nevertheless, this intensity is not
n⊥ n⊥ pn ± −
denotes a field cycle, the actions S , S corresponding sufficient to make the second electron overcome the ion-
L R
to the transitionamplitudes M andM obey the sym- ization potential and be freed by electron-impactioniza-
L R
metry S (p ,p ,t,t′,t′′) = S ( p , p ,t T/2,t′ tion. Finally, the highest driving-field intensity consid-
L 1 2 R − 1 − 2 ± ±
T/2,t′′ T/2). The distributions have also been sym- ered in this section is far above the recollision-excitation
±
metrized with respect to the exchange p p . To threshold, and slightly above the electron-impactioniza-
1 2
↔
a good approximation, the quantum-interference terms tion threshold. This implies that rescattering is classi-
[M (p ,p )]∗M (p ,p ),ν = µ, get washed out upon callyallowedforbothphysicalmechanisms. The compu-
ν 1 2 µ 1 2
6
the transverse-momentum integration, so that it is suf- tations in this section have been performed for Helium,
ficient to add the above-stated amplitudes incoherently. and the pertaining results are presented in Fig. 1.
Here,weconsideredVkg asconstantandweintegratethe As an overall feature, the distributions exhibit four
transition amplitude over the azimuthal angles ϕpn. peaks at (pj||,pν||) = (±2 Up,0), with j,ν = 1,2 and
We will now briefly discuss how the prefactors Vp2e j = ν. These peaks agree well with the constraints dis-
and Vp1e,kg behave with regard to the integration over cu6ssed in the previous secptions. This holds even if the
ϕpn. Obviously,if the second electron is excited from an driving-field intensity is just enough to excite the sec-
s state to an s state, the prefactors Vp2e and Vp1e,kg do ond electron and the only allowed momenta are 2 Up
±
not depend on this parameter. However, if a transition [Fig. 1.(a)]. Physically, this means that the first elec-
fromortoapstateisconsideredtherewillbeanangular tronwillreachitsparentionmostprobablyatacropssing
dependence in such prefactors. of the driving field, reaching the detector with the most
Forinstance,tunnelingionizationfromapstatewould probable momenta of 2 U , while the second electron
p
lead to the argument β(p˜2(t)) in Eq. (30). If excitation will reach it with vani±shing momentum.
p
fromans state to a p state or vice versatakes place,the The shapes of the distributions, however, differ con-
angular dependence of the prefactor Vp1e,kg is given by siderably. Indeed, at the RESI threshold intensity
β(κ) in Eq. (31)). When integrated over the azimuthal [Fig. 1.(a)], one observes ring-shaped distributions. As
anglesϕ , β(κ)2 and β(p )2 willyield2π,sothatthe the intensity increases, this distributions become more
pn | | | 2 |
7
4 (a) (b) (c) 4 (a) 4 (b)
2
1/2p/[U]2||p-02 1/2U]p 02 02
-4 /[2||-2 -2
p
-4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4 Contact Contact
p /[U]1/2 p /[U]1/2 p /[U]1/2 -4 -4
1|| p 1|| p 1|| p
-4 -2 0 2 4 -4 -2 0 2 4
FIG.1: ElectronmomentumdistributionsforHelium(E1g = 4 (c) 4 (d)
0.97 a.u., E2g = 2 a.u. and E2e = 0.5 a.u.) in a linearly 2 2
polarized, monochromatic field of frequency ω = 0.057 a.u..
2
In the picture, we considered all prefactors to be constant. 1/]p 0 0
U
Panels (a),(b)and(c) correspond toadriving-fieldintensity
I = 2.16×W/cm2, I = 2.5×W/cm2 and I = 3×W/cm2, /[2||-2 -2
p
respectively. The contour plots have been normalized to the Coulomb Coulomb
-4 -4
maximum probability in each panel.
-4 -2 0 2 4 -4 -2 0 2 4
p /[U ]1/2 p /[U ]1/2
and more elongated along the p axis [Fig. 1.(b)], un- 1|| p 1|| p
nk
til the maxima merge and cross-shapeddistributions are
observed [Fig. 1.(c)].
FIG. 2: Velocity-gauge electron momentum distributions for
This change of shape may be understood by analyz-
Helium (E1g = 0.97 a.u., E2g = 2 a.u. and E2e = 0.5 a.u.)
ing the momentum-space constraints. The widths of the
inalinearlypolarized, monochromaticfieldoffrequencyω=
distributions are determined by the tunnel ionization of 0.057 a.u. and intensity I = 2.16×W/cm2. In panels (a)
the second electron from an excited state. This process and (c), the first electron has been excited to 2s, while in
has no classical counterpart and leads to distributions panels(b),and(d)ithasbeenexcitedto2p. Theinteraction
peaked at p = 0 and which vanish at p = 2 U , employed is indicated in the figure. The contour plots have
2k 2k p
±
i.e.,atthedirectATIcutoff. Increasingtheintensitywill been normalized to themaximum probability in each panel.
p
onlymaketheeffectivepotentialbarriersmallerorwider,
andthusaffecttheoverallyield,butwillnotchangesuch
constraints. electron is either excited by a contact-type interaction
Theelongationsinthedistributionsaredeterminedby V(δ)(r r ) = δ(r r ) or by a long-range, Coulomb
12 1− 2 1− 2
the rescattering of the first electron. This rescattering, type interaction V(C)(r r )=1/(r r ). In the for-
in contrast,delimits a momentum regionwhich is highly mer case, V(δ)(p 12 k)1=−con2st., while1−in t2he latter case
dependent on the driving-field intensity. Therefore, its 12 1−
V(C)(p k) 1/(p k)2.
width in momentum space will vary. Specifically, at the 12 1− ∼ 1−
RESIthreshold,therewillbemaximainthedistributions In order to perform a direct comparison, we will take
at p = 2 U due to the fact that the first electron the same parameters as in Fig. 1, but incorporate the
1k p
rescatters±mopstprobably ata field crossing. However,as prefactors Vp(11es,→kg2s) and Vp(22es), or Vp(11es,→kg2p) and Vp(22ep),
these are the only classically allowed momenta, the dis- correspondingto the1s 2sor1s 2pexcitationwith
→ →
tributions will be fairly narrow around this value. With subsequent tunneling, respectively.
increasing driving-field intensity, the classically allowed In Fig. 2, we consider the lowest intensity in the pre-
region defined by Eq. (15) will become more and more vious figure and the velocity gauge. If the electron
extensive and this will cause the elongation. is excited to the 2s state [Fig. 2.(a)], we observe four
Note that the electrons are indistinguishable so that spots which are slightly elongated along the p axis.
nk
the above-stated arguments hold upon the exchange Hence, in comparison to its constant prefactor counter-
p p . Hence, the horizontal and vertical axis in part,i.e.,Fig. 1.(a), there was a narrowing. This narrow-
1k 2k
↔
the parallel momentum plane will be equally affected. ing is causedbythe interplayoftwo featuresin the pref-
actor V(2s). First, this prefactor exhibits two symmetric
p2e
nodes,which,forvanishingtransversemomentumarelo-
B. Bound-state signatures cated at p = U . As the transverse momentum
2k p
±
increases, these minima move towards vanishing paral-
p (2s)
We will now investigate how the shape of the bound lel momenta. Second, Vp2e decreases very steeply with
stateto whichthe secondelectronisexcitedisimprinted transverse momenta p2⊥. Hence, upon integration over
on the electron momentum distributions. We will also this parameter,the maincontributionswillbe causedby
employ different gauges and types of electron-electron small values of p2⊥ and will vanish near p2k = Up.
±
interaction. Explicitly, we will assume that the second If, on the other hand, one assumes that the second
p
8
electron is excited to 2p, there is both a broadening in
the distributions and a splitting in their peaks. These
features are depicted in Fig. 2.(b). The splitting occurs
at the axis p = 0,n = 1,2, and is caused by the fact 4 (a) 4 (b)
nk
tinhgatmVopm2eenetxa,hiib.ei.t,sexaavctelryywphreorneoounneceedxpneocdtseIamt[tv]atnoisbhe- 1/2U]p02 02
mveirnifiimedumbyanaddtirheectyiienlsdpetoctiboenmoafxtihmeurmad.iaTlhdisepheansdbeneecne p/[2||-2 -2
of Eq. (30), and omitting the Vp2e prefactor in our com- -4 Contact -4 Contact
putations. The latter procedure caused the additional
-4 -2 0 2 4 -4 -2 0 2 4
minima to disappear (not shown). The broadening in
X Axis Title X Axis Title
the distributions as compared to the 1s 2s case is a 4 (c) 4 (d)
→
coinofcntrsheeeaqsunineogndcetesraoanftsvtphe2rekse=mmu±cohmseUlnoptw.uemTrhpde2er⊥cereaaarnesdeaoilfnsotVhaepd(22adepbi)tsiweonnitcahel 1/2p/[U]2||p-202 -202
nodes atthe diagonalp =p andatthe anti-diagonal
1|| 2||
p
p1|| = p2|| of the p1||p2|| plane. -4 Coulomb -4 Coulomb
−
In this intensity regime, there seems to be little dif- -4 -2 0 2 4 -4 -2 0 2 4
ference in the shapes of the distributions if the electron p /[U]1/2 p /[U]1/2
1|| p 1|| p
is excited to the 2s state, regardless of whether the first
electroninteractswithitsparentionthroughacontactor
a Coulombinteraction[Figs. 2.(a)and(c), respectively].
This is possibly caused by the fact that the prefactor FIG. 3: Velocity-gauge electron momentum distributions for
V(2s), due to its fast-decaying behavior, delimits a very the same parameters as in the previous figure, but driving-
p2e
narrow region in momentum space. This adds up to the fieldintensityI =3×W/cm2. Inpanels(a)and(c),thefirst
very restrictive momentum constraints. In contrast, the electronhasbeenexcitedto2s,whileinpanels(b),and(d)it
hasbeenexcitedto2p. Theinteractionemployedisindicated
effectoftheCoulombtailismuchmorecriticaliftheelec-
in the figure. The contourplots havebeen normalized tothe
tron is promoted to the 2p state. Indeed, for a Coulomb
maximum probability in each panel.
type interaction [Fig. 2.(d)], the splitting of the peaks
at the axis p = 0 remain, but the nodes at p = p
nk 1|| 2||
and p = p disappear as compared to its contact-
1|| − 2|| contact-typeinteraction(seeFigs.3.(a)and 3.(b)). This
interaction counterpart [Fig. 2.(b)]. This is caused by
is expected, as V (p k) favors low momenta for the
the fact that the former minima are a characteristic of 12 1 −
the V(2p) prefactor, whereas the latter are mainly deter- former,whileitisconstantforthelatter. Physically,this
p2e reflects the fact that rescattering of the first electron is
mined by momentum-space effects. The Coulomb inter-
nowallowedtooccuroveranextensiveregioninmomen-
action introduces a further momentum bias, and washes
tum space. Hence, it does makea difference whether the
out the latter nodes.
second electron is excited by a long-range or zero-range
We will now discuss what happens if the intensity of
interaction.
the driving field is such that 3.17U >E E . These
p 2g− 2e We will now perform an analysis of the electron-
results are displayed in Fig. 3, for the highest intensity
momentum distributions in the length gauge. In this
in Fig. 1. As expected, all distributions are much more
case, the prefactor V governing the tunneling of the
elongated along the axis p = 0, as compared to the p2e
nk secondelectronexhibitsasingularity,andmustbeincor-
low-intensitycase. Theimprint,however,ofthedifferent
porated in the action. The modifications in the action
bound states to which the secondelectron is excited and
read, for the 2s and 2p bound states,
from which it subsequently tunnels are the same as in
thebelow-thresholdregime. Indeed,wenoticethatthere
2E(t)p˜ (t) ( p˜ (t) 2+p2 6E )
is a narrowing in the distributions for the 1s 2s case i∂ lnV(2s) = i 2 k 2 k 2⊥− 2e ,
[Figs. 3.(a) and (c)], and a splitting in the pe→aks at the − t p2e − χ (p˜ (t))χ (p˜ (t))
+ (cid:2)2 (cid:3)− 2
fouraxisinthe1s 2pcase[Figs.3.(b)and(d)]. Thisis (40)
→
not surprising, as the prefactors V(2s) and V(2p) exhibit and
p2e p2e
the same functional dependencies as before.
E(t)(p2 +2E 3 p˜ (t) 2)
The shapes of distributions, however, change much i∂ lnV˜(2p) =i 2⊥ 2e− 2 k , (41)
more critically in this intensity regime, with regard − t p2e p˜2(t)k p˜2(t)k 2+p22⊥(cid:2) +2E(cid:3)2e
to the type of electron-electron interaction, than for
the intensity used in Fig. 2. For all cases, the dis- (cid:16)(cid:2) (cid:3) (cid:17)
respectively, with
tributions computed using the Coulomb-type interac-
χ (p˜ (t)) = p˜ (t) 2+p2 2E and E(t) =
tion (Figs. 3.(c) and 3.(d)) are much more localized in ± 2 2 k 2⊥± 2e
the low-momentum regions than those computed with a ∂tA(t). (cid:16)(cid:2) (cid:3) (cid:17)
−
9
4 (a) 4.58E-18 (b) 2.75E-18 unexpected, aswe areassuming thatthe secondelectron
4.007E-18 2.406E-18 istunnelingfroma2sstate. Aspreviouslydiscussed,the
2 3.435E-18 2.062E-18 prefactor V(2s) exhibits a node in p = U , which
1/2p/[U]2||p-02 1122....1728419657E2EEE-1---1181888 6111....803773715159EEEE----11119888 laietxanidso.stAtfoonraiitnsnpsspa2ierenrcgotuwioliannrgiotfyo,Eftqhth.eel(e2dn4ig)sttahril-bsgoua2tuskiugoegngsper±asetlpfosanctgthopatrth,wewopuenrldke
5.725E-19 3.437E-19 be very similar to the velocity-gauge prefactor. This is
-4
0 0 a consequence of the fact that the second electron is
-4 -2 0 2 4 -4 -2 0 2 4
leaving when the field E(t) is near its maximum. For
(c) 1.22E-16 (d) 7.88E-17
4 a monochromatic field, this implies that the vector po-
1.068E-16 6.895E-17
tential A(t) is practically vanishing.
2 9.15E-17 5.91E-17
1/2U]p0 67..16E25-1E7-17 34..9942E5E-1-717 theOinnedsivhioduuladlncootnet,rhibouwteiovners,ftrhoamtweaecahreofcothnesidorebriintsgoornigly-
p/[2||-2 4.575E-17 2.955E-17 inating from the modification of the action. It is very
3.05E-17 1.97E-17 likely that, in order to recover the structure determined
-4 1.525E-17 9.85E-18 bytheprefactorV(2s),onemustconsiderthecoherentsu-
0 0 p2e
-4 -2 0 2 4 -4 -2 0 2 4 perpositionofalltheorbitsoriginatingfromthesplitting
p /[U]1/2 p1||/[Up]1/2 ofIm[t]when computing the yield. Since these contribu-
1|| p
tions are comparable, one expects the above-mentioned
nodes to be recovered due to quantum-interference ef-
FIG. 4: Length-gauge electron momentum distributions fects.
for Helium in a linearly polarized, monochromatic field.
Throughout, we assumed Vp1e,kg = const., V12 to be a
contact-type interaction, and incorporated Vp(22se) in the ac- C. Comparison with experiments
tion. Panels (a) and (c) correspond to the trajectories for
which the barrier has been narrowed by the modifications in
the action, while Panels (b) and (d) correspond to those for We will now perform a direct comparison with the re-
which it has been widened. The upperand lower panels cor- sults in Ref. [14]. In particular, in this reference, the
respondtodrivingfieldintensitiesI =2.16×1014W/cm2 and distributions encountered have been modeled employing
I =3×1014W/cm2,respectively. Inordertoperformaquan- the electron-impact ionization physical mechanism and
titative comparison, we are providing the explicit values for
a modified ionization threshold for the second electron.
theNSDI yield.
Apartfromthat,however,inviewofthe driving-fieldin-
tensities involved, one expects recollision-excitationtun-
neling to be present.
For each orbit, the tunneling time of the second elec-
Forthatpurpose,we willconsiderargonandthe same
tron will split into two values, as compared to the non-
laser-fieldparametersas in Ref. [14](c.f. Fig. 2 therein).
modified action. This has particularly important conse-
We will assume, however, that, when the first electron
quences as far as Im[t] is concerned, since it provides a
recollides,itexcitesthesecondelectronfromthe3pstate
roughmeasureofthe width ofthe barrierthroughwhich
either to the 4s or to the 4p state. Thereby, we took the
the secondelectrontunnels. Physically,this meansthere
velocitygauge,andassumedthatthe firstelectroninter-
will be one set of orbits for which the effective potential
acts with the ion by a Coulomb or contact interaction.
barrierwillbewidened,andanotheroneforwhichitwill
be narrowed.
The results for the 3p 4s excitation are presented
In Fig. 4, we present the contributions of each of the →
in Fig. 5. An overall feature in the distributions are two
orbitsresultingfromthissplittingforafinal2sstate,for
main maxima along the p , n = 1,2, axis. These fea-
two different driving-field intensities. For simplicity, in nk
order to single out the effect of the modified action, we tures are mainly caused by the Vp(24es) prefactor for the
took the rescattering prefactor V to be constant. tunnel ionization of the second electron, which decays
p1e,kg
In general, the distributions differ quantitatively in a veryrapidlywithincreasingtransversemomentaandex-
factor between 1.5 and 1.7, depending on whether Im[t] hibit nodes near p2k = 0.5 Up. In general, we have
±
decreased[Figs.4(a)and(c)]orincreased[Figs.4(b)and verified that this prefactor determines the shape of the
p
(d)]. This shows that the splitting in this quantity is electron-momentum distributions. Secondary maxima,
small, and therefore both contributions are comparable. aroundone orderofmagnitude smaller,occurdue to the
Furthermore, the distributions displayed in Fig. 4 are rescattering prefactor V(3p→4s). This prefactor exhibits
p1e,kg
strikingly similar to those observed in Fig. 1 (see panels an annular shape around p =p =0.
1k 2k
(a) and (c) therein), for which only constant prefactors The existing experiments, however, do not lead to
have been considered. Indeed, the width of all distribu- distributions concentrated along the axis of the p p
1k 2k
tions, along the axis, is determined by the direct ATI plane. The results for Helium in the previous section
cutoff, i.e., 2 U p 2 U . At first sight,this is suggestthat a p state may lead to broaderdistributions.
p nk p
− ≤ ≤
p p
10
4 (a) 4 (b) 4 (a) 4 (b)
1/2U]p 2 2 1/2U]p 2 2
p/[2|| 0 0 p/[2|| 0 0
-2 -2 -2 -2
-4 Contact -4 Contact -4 Contact -4 Contact
-4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4
(c) (d) 4 (c) 4 (d)
4 4
1/2p/[U]2||p02 02 1/2p/[U]2||p -022 -202
-2 -2
Coulomb Coulomb
-4 Coulomb -4 Coulomb -4 -4
-4 -2 0 2 4 -4 -2 0 2 4
-4 -2p /0[U ]12/2 4 -4 -2p /[0U ]1/22 4 p1||/[Up]1/2 p2||/[Up]1/2
1|| p 1|| p
FIG. 6: Velocity-gauge electron momentum distributions for
FIG. 5: Velocity-gauge electron momentum distributions for
Argon in a linearly polarized, monochromatic field of fre-
Argon in a linearly polarized, monochromatic field of fre-
quency ω = 0.057 a.u. The electron is excited from 3p to
quency ω = 0.057 a.u. The electron is excited from 3p to
4s, i.e., E1g = 0.58 a.u., E2g = 1.02 a.u. and E2e = 0.40 4ap.u,.i.ien.,ouEr1cgal=cu0la.5ti8onas..u.T,hEe2lgas=er-1fi.e0l2dain.ute.nsaintydinEp2ean=els0(.3a1)
a.u. in our calculations. The laser-field intensity in panels
and (c), and panels (b) and (d) is I = 9×1013W/cm2 and
(a) and (c), and panels (b) and (d) is I = 9×1013W/cm2
I = 1.5×1014W/cm2, respectively. The type of interaction
and I =1.5×1014W/cm2, respectively. The typeof interac-
V taken is indicated in the figure. The contour plots have
tion V taken is indicated in the figure. The contour plots 12
12 been normalized to the maximum probability in each panel.
have been normalized to the maximum probability in each
Wehaveverified,however,thatthehighestyieldsonleft-hand
panel. We have verified, however, that the highest yields on
panels are between one and two orders of magnitude smaller
left-handpanelsarebetweenoneandahalfandtwoordersof
than those on theright-hand side.
magnitude smaller than those on theright-hand side.
cipal quantum number, this prefactor always exhibits
Forthatreason,wewillassumethat,instead,thesecond
nodes at lower absolute momenta than the ATI cutoff
electron is excited to the 4p state.
of p = 2 U . In fact, if V is taken to be constant,
Fig.6depictstheelectron-momentumdistributionsfor 2k p p2e
thedistributionsbecomeconsiderablybroaderandabet-
Argon under the assumption that the electron was ex- p
ter agreement with the experiments is obtained. This is
citedfrom3pto4p. Alldistributionsinthefigureexhibit
shown in Fig. 7, as ring-shaped distributions with four
fourmainmaxima,whicharebroaderthanthoseinFig.5
symmetric maxima at p = p and p = p . Such
1k 2k 1k 2k
and almost split at the axis p = 0. These maxima are −
nk maxima are mainly determined by the V(3p→4p) pref-
mainly determined by the prefactor V(4p), which has a p1e,kg
p2e actor. One should note, however, that this procedure
node at the axis for low transverse momenta and nodes
is inconsistent from a theoretical perspective: Since the
aroundp = U acrossawidetransverse-momentum
2k ± p electronhasbeenexcitedtothe4pstate,itshouldsubse-
range. Apart from that, the prefactors V(3p→4p) decay quently tunnel from it. Hence, the pertaining prefactor
p p1e,kg
moreslowlywithregardtothetransversemomenta. This must be taken.
impliesthat,uponintegration,alargermomentumregion
will be contributing to the NSDI yields. As in the previ-
ous case, this prefactor also leads to secondary maxima IV. CONCLUSIONS
(see Figs. 6.(c) and (d) for concrete examples). In all
cases, both in Figs. 5 and 6, a Coulomb type interaction
Our analysis of the rescattering-excitation ionization
mainly introduces a bias towards lower momenta.
(RESI) mechanism shows that the NSDI electron mo-
Despitetheabove-mentionedbroadening,theelectron- mentum distributions depend on the interplay between
momentum distributions in Fig. 6 are still considerably the relevant momentum-space regions, the type of in-
narrower than those observed in Ref. [14]. Within our teraction exciting the second electron, and the spatial
(4p)
framework, this constraint is imposed by the V pref- dependence of the bound states involved. We will com-
p2e
actor. In fact, we have verified that, for large prin- mence by discussing each of these issues separately.