Energy transfer among distant quantum systems in spatially shaped laser fields: Two H atoms with the internuclear separation of 5.29 nm (100 a.u.)

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by  Guennaddi K. Paramonov

Energy transfer among distant quantum systems in spatially shaped laser fields:
Two H atoms with the internuclear separation of 5.29 nm (100 a.u.)
Guennaddi K. Paramonov, Oliver Ku¨hn
Institut fu¨r Physik, Universita¨t Rostock, 18051 Rostock, Germany
and Andr´e D. Bandrauk
Laboratorie de Chimie Th´eorique, Facult´e des Sciences,
Universit´e de Sherbrooke, Sherbrooke, Qu´ebec, Canada J1K 2R1
The quantum dynamics of two distant H atoms excited by ultrashort and spatially shaped
laser pulses is studied by the numerical solution of the non-Born-Oppenheimer time-dependent
Schr¨odingerequationwithin athree-dimensional(3D)model, includingtheinternucleardistanceR
1 and the two z coordinates of the electrons, z1 and z2. The two 1D hydrogen atoms, A and B, are
assumed to be initially in their ground states with a large (but otherwise arbitrary) internuclear
1
0 separation of R= 100 a.u. (5.29 nm). Two types of a spatial envelope of a laser field linearly po-
2 larized along thez-axisare considered: (i) abroad Gaussian envelope,such that atom A is excited
by the laser field predominantly, and (ii) a narrow envelope, such that practically only atom A is
n excited by the laser field. With the laser carrier frequency ω = 1.0 a.u. and the pulse duration
Ja tp = 5 fs, in both cases an efficient energy transfer from atom A to atom B has been found. The
ionization of atom Bachieved mostlyafter theendof thelaserpulseisclosetoorevenhigherthan
3
that of atom A. It is shown that with a narrow spatial envelope of the laser field, the underlying

mechanismsof theenergy transferfrom AtoBandtheionization ofBare theCoulomb attraction
]
h of the laser driven electron by the proton of atom B and a short-range Coulomb repulsion of the
p two electrons when their wave functions significantly overlap in the domain of atom B. In the case
- of a broad Gaussian spatial envelope of the laser field, the opposite process also occurs, but with
t smaller probability: the energy is transferred from the weakly excited atom B to atom A, and the
n
a ionizationofatomAisalsoinducedbytheelectron-electronrepulsioninthedomainofatomAdue
u toa strong overlap of theelectronic wave functions.
q
[
I. INTRODUCTION interatomic Coulombic decay (ICD). Nowadays, ICD is
2
wellestablishedalsoexperimentally forinner-valenceex-
v
citation of many electron systems [2–6]. In recent work
6
Thelaserdrivendynamicsofdistantquantumsystems
3 [7], ICD was demonstrated experimentally for a helium
can depend in general on the spatial and temporal en-
2 dimer. Since helium atoms have no inner-valence elec-
velope of the applied laser field. For example, at a gas
6 trons, a different type of ICD is operative for this case.
. pressure of 1 atm., the interparticle distance is about It was thus concluded in [7] that since ICD in a he-
1
100 a.u. (5.29 nm). If such a system, e.g. composed
1 lium dimer takes place at interatomic distances up to
of H atoms, is excited by a laser field with the carrier
0 ≈ 12 a.u., no overlap of the electronic wave functions is
1 frequency ω = 1.0 a.u., corresponding to the ground- required for the process.
v: stateenergyofH2 atalargeinternucleardistanceR,the The present work is addressed to a quantum system
wavelength is λ= 861 a.u. (45.56 nm). If the laser field
i composed of two H atoms with the initial internuclear
X is focused within the diffraction limit onto a spot with
separationof100a.u. (5.29nm)whichisexcitedbyspa-
r the width λ, the Gaussian spatial envelope of the field tiallyshapedlaserpulses: spatiallybroadpulsesexciting
a may result in quite different electric field strengths for
both H atoms, and spatially narrow pulses exciting only
H atoms separated by about 100 a.u., especially at the
one H atom of the entire H-H system. The relative sim-
edges of the Gaussian spatial envelope. Although the
plicityofthe H-Hsystemunderconsideration(similarto
H atoms are far away from each other, their electron-
thatusedin[10])makesitpossibletotreatthelong-range
electron interaction should not be a priori neglected, es-
electronicmotionexplicitlytogetherwiththenuclearmo-
pecially upon their excitation by the laser field, because
tion such as to reveal the role played by the electron-
the electronic wave functions extend and vanish, strictly
electron interaction and by the overlap of the electronic
speaking, only at infinity. Therefore, the energy transfer
wave functions. An example of long-range laser-induced
among distant quantum systems, similar to that studied
electron transfer (LIET) in the one electron linear H+-
in [1–8], can be anticipated to occur in spatially shaped H+ atom-molecule system has been treated previously
laserfieldsaswell. Forultrashortlaserpulses,containing 2
withintheBorn-Oppenheimerapproximation[11]. Long-
onlyfewopticalcycles,onemustalsoconsiderthecarrier
range charge and energy transfer can occur also in large
envelope phase (CEP) of the pulse [9].
molecularsystems,asdescribedrecentlyinRef. [12]and
The long-range energy transfer from an excited atom references therein.
to its neighbor has been recently studied by Cederbaum The following two types of H-H systems will be dis-
et al. for molecular clusters [1] and is known as the tinguished in the present work: (i) a ‘molecular’ H-H
2
system, representing an elongated configuration of the polarization terms are neglected. The Coulomb poten-
H molecule, similar to that studied recently in [13] for tials in Eq. (2) read
2
long-range entanglement, and (ii) an ‘atomic’ H-H sys-
1 1
tem, representingtwo distant H atoms. Accordingly,the V (R)= , V (z ,z )= ,
pp ee 1 2
initial state of a molecular H-H system is assumed to R (z1−z2)2+α
be entangled by spin exchange and represented by the p
Heitler-London symmetric product of atomic wave func- 1 1
V (z ,R)=− − ,
tions, while the initial state of an atomic H-H system ep k
(z −R/2)2+β (z +R/2)2+β
k k
is not entangled – it is a direct-product state of atomic
p p (3)
wave functions. In both cases the excitation of H-H is
where k = 1,2, and the regularization parameters, α =
accomplished by laser pulses with (i) a broad Gaussian
0.1 × 10−3 and β = 1.995, have been chosen (similar
spatial envelope, such that both H atoms are excited by
to previous work [10]) such as to reproduce the ground-
thelaserfield,withatomAbeingexcitedpredominantly,
state (GS) energy of the H-H system at R = 100 a.u.
and (ii) with a narrow spatial envelope, such that only
(EGS =−1.0 a.u.).
atom A is excited by the laser field. H−H
The interaction of the H-H system with the laser field
The paper is organized as follows. The model of the
istreatedwithinthe semiclassicalelectricdipole approx-
H-HsystemandtechniquesusedaredescribedinSec.II.
imation by the Hamiltonian
Excitation, energy transfer, and ionization of an unen-
tangledatomicH-HsystemarepresentedinSec.III.Sec- 2
1∂A(t)
tion IVis devotedto the laser-drivendynamics ofan en- HˆSF(z1,z2,t)=− (1+γ) F(zk)zk, (4)
c ∂t
tangled molecular H-H system. The results obtained are Xk=1
summarized and discussed in the concluding Section V.
where γ =(1+2m )−1, A(t) is the vector potential, c is
p
thespeedoflight,andF(z)isthespatial-shapefunction,
or envelope, of the laser field. In the general case we
II. MODEL, EQUATIONS OF MOTION, AND
set F(z ) = F(z ), except for some model simulations
1 2
TECHNIQUES
detailed in Sec. IV.
The vector potential, A(t), is chosen in the following
Within the 3D four-body model of H-H excited by form:
the temporally and spatially shaped laser field the total c
Hamiltonian HˆT is divided into two parts, A(t)= ωE0sin2(πt/tp)cos(ωt+φ), (5)
HˆT(R,z1,z2,t)=HˆS(R,z1,z2)+HˆSF(z1,z2,t), (1) whereE0 istheamplitude, tp isthe pulsedurationatthe
base,ω isthelasercarrierfrequency,andφisthecarrier-
where Hˆ (R,z ,z ) represents the H-H system and envelope phase (CEP).Note that it has been shownpre-
S 1 2
Hˆ (z ,z ,t)describestheinteractionofthesystemwith viously [9] that the carrier phase of the laser pulse is
SF 1 2
important only for pulses having less than 15 optical cy-
the laser field. The applied laser field is assumed to be
cles. The definition of the system-field interaction by
linearly polarized along the z-axis, the nuclear and the
Eq. (4) via the vector potential, suggested in [13], as-
electronicmotionarerestrictedtothepolarizationdirec-
sures that the electric field E(t) = −1∂A(t)/∂t has a
tion of the laser electric field. Accordingly, two z coor- c
dinates of electrons, z andz , measuredwith respect to vanishing direct-current component, tpE(t)dt = 0, and
1 2 0
thenuclearcenterofmass,aretreatedexplicitlytogether satisfies Maxwell’s equations in the pRropagationregion.
with the internuclear distance R. A similar model has Itissuitabletodefine,onthebasisofEqs.(4)and(5),
been used previously in [10] for the H molecule, where the local effective-field amplitudes for atoms A and B as
2
each particle, electron or proton, is treated in 1D, i.e., z follows:
and R.
EA =E F(z ), EB =E F(z ), (6)
Thetotalnon-Born-OppenheimersystemHamiltonian 0 0 A 0 0 B
(employing a. u.: e=¯h=me =1) reads where zA =−50a.u., zB =50 a.u.. The respective time-
dependent electric fields acting on atoms A and B read
1 ∂2
HˆS(R,z1,z2)=−mp∂R2 +Vpp(R) EA,B(t)=E0A,B[sin2(πt/tp)sin(ωt+φ)
π
− sin(2πt/t )cos(ωt+φ)]. (7)
2 1 ∂2 ωtp p
+ − +V (z ,R) +V (z ,z ), (2)
Xk=1(cid:20) 2µe∂zk2 ep k (cid:21) ee 1 2 ThefirstterminEq.(7)correspondstoalaserpulsewith
a sin2-type temporal envelope, while the second, the so-
where m is the proton mass, µ = 2m /(2m + 1) called ‘switching’ term, appears due to the finite pulse
p e p p
is the reduced electron mass, and non-diagonal mass- duration [13, 14].
3
The 3D time-dependent Schr¨odinger equation for 1D Theimaginary-timepropagationshavebeenperformed
electrons (z , z ) and 1D protons (R), with a reduced, ‘non-interacting’ atoms, version of the
1 2
system Hamiltonian (2) wherein the nuclear V (R) and
∂ pp
i Ψ= Hˆ (R,z ,z )+Hˆ (z ,z ,t) Ψ, (8) electronic V (z ,z ) terms were omitted and only the
∂t S 1 2 SF 1 2 ee 1 2
(cid:2) (cid:3) CoulombicinteractionVep(z1(2),R)ofeachelectronwith
has been solved numerically with the propagation tech- its nearestprotonwas taken into accountalong with the
nique adapted from [15] for both electron and proton
three kinetic-energy terms of Eq. (2). The results ob-
quantummotion. Inparticular,calculationsfor the elec- tainedgaveadirect-productinitialstate. Forthesakeof
tronmotionhavebeenperformedbyusing200-pointnon-
comparison, we have also performed the imaginary-time
equidistant gridsfor the Hermite polynomials andcorre- propagationswith the complete systemHamiltonian(2).
spondingweightsforthenumericalintegrationonthein-
The results obtained proved to be practically identical
terval(−∞,∞)forthez1andz2coordinates. Forthenu- to those obtained in the case of two non-interacting H
clearcoordinateR,a256-pointequidistantgridhasbeen
atoms. The energy difference, for example, is less than
used on the interval [75 a.u., 125 a.u.]. The time-step of 1.8×10−5 a.u., implying that the Coulombic interaction
the propagationwas ∆t=0.021 a.u. (1 a.u.=24 asec).
V (z ,z ) of two H atoms in their ground states is neg-
ee 1 2
The wave functions of the initial states have been
ligible at R=100 a.u..
obtained by numerical propagation of the equation of
motion (8) in imaginary time without the laser field
(a)
(E =0).
0
Upon excitation of the H-H system by the laser field, A B
0.3−
theelectronicwavefunctionsofitsatomicAandBparts S
E
may overlap. In order to study the energy transfer from TI
AtoB,therespective‘atomic’energies,EA(t)andEB(t), BILI 0.2− P(z1) P(z2)
A
aredefinedonthebasisofEqs.(2)and(3)suchthattheir B
sum always gives the correct total energy of the entire RO 0.1−
P
H-H system. The definitions of ‘atomic’ energies E (t)
A
and E (t) for the H-H systems of atomic and molecular 0 . 0
B -100 -50 0 50 100
origin are different due to the different symmetry and z and z grids (a.u.)
1 2
entanglement of the respective wave functions and will
(b)
be specified in the following sections.
u.)0.15−
The ionization probabilities for atoms A and B have a.
beencalculatedfromthetime-andspace-integratedout- S (
E
going fluxes separately for the positive and the negative OP0.10−
L Gaussian
directions of the z and z axes at z = ±91 a.u.. E
1 2 1,2 V
Specifically, we calculated four ionization probabilities: N
I (z = −91a.u.) and I (z = −91a.u.) for atom A, L E0.05−
A 1 A 2 A Narrow
and IB(z1 = 91a.u.) and IB(z2 = 91a.u.) for atom B. ATI
At the outer limits of the z-grids, absorbing boundaries SP0.0 0 -100 -50 0 50 100
have been provided by imaginary smooth optical poten-
z and z grids (a.u.)
1 2
tials adapted from that designed in [16]. Similar optical
potentials have been also provided for the R-axis but,
FIG. 1. The initial unentangled direct-product state of the
in practice, the wave-packet never approached the outer
H-H system and the spatial envelopes of the applied laser
limits of the R-grid. pulses. (a) - electron probabilities: P(z1) is the probability
tofindelectrone1initiallybelongingtoatomAatz=z1with
anyvaluesoftheothertwocoordinates,Randz2;probability
III. EXCITATION OF H-H FROM AN P(z2)hasasimilarmeaningforelectrone2initiallybelonging
UNENTANGLED DIRECT-PRODUCT INITIAL to atom B; (b) - spatial envelopes of applied laser fields: the
STATE broadGaussianenvelopeofEq.(14)andthenarrowenvelope
of Eq. (16).
The spatial part of the initial unentangled direct-
product ground-state wavefunction of H-H of an atomic Theinitialunentangleddirect-productstateoftheH-H
configuration used in the imaginary time propagation is systemrepresentingtwonon-interactingHatoms,Aand
definedwiththe unsymmetrizedHeitler-Londonelectron B, is presented in Fig. 1(a) by the electron probabilities
wave function as follows: P(z ) and P(z ), which are defined as follows:
1 2
Ψ(R,z ,z ,t=0)=Ψ (z )Ψ (z )Ψ (R), (9)
1 2 1SA 1 1SB 2 G
where Ψ1SA(z1) = e−|z1−zA| at zA = −50 a.u., P(z1)=Z dRZ dz2|Ψ(R,z1,z2)|2 (10)
Ψ1SB(z2) = e−|z2−zB| at zB = 50 a.u., and ΨG(R) is
a proton Gaussian function centered at R=100 a.u.. for electrone initially belonging to atomA with proton
1
4
p , and similarly, of the proton-proton and the electron-electron interac-
A
tion are assumed to be equally shared between atoms A
P(z )= dR dz |Ψ(R,z ,z )|2, (11) andB.Thekineticenergyofelectrone1andtheenergyof
2 1 1 2
Z Z Coulombic interaction of both electrons, e and e , with
1 2
protonp areentirelyassignedto atomA.Similarly,the
A
for electrone initially belonging to atom B with proton
2 kineticenergyofelectrone andtheenergyofCoulombic
2
p . Theseelectronprobabilitiesgivetheoverallprobabil-
B interaction of both electrons with proton p are entirely
B
itytofindanelectronataspecifiedpointofthez-axisat
assignedto atomB.The sum of‘atomic’energiesalways
anypositionoftheotherelectronandatanyinternuclear
gives the correct total energy of the entire H-H system.
distance.
Notice,inparticular,thatthefactthatthekineticenergy
Aftercalculationoftheunentangledinitialstateofthe
of electron e /e is entirely assignedto atom A/B corre-
1 2
atomic H-H system, its laser-driven quantum dynamics
sponds to the initial electron probabilities P(z )/P(z )
1 2
in realtime has been exploredwith the complete system
inthe unentangledatomicstate [Fig.1(a)]. Indeed, elec-
Hamiltonian of Eq. (2). The spatial envelopes of the
trone islocalizedinthevicinityofprotonp ofatomA
1 A
applied laser pulses, Gaussian and narrow, defined by
andelectrone islocalizedinthevicinityofprotonp of
2 B
Eqs. (14) and Eq. (16) below, are presented in Fig. 1(b)
atom B. Therefore, if e.g. the laser pulse with a narrow
to illustrate the local effective-field strengths acting on
spatialshape[Fig.1(b)]isusedtoexcitetheextendedH-
atoms A and B.
H system, only the ‘atomic’ energy E (t) will increase,
A
Twopossiblechoicesforthelasercarrierfrequencyare:
while E (t) will not be affected.
B
ω = 0.5 a.u. (corresponding to the ground-state energy
The left panel of Fig. 2 presents the dynamics of the
of an H atom), and ω = 1.0 a.u. (corresponding to the
atomic H-H system excited by the laser pulse with the
ground-state energy of H-H at large R). Our numerical
Gaussian spatial envelope
simulations have shown that at ω =0.5 a.u., an efficient
ionizationofAtakesplace,whiletheenergytransferfrom F (z)=exp{−[(z−z )/λ]2}, (14)
G 0
A to B and the ionization of B are not efficient. In con-
trast, at ω = 1.0 a.u., the energy transfer from A to B where λ = 861 a.u. (45.56 nm) and the laser field is
andthe ionizationofBareefficient. This implies the ex- assumed to be focused onto a spot centered at z , where
0
istenceofanoptimallaserfrequencysuitableforthemost z =−1.5λ[z =-1291.5a.u. (-68.34 nm)]: cf. Fig. 1(b).
0 0
efficient energy transfer. This issue will be addressed in The laser pulse parameters are: ω = 1.0 a.u., t = 5 fs,
p
a forthcoming work; below we present the results ob- andE =1.0a.u. (E =5.14×109 V/cm, intensity IA =
0 0 0
tained at ω = 1.0 a.u.. With the laser pulse duration 3.5 × 1016 W/cm2). Accordingly, the spatial envelope
tp = 5 fs, as used throughout this work, the number of function F(z) in the interaction Hamiltonian of Eq. (4)
optical cycles during the pulse, Nc = ωtp/2π, is about is set equal to FG(z) of Eq. (14), and the effective-field
33. Therefore,the carrierphaseofthe laserfieldφisnot amplitudes for atoms A and B defined by Eq. (6) are:
important [9] and set equal to zero in our simulations EA =0.125a.u. (intensityIA =5.48×1014W/cm2)and
0 0
below [φ=0 in Eqs. (5) and (7)]. EB = 0.088 a.u., (IB = 2.72×1014 W/cm2), implying a
0 0
In orderto study the energy transferfrom A to B, the dominantexcitationofatomA,whichisclearlyobserved
respective ‘atomic’energies,EA(t) andEB(t), havebeen in Fig. 2(a).
defined on the basis of Eqs. (2) and (3) as follows: From Fig. 2(a) we see that the ‘atomic’ energy E (t)
A
is controlled by the applied laser pulse: it increases in
1 ∂2 1 1
the first half of the pulse and decreasesto the end of the
E (t)= Ψ(t) − + V (R)+ V (z ,z )
A (cid:28) (cid:12)(cid:12) 2mp∂R2 2 pp 2 ee 1 2 pulse. In contrast, the ‘atomic’ energy EB(t) does not
(cid:12) follow the applied laser pulse: it slowly increases up to
(cid:12)
theendofthelaserpulseandevenexceedsE (t)shortly
A
1 ∂2 2 1 aftertheendofthepulse. Asimilarbehaviorisfoundfor
− − Ψ(t) (12)
2µe∂z12 Xk=1p(zk+R/2)2+β(cid:12)(cid:12)(cid:12) (cid:29) wthheil‘eattohmeilca’seiorn-iinzdauticoendsiIoAniaznadtioInBIpArersiesnestefdasitninFitgh.e2s(bec)-:
(cid:12)
ondhalfofthepulse,theionizationprobabilityI sharply
for atom A and similarly, B
rises after the end of the pulse. Such a behavior will be
1 ∂2 1 1 referred to below as a ‘sequential’ ionization, implying a
E (t)= Ψ(t) − + V (R)+ V (z ,z )
B (cid:28) (cid:12) 2m ∂R2 2 pp 2 ee 1 2 delayedinteractionduetoLIET.Thetime-dependentex-
(cid:12) p pectationvalues hR(t)i, hz (t)i andhz (t)i are presented
(cid:12) 1 2
(cid:12) in Fig. 2(c). The time-dependent deviations of the ex-
1 ∂2 2 1 pectation values hz1(t)i and hz2(t)i,
− − Ψ(t) , (13)
2µe∂z22 Xk=1 (zk−R/2)2+β(cid:12)(cid:12) (cid:29) ∆z1(t)=hz1(t)i−hz1(t=0)i,
p (cid:12)
(cid:12)
for atom B. From Eqs. (12) and (13) we see that the ki-
neticenergyofnuclearmotionandthepotentialenergies ∆z (t)=hz (t)i−hz (t=0)i, (15)
2 2 2
5
u.) (a) u.) (e)
S (a.-0.48− EA S (a.-0.48− EA
E E
GI GI
R R
E E
N N
E-0.49− E-0.49−
C' C'
MI MI
O EB O EB
T T
`A-0.50− `A-0.50−

0 5 10 15 20 0 5 10 15 20
TIME (fs) TIME (fs)
NS 0.012−(b) NS 0.012−(f) IB(z2=91 a.u.)
O O
TI IA(z1=-91 a.u). TI IA(z1=-91 a.u.)
A A
Z Z
NI0.008− IB(z2=91 a.u). NI0.008−
O O
C' I C' I
MI0.004− MI0.004−
O O
T T
A A
` `
0.0 0 0.0 0
0 5 10 15 20 0 5 10 15 20
TIME (fs) TIME (fs)
S (a.u.) 100−(c) < R > S (a.u.) 100−(g) < R >
E E
U U
L 50− L 50−
A < z > A < z >
V 2 V 2
N Δz (×2) N Δz (×2)
TIO 0 − Δz1 (×2) TIO 0 − Δz1 (×2)
A 2 A 2
PECT -50− < z1 > PECT -50− < z1 >
X X
E E
0 5 10 15 20 0 5 10 15 20
TIME (fs) TIME (fs)
V/cm) 0.60−(d) V/cm) 0.10−(h)
G G
D ( 0.30− D ( 0.05−
L L
FIE 0.0 0 FIE 0.0 0
C C
RI-0.30− RI-0.05−
T T
C C
LE-0.60− LE-0.10−
E E
0 5 10 15 20 0 5 10 15 20
TIME (fs) TIME (fs)
FIG. 2. Quantumdynamics of H-Hexcited from an unentangled direct-product initial state byspatially shaped laser fields: a
Gaussian spatial envelope (a-d) and a narrow spatial envelope (e-f). (a) and (e) - the energy transfer; (b) and (f) - ‘atomic’
ionizations[ionizationprobabilitiesIA(z1 =91a.u.)andIB(z2=−91a.u.)areoftheorderof10−6 andarenotshown];(c)and
(g) - expectation valueshRi,hz1i and hz2i (deviations of theexpectation values,∆z1 and∆z2,definedbyEqs. (15)are scaled
up by a factor of two); (d) and (h)- local effective laser fields EA(t) [Eqs. (6) and (7)] acting on atom A.
plotted in Fig. 2(c) are scaled up by a factor of two. We In order to clarify these findings further, we have per-
seefromFig.2(c)that∆z (t)>0,implyingattractionof formedasimilarnumericalsimulationwithanarrowspa-
1
electrone byprotonp . Incontrast,∆z (t)<0,imply- tial envelope of the laser pulse acting practically only on
1 B 2
ing attractionofelectrone by protonp . It is alsoseen atom A of the entire H-H system. Specifically, the fol-
2 A
fromFigs.2(a),2(b)and2(c)thatboththeenergytrans- lowing sin2-type spatial envelope of the laser pulse was
ferfromAtoBandthe‘sequential’ionizationofBinthe used for z =z and z =z :
1 2
positivedirectionofthez -axiscorrelatewithasmallde-
2
π(z−z )
creaseofthe spatialseparationofelectrons,|hz2i−hz1i|, FN(z)=sin2 a , za ≤z ≤zb, (16)
suchthatthe electron-electronrepulsion(EER)becomes (cid:20) zb−za (cid:21)
effective.
where z =−60 a.u., z =−40 a.u., and F (z)=0 oth-
a b N
6
erwise, as illustrated in Fig. 1(b). Accordingly, the spa-
0.003
tialenvelopefunctionF(z)inEq.(4)issetequaltoF (z)
N
P(z) P(z)
of Eq. (16). The amplitude of the pulse, E0 = 0.02 a.u. ES 1 2
(I0 =1.4×1013W/cm2),ischosensuchthatthe‘atomic’ LITI 0.002
energy EA(t) gained in the field with the narrow spa- BI
A
tial envelope [Fig. 2(e)] is close to that gained with the B
O
Gaussian spatial envelope [Fig. 2(a)]. Note that, defined R 0.001
P
by Eq. (7), the effective-field strength EA(t) of the laser P(z)
P(z) 2
pulse with the narrow spatial envelope [Figs. 1(b) and P(z1) 1
0
2(h)] is much smaller than that of the laser pulse with -100 -50 0 50 100
the Gaussian spatial envelope [Figs. 1(b) and 2(d)], but z1 and z2 grids (a.u.)
the energy transfer from A to B is by about 30% more
efficient [Fig. 2(e)]. Also note that, due to the effective- FIG. 3. Electron probabilities P(z1) (solid line) and P(z2)
field amplitude acting on atom B in the current case of (dashed line) at theend of the5 fs laser pulse with a narrow
thenarrowspatialenvelopeofthepulseisEB =0,theef- spatial envelope [Eq. (16) and Fig. 1(b)] which excites only
ficientenergytransfertoatomBisattribut0edentirelyto electron e1 (coordinate z1) in thedomain of atom A.
the electron-electron repulsion V (z ,z ) and electron-
ee 1 2
proton attraction V (z ,R) in the vicinity of atom B
ep 1(2)
(z1(2) =zB). ThisalsoappliestotheionizationofatomB e1 reaches the domain z1 > 0 where it is attracted and
inthepositivedirectionofthez2-axis[Fig.2(f)],whichis acceleratedby proton pB, localized at zB =50 a.u.. Due
not only by about 30% more efficient than that induced to the electron-proton Coulombic attraction, the prob-
by the laser pulse with the broad Gaussian spatial enve- ability P(z1) has its local maximum at z1 = zB, cor-
lope [Fig. 2(b)], but I is even larger than I at t>7 fs responding to LIET of Ref. [11]. Simultaneously, when
B A
[Fig. 2(f)], implying that the ‘sequential’ ionization IB, electrone1 approachesthedomainofzB =50a.u.,where
induced by EER,is more efficientthan the laser-induced the initial probabilityP(z2) of electrone2 has the global
ionization I . maximum[Fig.1(a)],theelectron-electronCoulombicre-
A
pulsion becomes very strong. Therefore, the probability
Taking into account that in the unentangled direct-
P(z ) is extended into the domain of z > z (Fig. 3),
product state (9), electrons e and e are well localized 2 2 B
1 2 givingrisetotheionizationI inthepositivedirectionof
on the z-axis [Fig. 1(a)], one can assume that the time- B
thez -axis. Finally,thewavefunctionisabsorbedbythe
dependent expectation values hz (t)i and hz (t)i repre- 2
1 2 imaginary optical potential at z ≥ 91 a.u.. Note that
sent the positions of electrons. A closer look at the data 2
in the current case of a narrow spatial envelope of the
presented in Figs. 2(c) and 2(g) for hz (t)i and hz (t)i
1 2 laser field [Fig. 1(b)] one can clearly distinguish between
shows that the two electrons of H-H do not approach
the laser-induced ionization on the negative direction of
each other closer than 98 a.u. (5.18 nm). Therefore, at
thez -axis,representedinFig.2(f)byI (z =−91a.u.),
a first glance, the energy transfer from A to B and the 1 A 1
and the ‘sequential’ EER-induced ionization on the pos-
‘sequential’ ionization of B take place, similarly to ICD
itive direction of the z -axis, represented in Fig. 2(f) by
[1–8], without any noticeable overlap of the respective 2
I (z =91a.u.). Also note that the ionizationprobabili-
electronic wave functions. Nevertheless, a closer look at B 2
ties I (z = 91a.u.) and I (z = −91a.u.) are less than
spatialdistributionsoftheelectronicwavefunctionsillus- A 1 B 2
10−6 at t=5 fs.
tratedin Fig. 3 revealsthat EERis in fact a short-range
process which takes place in the vicinity of atom B at
z =50 a.u.. Here, the Coulombic attractionof electron
B
e by protonp effectively localizes electrone . Accord-
1 B 1
ingly, the overlap of the electronic wave functions has a 0.0004 P(z) P(z)
S 1 2
sharp narrow maximum at z = 50 a.u. (Fig. 3), which E
B TI
results in a strong EER. LI
BI
A
Electron probabilities P(z1) and P(z2) presented in OB 0.0002
Fig. 3 correspond to the end of the 5 fs laser pulse with R
P
the narrow spatial envelope defined by Eq. (16). This P(z) P(z)
1 2
pulse excites practically only electron e1 in the vicinity P(z1) P(z2)
0
of zA = −50 a.u., therefore the probability distribution -100 -50 0 50 100
P(z1) at t = 5 fs is broadened in comparison to the ini- z1 and z2 grids (a.u.)
tial one [Fig. 1(a)] in both z < z and z > z di-
1 A 1 A
rections. The laser-induced extension of P(z1) into the FIG. 4. Electron probabilities P(z1) (solid line) and P(z2)
domain of z < z gives rise to the ionization I on the (dashedline)at2.5fsaftertheendofthe5fslaserpulsewith
1 A A
negative direction of the z -axis. At z ≤ −91 a.u., the aGaussianspatialenvelopewhichexcitesbothelectrons,with
1 1
wave function is absorbed by the imaginary optical po- the excitation of electron e1 (coordinate z1) being dominant
tential. In contrast, at z >z , the laser-drivenelectron in comparison to that of electron e2 (coordinate z2).
1 A
7
In the case of the broad Gaussian spatial envelope of IV. EXCITATION OF H-H FROM AN
thelaserpulsecenteredatz =−1291.5a.u. (-68.34nm), ENTANGLED INITIAL STATE
0
both electrons, e and e , are excited by the laser field
1 2
andthereforeionizationinbothpositiveandnegativedi- The H-H molecular system represents an elongated
rectionsofthez-axisisinducedbyboththelaserfieldand configuration of the H molecule. Therefore, its ini-
2
subsequently by EER. The electron probabilities P(z1) tial electronic ground state is entangled via exchange
(solidline)andP(z2)(dashedline)fortheGaussianspa- [7, 10, 20]. The spatial part of the initial entangled
tial envelope of the 5 fs laser pulse are plotted in Fig. 4 ground-statewavefunctionofH-Hforasingletelectronic
at t = 7.5 fs, when both ionization probabilities plotted state is given by
in Fig. 2(b) approach their maximum values. Although
bothelectronsareexcitedbythelaserfield,theexcitation Ψ(R,z ,z ,t=0)=[Ψ (z )Ψ (z )
1 2 1SA 1 1SB 2
ofelectrone inthedomainofatomAisabout1.4times
1
strongerthanthe excitationofelectrone inthe domain
2
of atom B. Accordingly, the local maximum of P(z1) at +Ψ1SB(z1)Ψ1SA(z2)]ΨG(R), (17)
z = 50 a.u. (Fig. 4) is almost 4 times higher than the
B
where Ψ (z ) are defined by Eq. (9) and Ψ (R) is
local maximum of P(z2) at zA =−50 a.u. and therefore 1SA,B 1,2 G
a proton Gaussian function centered at R = 100 a.u..
the EER-induced ionization in the positive direction of
Imaginary time propagations have been performed with
the z -axis is stronger than the EER-induced ionization
2
the complete system Hamiltonian (2). In the entangled
in the negative direction of the z -axis. In contrast, the
1
initial state of the H-H system, electron probabilities
laser-induced ionization in the negative direction of the
P(z ) and P(z ) are identical to each other.
z -axis is stronger than that in the positive direction of 1 2
1
For the simulations with this entangled initial state
the z -axis. Indeed, it is seen from Fig. 2(b) that at the
2
(17), the ‘atomic’ energies E (t) and E (t) are defined
end of the 5 fs laser pulse, the ionization probability I A B
B
on the basis of Eqs. (2) and (3) as follows:
is more than twice smaller than I , while I ≈ I at
A B A
t>10 fs.
1 ∂2 1 2 1 ∂2 1
E (t)= Ψ(t) − − + V (R)
A (cid:28) (cid:12) 2m ∂R2 2 2µ ∂z2 2 pp
It is seen from Fig. 4 that both local maxima, P(z1) (cid:12)(cid:12) p kX=1 e k
at z = −50 a.u. and P(z ) at z = 50 a.u., are sig- (cid:12)
1 2 2
nificantly smaller than the local maximum of P(z ) at
1 2
z =−50a.u. producedbythe laserpulsewithanarrow 1 1
1 + V (z ,z )− Ψ(t) (18)
spatialenvelope(seeFig.3). ThereforetheEER-induced 2 ee 1 2 (z +R/2)2+β(cid:12) (cid:29)
kX=1 k (cid:12)
ionization in the case of the broad Gaussian spatial en- p (cid:12)
(cid:12)
velope of the laser pulse [Fig. 2(b)] is smaller than that for atom A, and
in the case of the narrow spatial envelope of the pulse
[Fig. 2(f)]. 1 ∂2 1 2 1 ∂2 1
E (t)= Ψ(t) − − + V (R)
B (cid:28) (cid:12) 2m ∂R2 2 2µ ∂z2 2 pp
(cid:12) p Xk=1 e k
It can be concluded from Fig. 4 that in the general (cid:12)
(cid:12)
case, when both distant atoms are excited by the laser
field, energy is transferred from A to B and from B to 1 2 1
A, and EER-induced ionization occurs in both A and B + 2Vee(z1,z2)− (z −R/2)2+β(cid:12)Ψ(t)(cid:29) (19)
parts of H-H along with the laser-induced ionization. If, kX=1 k (cid:12)
p (cid:12)
for example, the laser pulse has a wide Gaussian spatial (cid:12)
for atom B. From Eqs. (18) and (19) we see that the
envelope and/or is centered at z =0, the mutual energy
kinetic energies of nuclear and electronic motion as well
transfers and the EER-induced ionization probabilities
as the potential energies of the proton-proton and the
are substantial for both distant atoms A and B even at
electron-electron Coulombic interaction are assumed to
a large internuclear separation.
be equally shared between atoms A and B. The energy
of Coulombic interaction of both electrons with proton
Finally we note that the appearance of sharp local p is assignedto atomA, while the energyofCoulombic
A
maxima of P(z ) at z = 50 a.u. and of P(z ) at interaction of both electrons with proton p is assigned
1 1 2 B
z = −50 a.u. (see Figs. 3 and 4) confirm spreading, to atom B. The sum of these ‘atomic’ energies always
2
delocalization, and non-factorization of the initially fac- gives the correct total energy of the entire H-H system.
torized and well localized on the z-axis wave function The choice of the electron kinetic energies in Eqs. (18)
of H-H [Fig. 1(a)] and can therefore [17] be treated as and (19) corresponds to the initial electron probabilities
the emergence of long-range entanglement, or quantum P(z ) and P(z ) in the entangled molecular state. Both
1 2
non-local connection, at R = 100 a.u.. Recently this electronse ande arelocalizedwiththe50%probability
1 2
long-range entanglement attracted considerable interest inthevicinityofprotonp ofatomAandiftheextended
A
both in theory and experiment [13, 18–21]. H-H system is excited e.g. by the narrowly shaped laser
8
ES (a.u.)-0.48−(a) EA ES (a.u.)-0.48−(d) EA
GI GI
R R
E E
N-0.49− N-0.49−
E E
C' C'
MI EB MI EB
O O
AT-0.50− AT-0.50−
` `

0 5 10 15 20 0 5 10 15 20
TIME (fs) TIME (fs)
NS 0.012−(b) NS 0.012−(e) IBTotal
TIO IATotal TIO IATotal
A A
Z Z
ONI0.008− IBTotal ONI0.008−
C' I C' I
MI0.004− MI0.004−
O O
T T
A A
' '

0 5 10 15 20 0 5 10 15 20
TIME (fs) TIME (fs)
V/cm) 0.60−(c) V/cm) 0.10−(f)
G G
D ( 0.30− D ( 0.05−
L L
FIE 0.0 0 FIE 0.0 0
C C
RI-0.30− RI-0.05−
T T
C C
LE-0.60− LE-0.10−
E E
0 5 10 15 20 0 5 10 15 20
TIME (fs) TIME (fs)
FIG. 5. Quantum dynamics of H-H excited from the entangled initial state (17) by spatially shaped laser fields [Fig. 1(b)]: a
broad Gaussian spatial envelope(left panel)andanarrow spatialenvelope(rightpanel). (a) and(d)-theenergy transfer; (b)
and (e) - ‘atomic’ ionizations; [the total ionization probabilities ITAotal and ITBotal are defined by Eq. (20)]; (c) and (f) - local
effective laser fields EA(t) [Eqs. (6) and (7)] acting on atom A.
pulseofFig.1(b),bothelectronsgiverisetothe‘atomic’ timescale of t = 20 fs from atom A to atom B in
energy E (t), each with 50% probability. the entangled state is very similar to that transferred
A
The quantum dynamics of H-H excited from the en- from A to B in the unentangled direct-product state:
tangled initial state (17) by spatially shaped laser pulses ∆EB ≈ 0.01 a.u. for the broad Gaussian spatial en-
is presented in Fig. 5. The left panel corresponds to velope, and ∆EB ≈ 0.013 a.u. for the narrow spatial
the broad Gaussian spatial envelope centered at z = envelope.
0
−1291.5a.u. (-68.34nm)andtherightpanelcorresponds (ii)Themaximum‘atomic’energyE (t)gainedduring
A
tothenarrowspatialenvelopecenteredatzA =−50a.u. thelaserpulsebyatomAintheentangledstateissmaller
(-2.65nm),seeFig.1(b). Forthesakeofcomparison,the thanthatgainedintheunentangleddirect-productstate.
laserfieldsusedarethesameasinthepreviouscaseofthe In contrast, the maximum ‘atomic’ energy E (t) gained
B
unentangled direct-product initial state [see Figs. 1(b) during the pulse by atom B in the entangled state is
and 2]. Specifically, the Gaussian spatial envelope is de- substantially larger than that gained in the unentangled
fined by Eq. (14), the narrow spatial envelope is defined direct-product state.
by Eq. (16), the carrier frequency of the laser pulse with
(iii)Moreover,inthecaseoftheentangledinitialstate,
thesin2-typetemporalenvelopeofEq.(7)isω =1.0a.u.,
the ‘atomic’energyE (t)is controlledbythe laserpulse
and the pulse duration at the base is t =5 fs. B
p similarly to E (t), even when only atom A is excited
A
From the comparisonof the quantum dynamics of the by the laser pulse with a narrow spatial envelope: the
unentangled state [Figs. 2(a) and 2(e)] to that of the energy E (t) increases in the first half of the laser pulse
B
entangledstate [Figs. 5(a)and 5(d)] the following obser- and decreases at the end of the pulse, in contrast to the
vations are made. caseoftheunentangleddirect-productinitialstate. This
(i) The overall energy ∆E transferred on a long observationimpliesthatthechangesmadebytheapplied
B
9
laser field to the entangled wave function in the domain (a)
of atom A at z ≈ −50 a.u. result in simultaneous
cohfatnhgeeswianvaetofumn1cB,t2ioantzb1,y2 ≈exc5h0aan.gue..duSeutcohtahneseynmtamngetlerdy S (a.u.)-0.498− EEAB -- dsoalsihded
behaviour is very important for a long-range quantum GIE
communication among distant quantum systems [19]. R
E-0.50 0 −
N
(iv) After the end of the laser pulse, the ‘atomic’ en- E
ergies E (t) and E (t) demonstrate out-of-phase oscil- C'
A B MI
lations: slow oscillations in the case of the unentangled O
T-0.502−
direct-productinitialstate [Figs.2(a)and2(e)]andslow A
'
oscillations modulated with very fast ones in the case of 0 0.2 0.4 0.6 0.8 1
TIME (fs)
the entangledinitialstate [Figs.5(a)and5(d)]. The am-
(b)
plitudes of both slow and fast oscillations of E (t) and
A
E (t) decrease with time, which may indicate the for-
mBationofaquasi-stableconfigurationoftheexcitedH-H a.u.) 0.006−
system. > ( 0.003−
2
It is instructive, before proceeding, to consider the <z
laser-driven dynamics of the entangled H-H system in d 0.000−
n
tahenainrriotiwalssptaagtiealofenitvseleoxpceita(t1i6o)nwbyhicthheelxacsietrespuellseectwroitnhs z> a1-0.003−
<
e and e only in the domain of atom A. In Fig. 6, the
1 2 -0.006−
laser-driven dynamics of entangled H-H is presented on

0 0.2 0.4 0.6 0.8 1
the timescale of 1 fs. Time-dependent ‘atomic’ energies,
TIME (fs)
E (t) and E (t), and expectation values of electronic
A B (c)
coordinates, hz (t)i and hz (t)i, are shown in Figs. 6(a)
1 2 m)
and 6(b) respectively. The time-dependent laser field is V/c 0.03−
shown in Fig. 6(c). G
It is seen from Fig. 6(a) that the ‘atomic’ energies D (
L
EA(t) and EB(t) oscillate out-of-phase with respect to FIE 0.0 0
each other, with EA(t) being in-phase and EB(t) being RIC
out-of phase with the applied laser field [Fig. 6(c)]. T
C
As seen from Figs. 6(b) and 6(c) the electrons also LE-0.03−
E
follow the applied laser field. Note that expectation val- 0 0.2 0.4 0.6 0.8 1
ues hz (t)i and hz (t)i in the entangled H-H system are TIME (fs)
1 2
identical and not distinguishable in Fig. 6(b). A perfect
electron-field following at the laser carrier frequency be-
ing as high as 1 a.u. is very interesting. Previously, the FIG. 6. The initial (1 fs) stage of excitation of H-H from
electron-fieldfollowing on the level of expectation values the entangled initial state [Eq. (17)] by the laser pulse with
a narrow spatial envelope [Eq. (16)] acting on atom A. (a) -
of electronic coordinates have been explored only in the
infrared [15] and near-infrared [22] domains of the laser time-dependent‘atomic’energiesEA(t)(solidline)andEB(t)
(dashed line); (b) -electron-field in-phasefollowing; (c) -the
carrier frequency. The electron-field following at high
laser field acting on atom A.
laserfrequencies is reminiscentof the well-knownrecolli-
sion model of Corkum [23]. However, an important fea-
ture ofFig.6(b) isthatelectronsfollowthe appliedlaser
field in-phase, while according to the theoretical model as follows. At ω <0.1 a.u., the electron is in the ground
used in [23]and numericalresults of [15]electrons follow state, the polarizability is negative and, therefore, the
thefieldout-of-phase: hz(t)idecreaseswhenelectric-field electron follows the applied laser field out-of-phase. In
strength E(t) increases. A detailed study of the electron contrast,atω =1 a.u., the electronis wellabove the ex-
dynamics showed that, probably due to their finite al- cited state, the polarizability changes the sign, and elec-
beit very small mass, the electron do not react to the tron follows the field in-phase. Similar behaviour has
first half-cycle of the applied field at ω = 1 a.u. and beenobservedinarecentwork[24]formolecularionH+2
follows the field in-phase at t > 0.15 fs. On the other excited at wavelength λ =800 nm (ω = 0.057 a.u.). At
hand, the out-of-phaseelectron-fieldfollowingtake place fixed R=2 a.u., the electron is in the ground state and
at ω < 0.1 a.u.. Similar results have been obtained for follows the field out-of-phase,whereas at R=7 a.u., the
the unentangled H-H system excited by the laser pulse laser carrier frequency ω = 0.057 a.u. is larger than the
withanarrowspatialenvelope: theelectrone1ofatomA energydifference E1sσu−E1sσg, and the electronfollows
follows the applied field in-phase at ω =1 a.u. and out- the field in-phase.
of-phase at ω <0.1 a.u.. These results can be explained Comingbacktothelaser-drivendynamicsoftheentan-
10
gledH-Hsystemonthelongtimescaleof20fs(Fig.5),we the entangled initial state is required. To this end, we
note that ionization of atom B starts only in the second performed several model simulations with the entangled
halfofthelaserpulse[seeFigs.5(b)and5(e)]. Duetothe initialstateexcitedinthedomainofatomAbythelaser
symmetry of the entangled wavefunction, the ionization pulses with narrowspatial envelopes defined by Eq. (16)
probabilities of electrons e and e are identical to each for various model assumptions of the system-field inter-
1 1
other both for the positive and the negative directions action to be specified below. The results obtained are
of the z-axes: I (z = −91a.u.) = I (z = −91a.u.) presented in Figs. 7 and 8.
A 1 A 2
and IB(z1 = 91a.u.) = IB(z2 = 91a.u.). Therefore, in Electron probabilities P(z1) and P(z2) presented in
Figs. 5(b) and 5(e) the total ionization probabilities, Figs. 7(a) and 7(b) correspond to the end of the 5 fs
laser pulse with a narrow spatial envelope [Fig. 1(b)]
ITotal =I (z =−91a.u.)+I (z =−91a.u.),
A A 1 A 2 which excites both electrons e1 and e2 in the vicinity
of z = −50 a.u. (atom A) and does not affect elec-
A
trons e and e in the vicinity of z =50 a.u. (atom B).
ITotal =I (z =91a.u.)+I (z =91a.u.), (20) 1 2 B
B B 1 B 2 Apparently, electron probabilities P(z ) and P(z ), pre-
1 2
sented in Figs. 7(a) and 7(b) at t = 5 fs, are extended
are plotted. From the comparison of Figs. 2(b) and 2(f)
in comparison to their initial ones in both z < z
to Figs. 5(b) and 5(e) one can easily see that the total 1,2 A
time-dependent ionization probabilities ITotal and ITotal and z1,2 >zA directions. The laser-induced extension of
A B P(z ) and P(z ) into domains z <z gives rise to the
used in the case of the entangled initial state are almost 1 2 1,2 A
ionization I in the negative directions of the z -axes.
equal (but not identical) to the time-dependent ioniza- A 1,2
At z ≤−91 a.u., the wavefunction is absorbedby the
tion probabilities I (z = −91a.u.) and I (z = 91a.u.) 1,2
A 1 B 2
imaginary optical potentials. In contrast, at z > z ,
usedinthecaseofthe unentangleddirect-productinitial 1,2 A
the laser-driven electrons e and e reach the domains
state. Onecanconcludethereforethatthe entanglement 1 2
z > 0 where they are attracted and accelerated by
oftheinitialstateofH-H,includingitsexchangesymme- 1,2
proton p , localized at z = 50 a.u., more and more
try,doesnotchangeitsionizationprobabilityincompari- B B
efficiently. Due to the electron-proton Coulomb attrac-
sontothenon-symmetricunentangleddirect-productini-
tion, electron probabilities P(z ) and P(z ) increase in
tial state. 1 2
thevicinityofz =z andthusdestroytheinitiallyper-
The other consequence of the exchange symmetry of 1,2 B
fect symmetry of the entangled wavefunction, as clearly
the entangled wave function is a very small spatial sep-
seen from Figs. 7(a) and 7(b). The extension of elec-
aration of electrons all over the z-grid. The time-
tron probabilities P(z ) and P(z ) into the domains of
dependent expectation values hz (t)i and hz (t)i are al- 1 2
1 2
z >z gives rise to the EER-induced ‘sequential’ ion-
most identical, both being close to 0 on the timescale of 1,2 B
izationI in the positive directions ofthe z -axes. The
20 fs. If the perfect symmetry of the wave functions is B 1,2
only problem to be clarified now is EER in the domain
even slightly changed due to the excitation of H-H by
of atom B, because at a first glance one could conclude
the laser field, the very small spatial separation of the
from Figs. 7(a) and 7(b) that the electrons coming from
electrons leads to their strong Coulombic repulsion all
atom A to atom B occupy the domains of z > z .
over the z-grid. This suggests the reason for the fast 1,2 B
In order to clarify this issue, we performed two model
out-of-phase oscillations of ‘atomic’ energies E (t) and
A
simulations with a 5 fs laser pulse having narrow spatial
E (t) after the end of the laser pulse [see observation
B
envelope, assuming that only one of the two electrons is
(iv) above and Figs. 5(a) and 5(d)]. The fast out-of-
excited by the laser field in the domain of atom A. The
phase oscillations of the ‘atomic’ energies occur in the
results obtained are presented in Figs. 7(c) and 7(d).
case of the entangled initial state and do not occur in
the case of the unentangled direct-product initial state First we assume that only electron e1 is excited by
[Figs. 2(a) and 2(e)]. In the case of the direct-product the laser field in the domain of atom A. To this end, we
initialstate,theminimumdifferencebetweenhz (t)iand chose the spatial envelope function F(z) in the interac-
1
hz2(t)i is about 98 a.u. (5.18 nm) [see Figs. 2(c) and tion Hamiltonian of Eq. (4) as follows: F(z1) = FN(z1)
2(g)], therefore EER is very weak all overthe z-grid,ex- and F(z2)=0, where the narrowspatial envelope FN(z)
cept for the special cases of very sharp overlaps of the is given by Eq. (16) and illustrated in Fig. 1(b). The
electronic wave functions in the vicinity of protons pA respective electron probabilities P(z1) and P(z2) at the
and p (see Figs. 3 and 4). end of the pulse (t = 5 fs) are shown in Fig. 7(c). Sec-
B
A very strong EER taking place in the case of the en- ondly, we assume that only electron e2 is excited by the
tangled initial state should also result in a very efficient laser field in the domain of atom A. Accordingly, the
EER-induced ‘sequential’ ionization, similar to that de- spatial envelope function F(z) in the interaction Hamil-
scribedinSec.IIIfortheunentangleddirect-productini- tonian of Eq. (4) is chosen as follows: F(z1) = 0, and
tialstateofH-H.Ontheotherhand,itwasshownabove F(z2) = FN(z2). The respective electron probabilities
that the entanglement of the initial state of H-H does P(z1) and P(z2) at the end of the pulse (t = 5 fs) are
not change its ionization probability in comparison to shown in Fig. 7(d).
the direct-product initial state. Therefore, a closer look It is seen from the results presented in Figs. 7(c) and
at the process of EER-induced ionization in the case of 7(d) that the EER-induced ionization of atom B with