The detection of spatial and temporal electronic motion by scattering of subfemtosecond pulses of 10 keV electrons from coherent superpositions of electronic states of both H and T+2 is investigated. For the H atom, we predict changes in the diffraction images that reflect the time-dependent effective radius of the electronic charge density. For an aligned T+2 molecule, the diffraction image changes reflect the time-dependent localization or delocalization of the electronic charge density

Shao, Hua-Chieh

Starace, Anthony F.

English

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Anthony F. Starace Publications Research Papers in Physics and Astronomy 
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Detecting Electron Motion in Atoms and Molecules 
Hua-Chieh Shao 
University of Nebraska-Lincoln, hshao@unl.edu 
Anthony F. Starace 
University of Nebraska-Lincoln, astarace1@unl.edu 
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Detecting Electron Motion in Atoms and Molecules
Hua-Chieh Shao and Anthony F. Starace
Department of Physics and Astronomy, The University of Nebraska, Lincoln, Nebraska 68588-0299, USA
(Received 24 July 2010; published 20 December 2010)
The detection of spatial and temporal electronic motion by scattering of subfemtosecond pulses of
10 keV electrons from coherent superpositions of electronic states of both H and Tþ2 is investigated. For
the H atom, we predict changes in the diffraction images that reflect the time-dependent effective radius of
the electronic charge density. For an aligned Tþ2 molecule, the diffraction image changes reflect the time-
dependent localization or delocalization of the electronic charge density.
DOI: 10.1103/PhysRevLett.105.263201 PACS numbers: 34.80.Bm
At present, attosecond physics is able to provide unpre-
cedented temporal resolution and control of electronic
processes [1,2]. Single, few-cycle attosecond pulses with
photon energies @!< 100 eV and temporal widths in the
range of 130–80 as [3,4] have been achieved, making
possible the new field of light-wave electronics in which
the electric field of the pulse (rather than its intensity
profile) is used to control electronic motion [2,5,6].
However, to obtain spatial resolution of temporal changes
in electronic charge densities, pulses having much shorter
wavelengths are necessary. Ultrafast electron diffraction
[7–9] and ultrafast electron microscopy [8,10–13] have
also made significant progress recently, enabling the ob-
servation of transient structures in chemical reactions and
of time-dependent phenomena in condensed matter. The
temporal widths of these electron pulses, however, range
from femtosecond to picosecond, which is inadequate for
studying electronic motion in atoms and molecules.
Recently, methods have been proposed for producing
single-electron, attosecond pulses [14–16]. The attosecond
temporal and subangstrom spatial resolutions of such short
keVelectron pulses (at the target location) would make the
four-dimensional (4D) study of electronic motion feasible.
We report here benchmark calculations for attosecond
electron pulse scattering from oscillating electronic charge
distributions in two prototypical systems: the H atom and
the Tþ2 molecule. We demonstrate the effect of such spatial
and temporal charge oscillations on the differential scat-
tering cross section (DSCS). For the H atom, a few-
femtosecond laser pulse is proposed to excite a coherent
superposition of two electronic states; for the Tþ2 molecule,
such a superposition state is assumed. In both cases the
charge distribution oscillates with the beat frequency. The
DSCSs for an electron energy of 10 keV are calculated in
the Born approximation. For the H atom, the charge dis-
tribution has an oscillating effective radius. For the Tþ2
molecule, the superposed states have an electronic charge
density that oscillates from one nucleus to the other. In both
cases, the DSCSs are predicted to exhibit the 4D motion of
the electronic charge density.
Figure 1 shows the setup [8,9] for which our H atom
calculations are performed. The few-femtosecond laser
pulse creates a superposition of electronic states which is
then probed by a time-delayed, 10 keV single-electron
attosecond pulse. The electron pulse may be generated
by photoemission [15,17] and then compressed to an atto-
second duration by a spatially dependent ponderomotive
potential [15,16]. By adjusting the time delay between the
pump and the probe pulses, the 4D motion of the oscillat-
ing charge distribution is manifested in the time-resolved
diffraction images of the scattered electrons.
For the H atom, the energy of the 8th harmonic of the Ti:
sapphire laser relative to the 1s ground state is between the
energies of the 3p and 4p states. The populations of these
and nearby states can be calculated by using first-order
time-dependent perturbation theory for a pulse whose vec-
tor potential has the form
A ðtÞ ¼ A0 exp½4ðln2Þt2=2 sinð!tþÞ; (1)
where  is the polarization vector, A0 is the amplitude,  is
the width at half height of the Gaussian envelope, ! is the
carrier frequency, and  is the carrier-envelope phase. In
the electric dipole approximation and by defining 2 
2=8 ln2, the transition probability for jii ! jfi is
FIG. 1 (color online). Proposed H atom experiment. The red
arrow indicates the polarization of the pump laser. For future
reference, the coordinates and the angles are defined in the
figure.
PRL 105, 263201 (2010) P HY S I CA L R EV I EW LE T T E R S
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0031-9007=10=105(26)=263201(4) 263201-1  2010 The American Physical Society
jhfj ~Tjiij2 ¼ 
2
2ðA0!fiÞ2jhfj  rjiij2 2½eð!!fiÞ2 2
þ eð!þ!fiÞ2 2  2 cos2eð!2þ!2fiÞ 2; (2)
where  is the fine-structure constant and !fi  Ef  Ei.
(Atomic units are used throughout this Letter.)
The populations of several np states produced by a
linearly polarized laser pulse with central energy
12.45 eV and peak intensity I ¼ 1 1012 W=cm2 as a
function of the pulse width  are shown in Fig. 2. (Since
there are many cycles per pulse, we have set  ¼ 0.) For
 ¼ 8:7 fs, the 3p and 4p states are equally populated,
while the populations of other states are negligible. The
charge density of the resulting coherent state Hð3pþ 4pÞ
is
ðtÞ ¼ 12½23p0 þ24p0 þ 23p04p0 cosð!43tÞ: (3)
The interference term oscillates with the beat frequency
!43 (with a period of 6.3 fs), resulting in an oscillating
effective radius of the charge distribution.
For a keVelectron pulse, the DSCSs for the H atom may
be calculated in the first-order Born approximation [18].
Since the temporal width of the ultrafast electron pulse ( 
110 as) is orders of magnitude shorter than the beat period
(6.3 fs), the atomic charge density is treated as frozen in
time during the scattering of the electron pulse. Moreover,
electron exchange is neglected owing to the high energy of
the electron pulse and its small momentum transfer [19].
The elastic scattering amplitude is
fðq; tÞ ¼  m
2
Z
dr0eiqr0Vðr0; tÞ; (4)
where q is the momentum transfer and m is the reduced
mass. The effective potential Vðr0; tÞ between the incident
electron and the H atom at the origin is
Vðr; tÞ ¼  1jrj þ
Z
dr0
ðr0; tÞ
jr r0j ; (5)
where the electron density  is defined in Eq. (3). The
DSCS is averaged over the momentum distribution jaðkÞj2
of the electron pulse, i.e.,
d
d
¼
Z
dkjaðkÞj2jfðq; tÞj2: (6)
For a well-collimated pulse moving in the x direction with
central momentum p0, the momentum distribution is
jaðkÞj2 ¼ Y
i¼x;y;z
1ffiffiffiffiffiffi
2
p
i
exp

ðki  p0iÞ
2
22i

; (7)
where 2i is its variance along each axis. We set x=p0x ¼
1:5 104, so that x=p0x  t ¼ ð2xp0xÞ1 ¼
110 as. The cylindrically symmetric transverse momentum
distribution corresponds to an angular divergence
103 rad. The 10 keV electron pulse (a case considered
in Ref. [15]) has a de Broglie wavelength of 0.12 A˚.
The elastic DSCSs in the detection plane (cf. Fig. 1) at
three delay times are shown on the right in Fig. 3; the
electron densities [cf. Eq. (3)] at those times are shown on
the left. As the effective radius of the charge density
increases, the DSCS increases accordingly, demonstrating
the ability of an ultrafast electron pulse to map both spatial
and temporal electronic motion in atoms. In the left column
of Fig. 4, the DSCSs at three delay times are plotted as a
function of the azimuthal angle ’ for two different polar
angles  (cf. Fig. 1). The variation of the elastic DSCSs
with pump-probe delay time is in each case over 100%.
Note that inelastic DSCSs are largest for small scattering
angles. At large polar angles, deexcitation to the 3s state
gives the largest contribution to the inelastic DSCS. In the
10-7
10-6
10-5
10-4
10-3
10-2
10-1
1.0 3.0 5.0 7.0 9.0 11.0
Po
pu
la
tio
n
Pulse Duration τ (fs)
2p
3p
4p
5p
FIG. 2 (color online). Populations of np states of the H atom
as a function of the duration of a laser pulse with peak intensity
I ¼ 1 1012 W=cm2, ! ¼ 12:45 eV, and  ¼ 0 [cf. Eq. (1)].
FIG. 3 (color online). Right column: Differential cross sec-
tions in the y-z plane (cf. Fig. 1) for scattering of a 110 as,
10 keV electron pulse from the coherent state Hð3pþ 4pÞ
(whose beat period is T ¼ 6:3 fs). Left column: Charge density
of the state Hð3pþ 4pÞ. The three rows correspond to pump-
probe time delays of 0T, T=4, and T=2. Owing to symmetry, we
show only the upper half of each diffraction image.
PRL 105, 263201 (2010) P HY S I CA L R EV I EW LE T T E R S
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right column of Fig. 4, we compare the elastic DSCS to the
DSCS for this most important inelastic process as a func-
tion of polar angle  for ’ ¼ 0	. For polar angles  >
0:3	, the contribution of inelastic scattering is insignificant
compared to the elastic DSCS for any pump-probe delay
time.
Attosecond extreme ultraviolet pulse photoionization of
coherent superpositions of electronic states of the Hþ2
molecule have been studied theoretically [20–22]. The
photoelectron probabilities exhibit asymmetries dependent
on the pump-probe time delay. Only one of these studies
[22] showed how to produce the coherent states and, by
using a one-dimensional (1D) approximation, how nuclear
motion affects the results: The electronic superposition
state coherence survived even for times comparable to
the beat period. Very recently, ultrashort electron pulse
diffraction from the Hþ2 molecule has been investigated
by using a 1D approximation [23]. Inelastic and exchange
processes were found to play minor roles.
To illustrate attosecond electron pulse diffraction from a
coherently oscillating molecular electronic state in a theo-
retically tractable system (that allows one to obtain an
analytic expression for the DSCS), we consider the super-
position of the g1s and u1s states of a transversely
aligned tritium molecular ion Tþ2 for a bond length R ¼
6 a:u: (where the potential curves are relatively flat) [22].
This superposition state gives alternately a localized or
delocalized electronic charge density that varies with a
beat period of 7:9 fs [24] and that shows the ‘‘hopping’’
of the electron from one nucleus to the other. The molecu-
lar states are constructed by employing the LCAO approxi-
mation using the 1s state of the T atom [20,21]. This
approximation is valid for large bond lengths owing to
little overlap of the atomic wave functions on different
nuclei. The molecular states are thus given by
c g1s ¼ Agðð1Þ1s þð2Þ1s Þ; (8a)
c u1s ¼ Auðð1Þ1s ð2Þ1s Þ; (8b)
where Ag and Au are normalization constants and the
superscripts are the nuclear indices. We assume the follow-
ing coherent superposition of these two states:
c ðtÞ ¼ cgei!gtc g1s þ cuei!utc u1s; (9)
where cg and cu are the amplitudes of the jg1si and
ju1si states, respectively. (In our model, we do not con-
sider the production of this state or nuclear motion effects.)
The elastic scattering amplitude and the elastic DSCS for
this superposition state of Tþ2 can be obtained in the Born
approximation by using procedures similar to those used
for the H atom. In particular, inelastic processes and ex-
change effects are neglected [19,23]. Since the temporal
scales of nuclear rotation and separation for the heavy
tritium nuclei are much larger than the temporal width of
the electron pulse, treating the nuclear separation as sta-
tionary should be a good approximation. The elastic scat-
tering amplitude for fixed R is
fðq; tÞ ¼ 4m
q2

1þ 16ðjcgj
2A2g þ jcuj2A2uÞ
ð4þ q2Þ2

cos

R
2
 q

þ ðjcgj2A2g  jcuj2A2uÞ
Z
dr0ð1Þ1s 
ð2Þ
1s e
iqr0
þ i 16AgAuðc

gcue
i!ugt þ c:c:Þ
ð4þ q2Þ2 sin

R
2
 q

;
(10)
where R points from nucleus 1 to nucleus 2. Uncertainties
in both the molecular alignment and the internuclear sepa-
ration require further consideration. We assume an align-
ment distribution of the form cos12 with a half-width of
19.3	; we assume the bond length distribution is Gaussian
with a half-width of 5%. The DSCSs are averaged over
these alignment and separation distributions.
0.0
1.1
2.2
θ = 0.3°
0 T
T/4
T/2
0.00
0.42
 0  60  120  180
dσ
/d
Ω
 
(10
4  
a
.u
.)
Angle ϕ (deg)
θ = 0.4°
0.0
2.1
4.2
0 T
elastic
3s
0.0
2.1
 0.2  0.3  0.4  0.5
Angle θ (deg)
T/2
FIG. 4 (color online). Left column: Elastic DSCSs as a func-
tion of the azimuthal angle ’ for two different polar angles  (cf.
Fig. 1); each panel shows results for three pump-probe time
delays. Right column: Comparison of the elastic DSCS and the
most important inelastic one (for deexcitation to the 3s state)
plotted vs the polar scattering angle .
FIG. 5 (color online). Charge density of the Tþ2 state in Eq. (9)
(with cg ¼ cu, R ¼ 6) in the y-z plane and the corresponding
averaged elastic DSCS for three delay times.
PRL 105, 263201 (2010) P HY S I CA L R EV I EW LE T T E R S
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For the Tþ2 superposition state in Eq. (9) having cg ¼ cu,
Fig. 5 shows the electronic charge density and the corre-
sponding averaged elastic DSCS for three time delays
[relative to production of the state (9)], where the parame-
ters of the attosecond electron pulse are the same as for the
H atom case. Owing to negligible overlap of the atomic
orbitals for largeR, the second term in Eq. (10) is omitted.
For t ¼ 0T and T=2 the electron charge density is localized
on one nucleus or the other, whereas for t ¼ T=4 the
charge density is delocalized. In the latter case, interfer-
ence between electron waves scattered from the charge
densities at each nucleus results in deep minima in the
DSCS, whose locations determine R [7]. For t ¼ 0 or T=2,
the DSCSs are the same in the asymptotic region; i.e., it is
not possible to determine on which nucleus the charge
density is localized. In Fig. 6, we plot the DSCS vs azimu-
thal angle ’ for two polar angles  as well as a temporal
asymmetry parameter
Aðt;’; Þ ¼ dð0Þ  dðtÞ
dð0Þ þ dðtÞ ; (11)
for a delay t (where t ¼ T=4 in Fig. 6). Clearly, the
maximum temporal asymmetries occur at particular azi-
muthal angles. Note also that significant temporal asym-
metries occur for large polar angles, where inelastic
scattering should be negligible [23].
In conclusion, the DSCSs for ultrafast electron pulse
scattering from coherent superposition states of the H
atom and the Tþ2 molecule were calculated. They show
clearly the ‘‘breathing’’ of the charge density in the H atom
and the localization and delocalization of the charge den-
sity in the Tþ2 molecule. The present simulations confirm
the feasibility of using attosecond electron pulses for 4D
investigations of electron dynamics [15,16,23]. Such in-
vestigations complement those using optical attosecond
pulses in providing insights into controlling electronic
motion in atoms and molecules.
We gratefully acknowledge informative conversations
with M. Centurion concerning experimental capabilities.
This work is supported in part by NSF Grant No. PHY-
0901673 and by a Nebraska Research Initiative grant.
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 0
 150
 300
θ = 1.2°
0 T
T/4
 0
 60
 0  60  120  180
dσ
/d
Ω
 
(a.
u.)
Angle ϕ (deg)
θ = 1.5°
 0
 0.5
 1
θ = 1.2°
0
0.25
 0  60  120  180
A(
t=T
/4;
ϕ,
θ)
Angle ϕ (deg)
θ = 1.5°
FIG. 6 (color online). Left column: Averaged elastic DSCS
(from the right column of Fig. 5) plotted vs azimuthal angle ’
for two polar angles . Right column: The corresponding tem-
poral asymmetry parameter Aðt ¼ T=4;’; Þ [cf. Eq. (11)].
PRL 105, 263201 (2010) P HY S I CA L R EV I EW LE T T E R S
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263201-4


Detecting Electron Motion in Atoms and Molecules

La présence musulmane en Europe, de l’Ancien régime aux Temps modernes Jocelyne Dakhlia, directrice d’études, EHESS-CR

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Detecting Electron Motion in Atoms and Molecules

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