Two-electron time-delay interference in atomic double ionization by attosecond pulses Page: 2 of 5
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underlying methods have been described in detail else-
where [8], and so we describe only the essential ideas
here. We solve the time-dependent Schrodinger equation
from t 0 to t T, where the second pulse ends at time
T, and where I(t 0) is the ground state of the atom,--------------------------,
0.005
- 2 0.5 0.75 1.25 t (fs)
-0.005i-4(t) [Ho +V]4(t).
(1)
The laser-atom interaction, in the dipole approximation
and length gauge, is t E(t) - (ri + r2), and Ho
is the atomic Hamiltonian. The electromagnetic field,
E(t) (Ewi (t) + Ew2 (t))E, corresponds to two pulses
with different central frequencies and possibly different
durations and intensities, but with the same polarization
vector e in this example. The individual pulses are spec-
ified by
E1 (t) E() f(1) (t) sin(wit), t E [0, T1]
E-2 (t) E2) f(2) (t - t2) sin(w2(t - t2)), t E [t2, T]
(2)
on the time intervals where they are nonzero, where
f (,) (t) is the pulse envelope, chosen here to be
sin2(irt/T). The pulse durations are T1 and T2; T is the
time delay between the centers of the pulses as sketched
in the top panel of Fig. 2, and t2 T + (T1 - T2)/2.
After the second pulse the electrons are still interact-
ing and the wave function continues to evolve under Ho.
Calculating the ejection amplitudes for a fixed total en-
ergy formally requires propagating for an infinite time
after the second pulse and Fourier transforming the re-
sult. However, it is exactly equivalent to solve the driven
equation (E - H) IFsc =I (T), for the function Ts, at a
particular total energy, E, shared by the two outgoing
electrons. By solving this driven equation using exterior
complex scaling (ECS) of the electronic radial coordi-
nates [8], we automatically impose pure outgoing bound-
ary conditions on the scattered wave function. Then from
Tsc we can extract the amplitude C(k1, k2) for double
ionization with electronic wave vectors k1 and k2, as we
have done in several previous studies of double ioniza-
tion [8, 9]. The numerical solution of Eq.(1) was per-
formed using products of radial basis functions (discrete
variable representation) and coupled spherical harmon-
ics, as described in ref. [8]. Convergence was achieved
using a maximum total angular momentum of L 2, in-
dividual angular momenta up to l 14, and radial grids
extending to 170 bohr.
The probabilities we report here correspond to a sine
squared envelope for the pulses, f(') (t) in equation (2).
Calculations with Gaussian envelopes show that the cal-
culated probabilities display the same oscillations. We
have also verified that the present results employ pulses
with sufficiently large numbers of oscillations to be es-
sentially independent of the carrier phases.
As an example of the two-electron interference phe-
nomenon, we solve the time-dependent Schrodinger equa-0.008 -
o . 5 0.fs
- 0.006
0.002 -
y 0.4
0.3
C 0.2
p 0.1
N2
2 --
. ti = 0.2 fs
S 1 _
0 20 40 60 80 10
Electron Ener3
t.= 0.5 fs
2
0
i . . I ' 30
3. ~~i = 1fs .
- - 2
g g% 8 1
0 20 40 60 80 100
gy Sharnng (%~)FIG. 2: Electron energy sharing distributions at different time
delays for two-color two-photon double ionization. The energy
shared by the photoelectrons is 25 eV. Central frequencies:
Wi - 35 and )2 - 69 eV. Pulse durations: T1 T2 500 as.
Intensities: h1 - 101 W cm-2 and 12 - 2 x 1012 W cm-2
tion for different time delays, using two pulses of 500 as:
one with a central energy of 35 eV and intensity 1012
W cm-2, and a second pulse of 69 eV and 2 x 1012 W
cm-2. In Fig. 2 we show the energy-sharing distributions,
k1k2 f dQ1 f dQ2IC(k1, k2)12, resulting from the double
ionization amplitudes for a total energy equal to the sum
of the central energies of the pulses (104 eV) less the total
binding energy of the helium atom. Positive time delays,
T correspond to the 35 eV pulse arriving first. For a neg-
ative time delay of = -0.5 fs the two-color sequential
process takes place through excitation ionization: the 69
eV pulse ionizes He leaving He+ in the 2p state, and the
35 eV photon ionizes the excited He+ atom. The vertical
lines in the corresponding panel of Fig. 2 indicate the en-
ergies of electrons ejected sequentially by the excitation
ionization pathway at the central frequencies of the two
pulses in that case.
When both pulses reach the target simultaneously
( = 0) the maximum ionization probility is centered
at 50% energy sharing. As the time delay increases up
to 1 fs in Fig. 2, an increasing number of oscillations ap-
pear in these electron distributions, their number being
in principle unlimited in the infinite energy resolution of2
t = 0.75 fs_
-- --'1
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Rescigno, Thomas N. Two-electron time-delay interference in atomic double ionization by attosecond pulses, article, October 4, 2009; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc1013014/m1/2/: accessed April 23, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.