Quantum-mechanical simulations of interaction of many-electron quantum systems with ionizing laser pulses
We develop parallel program for numerical simulation of the interaction of intense laser pulses with manyelectron atoms on the basis of the time-dependent Kohn-Sham equations. It is shown that the use of modern computer clusters makes it possible to solve these equations for a wide class of atoms in...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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| Cite this: | Quantum-mechanical simulations of interaction of many-electron quantum systems with ionizing laser pulses / A.A. Romanov, A.A. Silaev, D.A. Smirnova, T.S. Sarantseva, A.A. Minina, M.V. Frolov, N.V. Vvedenskii // Вопросы атомной науки и техники. — 2018. — № 4. — С. 277-281. — Бібліогр.: 25 назв. — англ. |
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Romanov, A.A. Silaev, A.A. Smirnova, D.A. Sarantseva, T.S. Minina, A.A. Frolov, M.V. Vvedenskii, N.V. 2019-02-14T18:53:29Z 2019-02-14T18:53:29Z 2018 Quantum-mechanical simulations of interaction of many-electron quantum systems with ionizing laser pulses / A.A. Romanov, A.A. Silaev, D.A. Smirnova, T.S. Sarantseva, A.A. Minina, M.V. Frolov, N.V. Vvedenskii // Вопросы атомной науки и техники. — 2018. — № 4. — С. 277-281. — Бібліогр.: 25 назв. — англ. 1562-6016 PACS: 52.38.-r, 31.15.ee, 32.80.-t https://nasplib.isofts.kiev.ua/handle/123456789/147453 We develop parallel program for numerical simulation of the interaction of intense laser pulses with manyelectron atoms on the basis of the time-dependent Kohn-Sham equations. It is shown that the use of modern computer clusters makes it possible to solve these equations for a wide class of atoms in a relatively small time, determined by the parameters of the laser pulse and electronic configuration of the atom. High accuracy of the numerical code is demonstrated on the example of calculating the high-frequency spectrum of electron current excited during ionization of noble gas atoms by few-cycle laser pulse. Розробляється паралельний програмний код для чисельного моделювання взаємодії інтенсивних лазерних імпульсів з багатоелектронними атомами на основі нестаціонарного методу функціонала щільності. Показується, що використання сучасних багатопроцесорних обчислювальних кластерів дозволяє розв’язати нестаціонарні рівняння Кона-Шема для широкого класу атомів за відносно невеликий час, що визначається параметрами лазерного імпульсу і електронною конфігурацією атома. Демонстрація роботи чисельного коду представлена на прикладі розрахунку спектра високочастотного електронного струму, що збуджується при іонізації атомів інертних газів коротким лазерним імпульсом. Разрабатывается параллельный программный код для численного моделирования взаимодействия интенсивных лазерных импульсов с многоэлектронными атомами на основе нестационарного метода функционала плотности. Показывается, что использование современных многопроцессорных вычислительных кластеров позволяет решить нестационарные уравнения Кона-Шэма для широкого класса атомов за относительно небольшое время, определяемое параметрами лазерного импульса и электронной конфигурацией атома. Демонстрация работы численного кода представлена на примере расчета спектра высокочастотного электронного тока, возбуждаемого при ионизации атомов инертных газов коротким лазерным импульсом. This work was supported by the Russian Science Foundation through Grant No. 17-12-01574. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Нелинейные процессы Quantum-mechanical simulations of interaction of many-electron quantum systems with ionizing laser pulses Квантово-механічне моделювання взаємодії багатоелектронних квантових систем з іонізуючими лазерними імпульсами Квантово-механическое моделирование взаимодействия многоэлектронных квантовых систем с ионизирующими лазерными импульсами Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Quantum-mechanical simulations of interaction of many-electron quantum systems with ionizing laser pulses |
| spellingShingle |
Quantum-mechanical simulations of interaction of many-electron quantum systems with ionizing laser pulses Romanov, A.A. Silaev, A.A. Smirnova, D.A. Sarantseva, T.S. Minina, A.A. Frolov, M.V. Vvedenskii, N.V. Нелинейные процессы |
| title_short |
Quantum-mechanical simulations of interaction of many-electron quantum systems with ionizing laser pulses |
| title_full |
Quantum-mechanical simulations of interaction of many-electron quantum systems with ionizing laser pulses |
| title_fullStr |
Quantum-mechanical simulations of interaction of many-electron quantum systems with ionizing laser pulses |
| title_full_unstemmed |
Quantum-mechanical simulations of interaction of many-electron quantum systems with ionizing laser pulses |
| title_sort |
quantum-mechanical simulations of interaction of many-electron quantum systems with ionizing laser pulses |
| author |
Romanov, A.A. Silaev, A.A. Smirnova, D.A. Sarantseva, T.S. Minina, A.A. Frolov, M.V. Vvedenskii, N.V. |
| author_facet |
Romanov, A.A. Silaev, A.A. Smirnova, D.A. Sarantseva, T.S. Minina, A.A. Frolov, M.V. Vvedenskii, N.V. |
| topic |
Нелинейные процессы |
| topic_facet |
Нелинейные процессы |
| publishDate |
2018 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Квантово-механічне моделювання взаємодії багатоелектронних квантових систем з іонізуючими лазерними імпульсами Квантово-механическое моделирование взаимодействия многоэлектронных квантовых систем с ионизирующими лазерными импульсами |
| description |
We develop parallel program for numerical simulation of the interaction of intense laser pulses with manyelectron atoms on the basis of the time-dependent Kohn-Sham equations. It is shown that the use of modern computer clusters makes it possible to solve these equations for a wide class of atoms in a relatively small time, determined by the parameters of the laser pulse and electronic configuration of the atom. High accuracy of the numerical
code is demonstrated on the example of calculating the high-frequency spectrum of electron current excited during
ionization of noble gas atoms by few-cycle laser pulse.
Розробляється паралельний програмний код для чисельного моделювання взаємодії інтенсивних лазерних імпульсів з багатоелектронними атомами на основі нестаціонарного методу функціонала щільності. Показується, що використання сучасних багатопроцесорних обчислювальних кластерів дозволяє розв’язати
нестаціонарні рівняння Кона-Шема для широкого класу атомів за відносно невеликий час, що визначається
параметрами лазерного імпульсу і електронною конфігурацією атома. Демонстрація роботи чисельного коду
представлена на прикладі розрахунку спектра високочастотного електронного струму, що збуджується при
іонізації атомів інертних газів коротким лазерним імпульсом.
Разрабатывается параллельный программный код для численного моделирования взаимодействия интенсивных лазерных импульсов с многоэлектронными атомами на основе нестационарного метода функционала плотности. Показывается, что использование современных многопроцессорных вычислительных кластеров позволяет решить нестационарные уравнения Кона-Шэма для широкого класса атомов за относительно
небольшое время, определяемое параметрами лазерного импульса и электронной конфигурацией атома. Демонстрация работы численного кода представлена на примере расчета спектра высокочастотного электронного тока, возбуждаемого при ионизации атомов инертных газов коротким лазерным импульсом.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/147453 |
| citation_txt |
Quantum-mechanical simulations of interaction of many-electron quantum systems with ionizing laser pulses / A.A. Romanov, A.A. Silaev, D.A. Smirnova, T.S. Sarantseva, A.A. Minina, M.V. Frolov, N.V. Vvedenskii // Вопросы атомной науки и техники. — 2018. — № 4. — С. 277-281. — Бібліогр.: 25 назв. — англ. |
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| fulltext |
ISSN 1562-6016. ВАНТ. 2018. №4(116) 277
QUANTUM-MECHANICAL SIMULATIONS OF INTERACTION
OF MANY-ELECTRON QUANTUM SYSTEMS
WITH IONIZING LASER PULSES
A.A. Romanov1,2, A.A. Silaev1,2, D.A. Smirnova1, T.S. Sarantseva1,3, A.A. Minina1,3,
M.V. Frolov3, N.V. Vvedenskii1,2
1Institute of Applied Physics, Russian Academy of Sciences, Nizhny Novgorod, Russia;
2University of Nizhny Novgorod, Nizhny Novgorod, Russia;
3Department of Physics, Voronezh State University, Voronezh, Russia
E-mail: romanov.alexander.al@gmail.com
We develop parallel program for numerical simulation of the interaction of intense laser pulses with many-
electron atoms on the basis of the time-dependent Kohn-Sham equations. It is shown that the use of modern com-
puter clusters makes it possible to solve these equations for a wide class of atoms in a relatively small time, deter-
mined by the parameters of the laser pulse and electronic configuration of the atom. High accuracy of the numerical
code is demonstrated on the example of calculating the high-frequency spectrum of electron current excited during
ionization of noble gas atoms by few-cycle laser pulse.
PACS: 52.38.-r, 31.15.ee, 32.80.-t
INTRODUCTION
Interaction of intense laser field with atoms and
molecules is accompanied by many different phenome-
na interesting from the scientific and applied points of
view. These phenomena include, in particular, above-
threshold ionization [1], which consists in the possibility
of absorption of more photons than are required for the
ionization; high-order harmonic generation, which is the
result of the acceleration of freed electrons and their
collisions with the parent ions [2 - 8]. The excited elec-
tron currents can also contain a low-frequency compo-
nent responsible for the generation of radiation with a
frequency much lower than the optical ones, in particu-
lar, in the terahertz and mid-IR range [9 - 14].
Despite the fact that most of the atoms and mole-
cules are essentially many-electron quantum systems,
numerical studies of the above-mentioned ionization-
induced phenomena are traditionally based on the sin-
gle-active-electron approximation. Within the frame-
work of this approximation all the electrons except one
are frozen in their orbitals and the field of the parent ion
is described by a static potential well. Such models can
have high accuracy under certain conditions, but in
many problems single-active-electron approximation is
inapplicable, since it does not describe a number of es-
sentially many-electron effects, such as the polarization
of the atomic system. One of the methods for describing
many-electron effects, which is close to the many-
electron time-dependent Schrödinger equation, is the
time-dependent density functional theory, which has
recently been increasingly used in atomic physics and
nonlinear optics [15 - 18]. This approach is based on the
system of time-dependent Kohn-Sham (TDKS) equa-
tions, in which the Hamiltonian for individual orbitals
includes the interaction with the atomic nucleus, elec-
tron-electron interaction, as well as interaction with the
electric field of the laser pulse [17].
This paper is devoted to the development of parallel
program for the numerical solution of TDKS equations
for ab initio simulation of the evolution of many-
electron atoms (such as neon, argon, krypton, and other
noble gas atoms) during the interaction with an intense
laser fields. The algorithm for solving the three-
dimension TDKS equations is based on the spherical
harmonics expansion of the potential energy and wave
functions of orbitals. The increase in performance of the
program related to the parallelization of its individual
modules is analyzed and the high accuracy of the pro-
gram is demonstrated.
1. STATEMENT OF THE PROBLEM
Suppose that a many-electron atom interacts with a
linearly polarized laser-pulse electric field of strength
)(ˆ=)( tEt zE , where t is the time. We assume that the
intensity and wavelength of the laser pulse correspond
to the dipole approximation in which the action of the
magnetic field in the calculation of the atom dynamics
can be neglected [4]. We also limit ourselves here to the
consideration of atoms in which orbitals initially are
occupied by electrons of the opposite spin. Due to the
weak influence of the magnetic field, the zero spin po-
larization is conserved during interaction with the laser
pulse, therefore TDKS equations describing the dynam-
ics of a many-electron system are written down as fol-
lows (here and below, the atomic system of units is used
in which 1=== eme , where is the reduced Planck
constant, ||= ee − is the charge and me is the mass of
the electron):
.)],([)(
2
1=ˆ
/2,1,...,=),,(ˆ=),(
2 tVtzE
r
ZH
NntHt
t
i
ee
nn
r
rr
r
ψψ
++−∇−
∂
∂
(1)
Here, nψ is the wave function of the n-th TDKS or-
bital, Z is the nuclear charge, N is the (even) number of
electrons, r is the electron density, )],([ tVee rr is the
potential of electron-electron interaction. The electron
density is related to TDKS orbitals by equality
.),(2=),( 2/2
1=
tt n
N
n
rr ψr ∑ (2)
The electron-electron interaction potential consists
of the Hartree potential
,)(=)],([ 3
rr
rr
′−
′
′∫
r
r rdtVH
(3)
ISSN 1562-6016. ВАНТ. 2018. №4(116) 278
describing the electron repulsion in the framework of
the mean field, and the exchange-correlation potential
xcV for which the spin unpolarized form of LB94 ap-
proximation [19] is used
[ ].)(2sinh)(231
)()(2
)],([=)],([
1/311/3
1/321/3
94
rr
rr
rr
χβχ
rβχ
rr
−⋅+
−
tVtV LDA
xc
LB
xc
(4)
Here, )],([ tV LDA
xc rr is the exchange-correlation po-
tential in the local density approximation [17],
)(/)(=)( 4/3 rrr rrχ ∇ , and 05.0=β . The initial condi-
tion corresponds to the ground state of the atom, which
is described by the stationary Kohn-Sham orbitals
)(,0 rnψ :
)],([
2
1=ˆ
,=)(ˆ
0
2
0
,0,00
r
r
r
ψψ
ee
nnn
V
r
ZH
EH
+−∇−
(5)
where nE is the energy of the n-th orbital.
Using TDKS orbitals and Ehrenfest's theorem one
can find the time-dependent dipole acceleration of the
atomic system [18]:
).,()(=)( 3
3 t
r
ZrdtNt rrEa r∫−− (6)
Then, the macroscopic electron current density in
the produced plasmas can be found as
tdtNt t
g ′′− ∫ ∞− )(=)( aj , where gN is the gas density be-
fore the start of the ionization process [20]. In addition,
using the wave functions of TDKS orbitals after the
passage of the laser pulse, one can find the final photoe-
lectron momentum distribution [15]
,)(~2=)( 2/2
1=
pp n
N
n
P ψ∑ (7)
where p is the momentum vector and )(~ pnψ is the spa-
tial Fourier transform of the n-th TDKS orbital except
the projections on atomic bound states.
2. THE NUMERICAL IMPLEMENTATION
2.1. THE NUMERICAL METHOD
For the numerical solution of TDKS equations, or-
bitals ),( tn rψ are decomposed into spherical harmonics
,)cos(
)!(4
)!1)((2=),( ϕθ
π
ϕθ imm
llm eP
ml
mllY
+
−+ (8)
where θ is the polar angle with respect to z-axis di-
rected along the external field, ϕ is the azimuthal angle,
m
lP are the associated Legendre polynomials. The ini-
tial Kohn-Sham orbitals correspond to the electronic
configuration of the unperturbed atom and are given by
),,()(=),,( ,0
1
,0 ϕθϕθψ
nnlnn mYrrr Y− (9)
where nl and nm are orbital and magnetic quantum
numbers of the n-th orbital. Since the external electric
field has linear polarization, the magnetic quantum
number of each orbitals is conserved in time, therefore
in the decomposition of TDKS orbitals only the term
with m = mn exists:
).,(),(=),,,(
0=
1 ϕθϕθψ
nlmnl
l
n Ytrrtr Y∑
∞
− (10)
To find the time dynamics of the radial component
of the n-th TDKS orbital, the following method is used
(the modification of which is also used to calculate sta-
tionary orbitals). The potential of the electron-electron
interaction eeV is decomposed into spherical harmonics
up to the quadrupole term:
( ).1cos3
2
1),(
cos),(),(),,(
2(2)
(1)(0)
−+
+≈
θ
θθ
trV
trVtrVtrV
ee
eeeeee
(11)
Substituting (10), (11) into (1), multiplying by the
spherical harmonic
nlmY , and integrating over the solid
angle, we obtain a system of equations on nlY , which is
conveniently represented in the matrix form
( ) ).,(ˆˆ=),( trtr
t
i nnnn ΨWRΨ +
∂
∂ (12)
Here nΨ is the coulomn vector whose l-th element
is nlY , nR̂ is the diagonal matrix with elements
[ ]
).,(
3)1)(2(2
31)(
),(
2
1)(
2
1=ˆ
(2)
2
(0)
22
2
trV
ll
mll
trV
r
Z
r
ll
r
ee
n
ee
lln
+−
−+
++
−
+
+
∂
∂
−R
(13)
The elements of the pentadiagonal matrix nŴ ,
which represents the dipole interaction electrons with
the electric field and dipole and partly quadrupole terms
of electron-electron interaction, are given by
[ ]
( )
( )
+
+
−
−
+
+−
−
,otherwise
2=
1=
1=
2=
0
),,(
,),()(
,),()(
),,(
=ˆ
(2)
,
(1)
,
(1)
,1
(2)
,2
lk
lk
lk
lk
trVd
trVtrEc
trVtrEc
trVd
eeml
eeml
eeml
eeml
kln
n
n
n
n
W (14)
where
.
5)3)(2(2
2)(
2
3=
,
3)1)(2(2
1)(=
22
,,
22
,
++
−+
++
−+
ll
mlcd
ll
mlc
mlml
ml
(15)
The evolution of TDKS orbitals to the time step t∆
is performed using the following propagator:
.)](ˆ)(ˆ[exp=),(ˆ
),,(),(ˆ=),(
′+′′−∆+
∆+∆+
∫
∆+ tttdittt
trtttttr
nn
tt
tn
nnn
WRU
ΨUΨ
(16)
The propagator ),(ˆ tttn ∆+U is approximated up to
the second order in t∆ as follows:
( )
( ) ( ).2/)(ˆexp)(ˆexp
2/)(ˆexp),(ˆ
ttitti
ttittt
nn
nn
WR
WU
∆−∆−×
∆−≈∆+
(17)
The application of the exponential operator
( )/2ˆexp nti W∆− is performed using diagonalization of the
matrix nŴ by the unitary transformation
†
diag,
ˆˆˆ=ˆ
nnnn SWSW , where the sign " † " corresponds to
the Hermitian conjugation, resulting in
( ) ( ) .ˆ2/)(ˆexpˆ=2/)(ˆexp †
diag, nnnn ttitti SWSW ∆−∆− (18)
ISSN 1562-6016. ВАНТ. 2018. №4(116) 279
The operator ( )nti R̂exp ∆− can be applied using the
Crank-Nicholson approximation
/2).ˆ(1/2)ˆ(1)ˆ(exp 1
nnn tititi RRR ∆−∆+≈∆− − (19)
The second derivative in nR̂ is computed using the
Numerov approximation with the equidistant step along
r . To calculate the radial part )(,0 rnY of the stationary
Kohn-Sham orbitals (the initial condition for the time
problem), we use a similar algorithm in which the time
has purely imaginary values [21].
2.2. PARALLELIZATION OF NUMERICAL
CODE
The method used to solve TDKS equations is well
suited for execution on computer clusters, since it has
two levels of parallelism. The first level is related to the
properties of the matrices nR̂ and nŴ . The matrix nR̂
is diagonal and the application of the operator (19) is
calculated independently for each l-th component of the
vector nΨ (l parallelism). The operator (18) is applied
independently for each point of the radial grid (r paral-
lelism). The first level of parallelism is convenient for
implementing on systems with shared memory using
OpenMP technology. The second level of parallelism is
ensured by the fact that some operations for different
orbitals are performed independently (n parallelism).
This level of parallelism can be implemented for sys-
tems with a distributed memory using MPI technology.
For cluster systems having many nodes consisting of
several sockets the parallel propagation algorithm for
one time step can be implemented as follows:
• For each orbital )(tnΨ the action operator
),(ˆ tttn ∆+U is performed in parallel inside differ-
ent sockets of different nodes of the cluster.
• The calculated values )( ttn ∆+Ψ are sent to the
one (main) node.
• On the main node the electron-electron potential
)]([ ttVee ∆+r is calculated on the basis of )( ttn ∆+Ψ
and is sent to all the involved cluster nodes.
The Table compares the propagation time for one
time step without using parallelization (on a single core)
and using different variants of parallelization of pro-
gram modules on a computer cluster. The latter consists
of a large number (more than 30) nodes connected by a
high-speed computer-network InfiniBand with the
bandwidth 14 Gbit/s between nodes. Each node repre-
sents a two-socket system with two 10-core Intel Xeon
E5-2680 v2 processors. Calculations are performed for
three different noble gas atoms: neon, argon, and kryp-
ton. To describe the dynamics of the atom (using the
symmetry in the spin and in the magnetic quantum
number) only 4, 7, and 13 independent TDKS orbitals
are used for Ne, Ar, and Kr, respectively. The radial
numerical grid has Nr=104 nodes and the maximum
value of the orbital angular momentum is lmax =512.
As one can see from the Table, the usage of l and r
parallelisms inside one socket leads to a multiple de-
crease of propagation time. The increase of the perfor-
mance approximately equals to the number of cores in
the socket (which is 10 on the used system).
Atom
One-step calculation time (s)
single core
(no parallelism)
single socket
(r,l parallelism)
cluster
(n,r,l parallelism)
Ne 26.6 3.31 1.22
Ar 46.48 5.86 1.32
Kr 113.74 16.12 1.5
The one-time-step propagation time for three atoms: Ne,
Ar, and Kr. The first column shows the result for the
single-CPU (parallelization-free) calculation. The sec-
ond column is propagation time for 10 cores of single
processor with the parallelization of operators (18) and
(19). The third column shows the propagation time with
the additional parallelization in orbitals. It is computed
on 2 cluster nodes for Ne, 4 nodes for Ar, and 7 nodes
for Kr (each node has 20 cores in total)
At the same time, the efficiency of n parallelism
strongly depends on the speed of data transfer between
cluster nodes, on the number of sockets on each node,
and on the number of independent orbitals in an atom.
For the used computer cluster with 2 sockets on each
node the increase of the performance is 2.7 for Ne atom,
4.4 for Ar, and 10.7 for Kr. The one-step propagation
time for different atomic systems is approximately the
same (1.2…1.5 s), which is associated with almost
complete parallelization in orbitals.
It should be noted, however, that in the considered
example the number of numerical grid nodes Nr is fixed.
At the same time, when solving some physical problem
with the use of TDKS equations, it is necessary to re-
duce the coordinate step near the nucleus as the nuclear
charge increases in order to ensure high accuracy of
calculations. Moreover, it is necessary also to decrease
the time step t∆ to resolve the natural frequencies of the
lowest orbitals [22]. This leads to a rather significant
decrease in the speed of numerical simulation with the
increase of the atom nuclear charge, which will be
demonstrated in the examples of calculation below.
2.3. EXAMPLES OF CALCULATION
In order to test the developed computational code,
we consider neon, argon, and krypton atoms interacting
with a few-cycle laser pulse with the electric field hav-
ing the sine-squared envelope located at pt t≤≤0 :
( )
−
∂−∂
).(sin/sin)/(=)(
,/=)(
0
2
00 ttEtA
tAtE
p ωtπω
(20)
Here, E0 is the peak amplitude of the electric field,
corresponding to the peak intensity 14103 ⋅ W/cm2,
0.0570 ≈ω au is the carrier frequency, corresponding to
the wavelength 800 nm, and 7.14≈pt fs corresponds to
two cycles at the full-width at half maximum of intensi-
ty.
Calculations are performed in the spatial region
max0 rr ≤≤ , where rmax =200 au with the maximum val-
ue of the orbital angular momentum lmax =512. The step
along the radial coordinate is 0.04=r∆ au for Ne,
0.02=r∆ au for Ar, and 0.01=r∆ au for Kr, the time
step is 2102.4= −⋅∆t au for Ne, 3108= −⋅∆t au for Ar, and
3102= −⋅∆t au for Kr. To absorb waves approaching the
boundary along r, a three-hump imaginary potential [23]
with a total width of the absorbing layer labs = 50 is
ISSN 1562-6016. ВАНТ. 2018. №4(116) 280
used. To perform the calculations we use 4 sockets/2
nodes for Ne, 7 sockets/4 nodes for Ar, and 13 sockets/7
nodes for Kr. The large difference in rrN ∆/= maxr and
in the time step t∆ for different atoms leads to a differ-
ence in calculation time which is 5 hours for Ne, 32
hours for Ar, and 12 days 6 hours for Kr.
The spectra of dipole acceleration excited during inter-
action of Ne (a), Ar (b), and Kr (c) atoms with short
laser pulse with the wavelength of 800 nm and peak
intensity of 14103 ⋅ W/cm2. The dashed line indicates
classical cutoff of frequency ppс UI 3.17+=ω , where
pI is the ionization potential of atom and pU is the
maximum electron ponderomotive energy
The squared modulus of the Fourier spectrum of the
dipole acceleration is shown in Figure. As can be seen
from the Figure, the spectrum for Ne, Ar, and Kr con-
tains a high-frequency part lying in the vacuum ultravio-
let range, which is associated with rescattering of the
photoelectrons on the parent ion [2 - 8]. The shape of
the spectrum depends strongly on the type of atom.
With increasing ionization potential pI ( =pI 13.99 eV
for Kr, 15.76 eV for Ar, and 21.55 eV for Ne), the max-
imum energy of the generated harmonics increases ac-
cording to the well-known formula for the plateau cutoff
position ppс UI 3.17+=ω , where pU is the maximum
electron ponderomotive energy [2, 3]. At the same time
the spectral intensity decreases with increasing of ioni-
zation potential, since ionization probability of the atom
decreases. The high-frequency spectrum for the case of
Ar contains a minimum located approximately at 51 eV.
This minimum is called Cooper minimum and is oc-
curred due to the interference of two electron recombi-
nation channels in the ground state: ps → and pd →
[6, 24]. The minimum position obtained in the presented
calculation is in good agreement with the results of ex-
periments on the high harmonics generation in Ar [24],
as well as with known results of numerical calculations
using the time-dependent density functional theory [18],
which confirms the high accuracy of the developed
code. The Cooper minimum for Kr is not observed in
Figure since it is located at 80 eV [25] that is higher
than the cutoff frequency ωc for the considered parame-
ters of the laser pulse.
CONCLUSIONS
The parallel program code has been developed for
the numerical solution of the time-dependent Kohn-
Sham equations for ab initio modeling of the evolution
of many-electron atoms during the interaction with an
intense laser field. The solution algorithm is based on
the decomposition of the wave functions of the Kohn-
Sham orbitals and the potential energy into spherical
harmonics. The high accuracy of the program is demon-
strated by the example of calculating the spectrum of a
high-frequency electron current excited during ioniza-
tion of noble gas atoms by a few-cycle laser pulse. It is
shown that the use of parallelization between the nodes
of modern computer clusters and between the CPUs of
individual socket makes it possible to perform calcula-
tions for typical parameters of laser pulses in a relative-
ly small time from several hours to several days depend-
ing on the atomic system under consideration.
ACKNOWLEDGEMENT
This work was supported by the Russian Science
Foundation through Grant No. 17-12-01574.
REFERENCES
1. W. Becker, F. Grasbon, R. Kopold, D. Milošević,
G. Paulus, H. Walther. Above-threshold ionization:
From classical features to quantum effects // Ad-
vances in Atomic, Molecular, and Optical Physics.
2002, v. 48, p. 35-98.
2. P.B. Corkum. Plasma perspective on strong field
multiphoton ionization // Physical Review Letters.
1993, v. 71, p. 1994.
3. M. Lewenstein, P. Balcou, M.Y. Ivanov,
A. L’huillier, P.B. Corkum. Theory of high-
harmonic generation by low-frequency laser fields //
Physical Review A.1994, v. 49, p. 2117.
4. A.V. Kim, M.Y. Ryabikin, A.M. Sergeev. From
femtosecond to attosecond pulses // Physics-
Uspekhi. 1999, v. 42, p. 54-61.
5. F. Krausz, M. Ivanov. Attosecond physics // Reviews
of Modern Physics. 2009, v. 81, p. 163.
6. M.V. Frolov, N.L. Manakov, T.S. Sarantseva,
M.Yu. Emelin, M.Yu. Ryabikin, A.F. Starace. Ana-
lytic description of the high-energy plateau in har-
monic generation by atoms: can the harmonic power
increase with increasing laser wavelengths? // Physi-
cal Review Letters. 2009, v. 102, p. 243901.
7. D. Shafir, B. Fabre, J. Higuet, H. Soifer, M. Dagan,
D. Descamps, E. Mével, S. Petit, H.J. Wörner,
B. Pons, et al. Role of the ionic potential in high
harmonic generation // Physical Review Letters.
2012, v. 108, p. 203001.
8. H.-C. Wu, J. Meyer-ter Vehn. Giant half-cycle atto-
second pulses // Nature Photonics. 2012, v. 6, p. 304.
a
b
c
ISSN 1562-6016. ВАНТ. 2018. №4(116) 281
9. K.-Y. Kim, A.J. Taylor, J.H. Glownia,
G. Rodriguez. Coherent control of terahertz super-
continuum generation in ultrafast laser-gas interac-
tions // Nature photonics. 2008, v. 2, p. 605-609.
10. A.A. Silaev, N.V. Vvedenskii. Residual-current exci-
tation in plasmas produced by few-cycle laser pulses
// Physical Review Letters. 2009, v. 102, p. 115005.
11. N.V. Vvedenskii, A.I. Korytin, V.A. Kostin,
A.A. Murzanev, A.A. Silaev, A.N. Stepanov. Two-
color laser-plasma generation of terahertz radiation
using a frequency-tunable half harmonic of a femto-
second pulse // Physical Review Letters. 2014,
v. 112, p. 055004.
12. T. Balciunas, D. Lorenc, M. Ivanov, O. Smirnova,
A. Zheltikov, D. Dietze, K. Unterrainer, T. Rathje,
G. Paulus, A. Baltuška, et al. CEP-stable tunable
THz-emission originating from laser-waveform-
controlled sub-cycle plasma-electron bursts // Optics
Express. 2015, v. 23, p. 15278-15289.
13. V.A. Kostin, I.D. Laryushin, A.A. Silaev,
N.V. Vvedenskii. Ionization-induced multiwave
mixing: Terahertz generation with two-color laser
pulses of various frequency ratios // Physical Review
Letters. 2016, v. 117, p. 035003.
14. A.A. Silaev, V.A. Kostin, I.D. Laryushin,
N.V. Vvedenskii. Ionization mechanism of the gen-
eration of tunable ultrashort pulses in the mid-infrared
range // JETP Letters. 2018, v. 107, p. 151-156.
15. U. De Giovannini, D. Varsano, M.A. Marques,
H. Appel, E.K. Gross, A. Rubio. Ab initio angle-
and energy-resolved photoelectron spectroscopy
with time-dependent density-functional theory //
Physical Review A. 2012, v. 85, p. 062515.
16. S.-I. Chu. Recent development of self-interaction-free
time-dependent density-functional theory for nonper-
turbative treatment of atomic and molecular multipho-
ton processes in intense laser fields // The Journal of
Chemical Physics. 2005, v. 123, p. 062207.
17. C.A. Ullrich. Time-dependent density-functional
theory: concepts and applications // Oxford graduate
texts, Oxford University Press. 2011.
18. D.A. Telnov, K.E. Sosnova, E. Rozenbaum,
S.-I. Chu. Exterior complex scaling method in time-
dependent density-functional theory: Multiphoton
ionization and high-order-harmonic generation of Ar
atoms // Physical Review A. 2013, v. 87, p. 053406.
19. R. Van Leeuwen, E. Baerends. Exchange-correlation
potential with correct asymptotic behavior // Physi-
cal Review A. 1994, v. 49, p. 2421.
20. A.A. Silaev, M.Y. Ryabikin, N.V. Vvedenskii.
Strong-field phenomena caused by ultrashort laser
pulses: Effective one- and two-dimensional quan-
tum-mechanical descriptions // Physical Review A.
2010, v. 82, p. 033416.
21. D. Bauer, P. Koval. Qprop: A Schrödinger-solver for
intense laser-atom interaction // Computer Physics
Communications. 2006, v. 174, p. 396-421.
22. A. Castro, M.A. Marques, A. Rubio. Propagators for
the time-dependent Kohn-Sham equations // The Jour-
nal of Chemical Physics. 2004, v. 121, p. 3425-3433.
23. A.A. Silaev, A.A. Romanov, N.V. Vvedenskii. Mul-
ti-hump potentials for efficient wave absorption in
the numerical solution of the time-dependent Schrö-
dinger equation // Journal of Physics B: Atomic, Mo-
lecular and Optical Physics. 2018, v. 51, p. 065005.
24. H.J. Wörner, H. Niikura, J.B. Bertrand,
P.B. Corkum, D.M. Villeneuve. Observation of elec-
tronic structure minima in high-harmonic generation
// Physical Review Letters. 2009, v. 102, p. 103901.
25. A.D. Shiner, et al. Probing collective multi-electron
dynamics in xenon with high-harmonic spectroscopy
// Nature Physics. 2011, v. 7, p. 464.
Article received 01.07.2018
КВАНТОВО-МЕХАНИЧЕСКОЕ МОДЕЛИРОВАНИЕ ВЗАИМОДЕЙСТВИЯ
МНОГОЭЛЕКТРОННЫХ КВАНТОВЫХ СИСТЕМ
С ИОНИЗИРУЮЩИМИ ЛАЗЕРНЫМИ ИМПУЛЬСАМИ
А.А. Романов, А.А. Силаев, Д.А. Смирнова, Т.С. Саранцева, А.А. Минина, М.В. Фролов, Н.В. Введенский
Разрабатывается параллельный программный код для численного моделирования взаимодействия интен-
сивных лазерных импульсов с многоэлектронными атомами на основе нестационарного метода функциона-
ла плотности. Показывается, что использование современных многопроцессорных вычислительных класте-
ров позволяет решить нестационарные уравнения Кона-Шэма для широкого класса атомов за относительно
небольшое время, определяемое параметрами лазерного импульса и электронной конфигурацией атома. Де-
монстрация работы численного кода представлена на примере расчета спектра высокочастотного электрон-
ного тока, возбуждаемого при ионизации атомов инертных газов коротким лазерным импульсом.
КВАНТОВО-МЕХАНІЧНЕ МОДЕЛЮВАННЯ ВЗАЄМОДІЇ БАГАТОЕЛЕКТРОННИХ КВАНТОВИХ
СИСТЕМ З ІОНІЗУЮЧИМИ ЛАЗЕРНИМИ ІМПУЛЬСАМИ
А.А. Романов, А.А. Силаєв, Д.А. Смирнова, Т.С. Саранцева, А.А. Мініна, М.В. Фролов, Н.В. Введенський
Розробляється паралельний програмний код для чисельного моделювання взаємодії інтенсивних лазер-
них імпульсів з багатоелектронними атомами на основі нестаціонарного методу функціонала щільності. По-
казується, що використання сучасних багатопроцесорних обчислювальних кластерів дозволяє розв’язати
нестаціонарні рівняння Кона-Шема для широкого класу атомів за відносно невеликий час, що визначається
параметрами лазерного імпульсу і електронною конфігурацією атома. Демонстрація роботи чисельного коду
представлена на прикладі розрахунку спектра високочастотного електронного струму, що збуджується при
іонізації атомів інертних газів коротким лазерним імпульсом.
INTRODUCTION
1. STATEMENT OF the problem
2. The numerical implementation
2.1. The numerical method
2.2. Parallelization of numerical code
2.3. Examples of calculation
Conclusions
Acknowledgement
References
Квантово-механическое моделирование взаимодействия многоэлектронных квантовых систем с ионизирующими лазерными импульсами
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