Optimization of single-hump imaginary potentials for efficient absorption of the wave function in numerical solution of the time-dependent Schrödinger equation
This work is devoted to the development of the imaginary potential method for efficient absorption of the wave function on the periphery of the computational domain in numerical solution of the time-dependent Schrödinger equation. The optimal relationships between the width and amplitude of single-h...
Збережено в:
| Опубліковано в: : | Вопросы атомной науки и техники |
|---|---|
| Дата: | 2015 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2015
|
| Теми: | |
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/112213 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Optimization of single-hump imaginary potentials for efficient absorption of the wave function in numerical solution of the time-dependent Schrödinger equation / A.A. Silaev, N.V. Vvedenskii // Вопросы атомной науки и техники. — 2015. — № 4. — С. 290-293. — Бібліогр.: 15 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860267884277661696 |
|---|---|
| author | Silaev, A.A. Vvedenskii, N.V. |
| author_facet | Silaev, A.A. Vvedenskii, N.V. |
| citation_txt | Optimization of single-hump imaginary potentials for efficient absorption of the wave function in numerical solution of the time-dependent Schrödinger equation / A.A. Silaev, N.V. Vvedenskii // Вопросы атомной науки и техники. — 2015. — № 4. — С. 290-293. — Бібліогр.: 15 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | This work is devoted to the development of the imaginary potential method for efficient absorption of the wave function on the periphery of the computational domain in numerical solution of the time-dependent Schrödinger equation. The optimal relationships between the width and amplitude of single-hump imaginary potentials and the de Broglie wavelength corresponding to the maximum of absorption efficiency are determined.
Робота присвячена розвитку методу уявного потенціалу для поглинання хвильової функції на периферії розрахункової області при чисельному розв’язанні нестаціонарного рівняння Шрьодінгера. Знайдено оптимальні співвідношення між шириною і амплітудою уявного потенціалу та довжиною хвилі де Бройля, які відповідають максимальній ефективності поглинання.
Работа посвящена развитию метода мнимого потенциала для поглощения волновой функции на периферии расчётной области при численном решении нестационарного уравнения Шрёдингера. Найдены оптимальные соотношения между шириной и амплитудой мнимого потенциала и длиной волны де Бройля, соответствующие максимальной эффективности поглощения.
|
| first_indexed | 2025-12-07T19:02:48Z |
| format | Article |
| fulltext |
ISSN 1562-6016. ВАНТ. 2015. №4(98) 290
OPTIMIZATION OF SINGLE-HUMP IMAGINARY POTENTIALS FOR
EFFICIENT ABSORPTION OF THE WAVE FUNCTION IN NUMERICAL
SOLUTION OF THE TIME-DEPENDENT SCHRÖDINGER EQUATION
A.A. Silaev1,2, N.V. Vvedenskii1,2
1Institute of Applied Physics, Russian Academy of Sciences, Nizhny Novgorod, Russia;
2Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, Russia
E-mails: silaev@appl.sci-nnov.ru; vved@appl.sci-nnov.ru
This work is devoted to the development of the imaginary potential method for efficient absorption of the wave
function on the periphery of the computational domain in numerical solution of the time-dependent Schrödinger
equation. The optimal relationships between the width and amplitude of single-hump imaginary potentials and the
de Broglie wavelength corresponding to the maximum of absorption efficiency are determined.
PACS: 02.70.-c, 31.15.-p
INTRODUCTION
The numerical solution of the time-dependent
Schrödinger equation (TDSE) is one of the main tools
for investigation of different phenomena, in particular,
ionization-induced phenomena caused by ultrashort
laser pulses [1-8]. In the latter case, the TDSE in the
length gauge and dimensionless variables is written as
,),(
2
1= 2 ψψψ tV
t
i r+∇−
∂
∂ (1)
where t is the time, ),( tV r is the time-dependent parti-
cle potential energy, and r is the particle radius vector.
In the mathematical formulation of the problem it is
assumed that the boundaries of the computational do-
main are located at infinity. However, due to the finite
size of the computational grid, the electronic wave func-
tion can reach the grid boundary and, depending on the
method of numerical solution of the TDSE, reflect on it
or move on to the opposite edge of the grid. Usually, in
order to avoid reflection and transmission of the wave
function, the absorbing layers are introduced near the
grid boundaries [9]. The different methods of absorption
are considered [6, 9 - 15], among which the simplest and
most popular method uses negative imaginary potentials.
The essence of this method consists in the introduction
of negative imaginary potential (NIP) )(NIP rU on the
periphery of the computational domain [9],
).(),(),( NIP rrr UtVtV +→ (2)
It is important that high efficiency of absorption is
achieved in a limited range of de Broglie wavelengths.
The width of this range increases with the width of the
absorbing layer. However, the greater is the width of the
layer, the more CPU time is required for numerical cal-
culation. Therefore, the actual problem is the construc-
tion of compact imaginary potentials with a wide range
of effective absorption.
One of the variants to construct the potential having
a large range of absorption is the use of a set of smooth
single-hump imaginary potentials located next to each
other. Each single-hump potential will absorb in a cer-
tain range of wavelengths. The key moment is to deter-
mine the optimal parameters of single-hump potential,
corresponding to the maximum absorption efficiency. In
this work we investigate the reflection and transmission
properties of two different single-hump imaginary po-
tentials. We determine the dependences of the reflection
and transmission coefficients on the de Broglie wave-
length, as well as on the parameters of single-hump po-
tentials. The dependences of optimal parameters of the
absorbing potentials on the incident wavelength are
found.
1. STATEMENT OF THE PROBLEM
Let us assume for simplicity that the boundary of the
computational domain is flat and the imaginary poten-
tial depends only on the coordinate x, which is directed
across the layer. Then the scattering of a plane wave on
the imaginary potential is described by the one-
dimensional stationary Schrödinger equation
ψψψ
2
=)(
2
1 2
NIP2
2 kxU
x
+
∂
∂
− , (3)
where k is the x component of the wavenumber. The
absorbing single-hump NIP is defined as
( )lxiufxU /=)(NIP − , (4)
where 0>u and l are the amplitude and characteristic
width of the potential, )(ξf is a real even bell-shaped
function that defines the shape of the hump. We consid-
er here two kinds of NIPs: cosine-squared potential,
corresponding to
( )
>
≤
1,||0,
1||,2/cos=)(
2
ξ
ξπξξf (5)
and Pöschl–Teller potential, corresponding to
( ),cosh=)( 2 αξξ −f ( ).2acosh2=α (6)
The coefficient α in the last formula is introduced
in such a way that width of the function )(ξf at the
level of 1/2 is equal to 1, as for the function (5).
The efficiency of absorption of a plane wave is char-
acterized by the so-called survival probability S , which
is equal to the sum of the reflection R and transmission
T coefficients [14]. Its value is less than or equal to
unity due to the decrease of probability density inside
the absorbing layer. The lower is S , the higher is the
efficiency of the absorption. In order to find the trans-
mission and reflection coefficients we solve Eq. (3) with
the boundary conditions
∞→
−∞→+ −
,,
,,=)(
xte
xreex ikx
ikxikx
ψ (7)
ISSN 1562-6016. ВАНТ. 2015. №4(98) 291
which correspond to a plane wave incident from the left.
Reflection and transmission coefficients are 2|=| rR
and 2|=| tT , respectively.
The problem of scattering of a plane wave on the
single-hump imaginary potential (4) has two independ-
ent parameters, namely, the normalized wavelength
l/= λν , where k/2= πλ , and the normalized ampli-
tude, ul 2=ε . In order to optimize the parameters of
imaginary potential it is necessary to find the values of
ν and ε , which correspond to minimum of survival
probability S .
2. RESULTS
The coefficients of reflection and transmission are
calculated numerically for cosine-squared and Pöschl–
Teller potentials using the reduction of Eq. (3) to the
system of two equations of the first order. This system
of equations is solved by the Runge-Kutta fourth order
method for 1=l .
Fig. 1 shows the dependences of the reflection R
and transmission T coefficients, as well as the survival
probability S on the normalized wavelength ν for
three fixed values of the normalized amplitude, 2=ε ,
20, and 80. It can be seen that for both considered po-
tentials there exists an optimal normalized wavelength
optν corresponding to the minimum of survival proba-
bility. When optνν << or optνν >> the survival prob-
ability tends to unity. It is explained by the fact that for
large ν waves are mainly reflected, while for small ν
they pass through the absorbing layer. The minimum
value of the survival probability )(= optmin SS ν de-
creases dramatically with increasing ε . In order to
achieve effective absorption one should use sufficiently
large values 20≥ε . Also note that for the same values
of ε , the minimum survival probability minS for the
Pöschl-Teller potential is significantly lower as com-
pared with the cosine-squared potential. Thus, for
80=ε , the values of minS for considered potentials
differ by more than two orders of magnitude.
Fig. 2 shows the dependence of R , T , and S on
the normalized amplitude ε for fixed values of normal-
ized wavelength 1=ν , 2, and 8. For both considered
imaginary potentials functions )(εS first decrease to
some minimum values )(=min optSS ε with increasing
ε . Further behavior of functions )(εS is significantly
different for the two considered imaginary potentials.
For the cosine-squared potential S asymptotically ap-
proaches unity, while for Pöschl-Teller potential it be-
comes a constant, which is approximately equal to
minS . The latter means that an increase in the ampli-
tude of the potential do not reduce the efficiency of ab-
sorption. Note, however, that an unlimited increase in
the amplitude of the Pöschl-Teller potential in the nu-
merical solution TDSE is impossible because of the
infinitely long tail of potential, which may distort the
wave function in the computational domain.
Fig. 1. Dependences of reflection (R) and transmission
(T) coefficients, as well as survival probability S=R+T
(short-dashed, long-dashed, and solid curves,
respectively) on the normalized wavelength l/= λν for
various fixed values of the normalized amplitude
2=ε , 20 , and 80 . Calculations are performed for (a)
the cosine-squared potential (Eqs. (4), (5)) and (b)
Pöschl-Teller potential (Eqs. (4), (6))
Fig. 2. Dependences of reflection (R) and transmission
(T) coefficients, as well as survival probability S=R+T
(short-dashed, long-dashed, and solid curves, respec-
tively) on the normalized amplitude ε = l2u for various
fixed values of the normalized wavelength
1=ν , 2 and 8 . Calculations are performed for (a)
the cosine-squared potential (Eqs. (4), (5)) and (b)
Pöschl-Teller potential (Eqs. (4), (6))
ISSN 1562-6016. ВАНТ. 2015. №4(98) 292
Next, we calculated the dependence of the optimal
normalized amplitude optε on the normalized wave-
length ν for the considered potentials. Quadratic inter-
polation of the obtained dependences using the least
square method in the range of 2<<0.5 ν gives the
following results. For the cosine-squared potential
20.671 κε +≈opt , νπκ /2= (8)
and for the Pöschl-Teller potential,
.1.172 2κε +≈opt (9)
Fig. 3. Dependences of the survival probability
corresponding to optimal normalized amplitude optε
on the normalized wavelength ν for the cosine-squared
potential (solid curve) and Pöschl-Teller potential
(dashed curve)
The results of numerical calculations of the wave-
length-dependence of survival probability correspond-
ing to the optimum amplitude are shown in Fig. 3. It can
be seen that the survival probability decreases sharply
with decreasing of the de Broglie wavelength. In order
to find the amplitude u of the negative imaginary po-
tential corresponding to high-efficient absorption of the
de Broglie wavelength λ one should use the relation
2)//(= llu optopt λε , (10)
with the use of Eq. (8) or Eq. (9).
CONCLUSIONS
To conclude, in this work we have calculated the co-
efficients of transmission and reflection of the plane
wave on the two different single-hump negative imagi-
nary potentials (NIPs) for wide range of the potential
parameters and de Broglie wavelengths. It is shown that
the Pöschl-Teller potential (Eqs. (4), (6)) provides more
efficient absorption of the entire range of de Broglie
wavelengths than the cosine-squared potential (Eqs. (4),
(5)). At the same time the advantage of using the co-
sine-squared potential in numerical calculations is the
finite interval of its location, as opposed to Pöschl-
Teller potential.
The relationships between the width and the ampli-
tude of the considered potentials and the de Broglie
wavelength corresponding to the maximum of absorp-
tion efficiency are determined. The obtained optimal
parameters of NIPs can be used to construct the absorb-
ing potential containing several humps of different
width and amplitude for high-efficient absorption in
wide range of de Broglie wavelengths.
ACKNOWLEDGEMENTS
The development of the wave function absorption
methods was supported by the Government of the Rus-
sian Federation (Agreement No 14.B25.31.0008) and
the Russian Foundation for Basic Research (Grants No.
14-02-00847 and No. 14-02-31044). The development
of the program codes was supported by the Russian Sci-
ence Foundation (Grant No 15-12-10033).
REFERENCES
1. P.B. Corkum. Plasma perspective on strong field
multiphoton ionization // Physical Review Letters.
1993, v. 71, p. 1994-1997.
2. F. Krausz, M. Ivanov. Attosecond physics // Reviews
of Modern Physics. 2009, v. 81, p. 163-234.
3. A.A. Silaev, N.V. Vvedenskii. Residual-current ex-
citation in plasmas produced by few-cycle laser
pulses // Physical Review Letters. 2009, v. 102,
p. 115005-1-4.
4. A.A. Silaev, N.V. Vvedenskii. Quantum-mechanical
approach for calculating the residual quasi-dc cur-
rent in a plasma produced by a few-cycle laser pulse
// Physica Scripta. 2009, v. T135, p. 014024-1-5.
5. 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. 033416-1-14.
6. V.V. Strelkov, M.A. Khokhlova, A.A. Gonoskov,
I.A. Gonoskov, M.Y. Ryabikin. High-order harmon-
ic generation by atoms in an elliptically polarized la-
ser field: Harmonic polarization properties and laser
threshold ellipticity // Physical Review A. 2012, v.
86, p. 013404-1-10.
7. A.A. Silaev, N.V. Vvedenskii. Analytical descrip-
tion of generation of the residual current density in
the plasma produced by a few-cycle laser pulse //
Physics of Plasmas. 2015, v. 22, p. 053103-1-14.
8. A.A. Silaev, O.V. Meshkov, M.Y. Emelin,
N.V. Vvedenskii, M.Y. Ryabikin. Control of the
photoelectron dynamics for the effective conversion
of short-pulse, frequency-modulated optical radia-
tion into X-ray radiation // Quantum Electronics.
2015, v. 45, p. 393-400.
9. J.G. Muga, J.P. Palao, B. Navarro, I.L. Egusquiza.
Complex absorbing potentials // Physics Reports.
2004, v. 395, p. 357-426.
10. D. Neuhasuer, M. Baer. The time-dependent Schrö-
dinger equation: Application of absorbing boundary
conditions // Journal of Chemical Physics. 1989,
v. 90, p. 4351-4355.
11. A. Vibok, G.G. Balint-Kurti. Parametrization of
complex absorbing potentials for time-dependent
quantum dynamics // Journal of Physical Chemistry.
1992, v.96, p.8712-8719.
12. U.V. Riss, H.-D. Meyer. Investigation on the reflec-
tion and transmission properties of complex absorb-
ing potentials // The Journal of Chemical Physics.
1996, v. 105, p. 1409-1419.
13. N. Moiseyev. Derivations of universal exact com-
plex absorption potentials by the generalized com-
plex coordinate method // J. Phys. B: At., Mol. Opt.
Phys. 1998, v. 31, p. 1431-1441.
ISSN 1562-6016. ВАНТ. 2015. №4(98) 293
14. J.P. Palao, J.G. Muga, R. Sala. Composite absorbing
potentials // Physical Review Letters. 1998, v. 80,
p. 5469-5472.
15. A. Nissen, G. Kreiss. An optimized perfectly
matched layer for the Schrödinger equation // Com-
munications in Computational Physics. 2011, v. 9,
p. 147-179.
Article received 02.06.2015
ОПТИМИЗАЦИЯ ПАРАМЕТРОВ КОЛОКОЛООБРАЗНЫХ МНИМЫХ ПОТЕНЦИАЛОВ
ДЛЯ ЭФФЕКТИВНОГО ПОГЛОЩЕНИЯ ВОЛНОВОЙ ФУНКЦИИ ПРИ ЧИСЛЕННОМ РЕШЕНИИ
НЕСТАЦИОНАРНОГО УРАВНЕНИЯ ШРЁДИНГЕРА
А.А. Силаев, Н.В. Введенский
Работа посвящена развитию метода мнимого потенциала для поглощения волновой функции на перифе-
рии расчётной области при численном решении нестационарного уравнения Шрёдингера. Найдены опти-
мальные соотношения между шириной и амплитудой мнимого потенциала и длиной волны де Бройля, соот-
ветствующие максимальной эффективности поглощения.
ОПТИМІЗАЦІЯ ПАРАМЕТРІВ ДЗВОНОВИДНИХ УЯВНИХ ПОТЕНЦІАЛІВ ДЛЯ ЕФЕКТИВНОГО
ПОГЛИНАННЯ ХВИЛЬОВОЇ ФУНКЦІЇ ПРИ ЧИСЕЛЬНОМУ РОЗВ’ЯЗАННІ
НЕСТАЦІОНАРНОГО РІВНЯННЯ ШРЬОДІНГЕРА
О.А. Силаєв, М.В. Введенський
Робота присвячена розвитку методу уявного потенціалу для поглинання хвильової функції на периферії
розрахункової області при чисельному розв’язанні нестаціонарного рівняння Шрьодінгера. Знайдено опти-
мальні співвідношення між шириною і амплітудою уявного потенціалу та довжиною хвилі де Бройля, які
відповідають максимальній ефективності поглинання.
Introduction
1. Statement of the problem
2. Results
Conclusions
ACKNOWLEDGEMENTS
references
|
| id | nasplib_isofts_kiev_ua-123456789-112213 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T19:02:48Z |
| publishDate | 2015 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Silaev, A.A. Vvedenskii, N.V. 2017-01-18T19:38:42Z 2017-01-18T19:38:42Z 2015 Optimization of single-hump imaginary potentials for efficient absorption of the wave function in numerical solution of the time-dependent Schrödinger equation / A.A. Silaev, N.V. Vvedenskii // Вопросы атомной науки и техники. — 2015. — № 4. — С. 290-293. — Бібліогр.: 15 назв. — англ. 1562-6016 PACS: 02.70.-c, 31.15.-p https://nasplib.isofts.kiev.ua/handle/123456789/112213 This work is devoted to the development of the imaginary potential method for efficient absorption of the wave function on the periphery of the computational domain in numerical solution of the time-dependent Schrödinger equation. The optimal relationships between the width and amplitude of single-hump imaginary potentials and the de Broglie wavelength corresponding to the maximum of absorption efficiency are determined. Робота присвячена розвитку методу уявного потенціалу для поглинання хвильової функції на периферії розрахункової області при чисельному розв’язанні нестаціонарного рівняння Шрьодінгера. Знайдено оптимальні співвідношення між шириною і амплітудою уявного потенціалу та довжиною хвилі де Бройля, які відповідають максимальній ефективності поглинання. Работа посвящена развитию метода мнимого потенциала для поглощения волновой функции на периферии расчётной области при численном решении нестационарного уравнения Шрёдингера. Найдены оптимальные соотношения между шириной и амплитудой мнимого потенциала и длиной волны де Бройля, соответствующие максимальной эффективности поглощения. The development of the wave function absorption methods was supported by the Government of the Russian Federation (Agreement No 14.B25.31.0008) and the Russian Foundation for Basic Research (Grants No. 14-02-00847 and No. 14-02-31044). The development of the program codes was supported by the Russian Science
 Foundation (Grant No 15-12-10033). en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Нелинейные процессы в плазменных средах Optimization of single-hump imaginary potentials for efficient absorption of the wave function in numerical solution of the time-dependent Schrödinger equation Оптимізація параметрів дзвоновидних уявних потенціалів для ефективного поглинання хвильової функції при чисельному розв’язанні нестаціонарного рівняння Шрьодінгера Оптимизация параметров колоколообразных мнимых потенциалов для эффективного поглощения волновой функции при численном решении нестационарного уравнения Шрёдингера Article published earlier |
| spellingShingle | Optimization of single-hump imaginary potentials for efficient absorption of the wave function in numerical solution of the time-dependent Schrödinger equation Silaev, A.A. Vvedenskii, N.V. Нелинейные процессы в плазменных средах |
| title | Optimization of single-hump imaginary potentials for efficient absorption of the wave function in numerical solution of the time-dependent Schrödinger equation |
| title_alt | Оптимізація параметрів дзвоновидних уявних потенціалів для ефективного поглинання хвильової функції при чисельному розв’язанні нестаціонарного рівняння Шрьодінгера Оптимизация параметров колоколообразных мнимых потенциалов для эффективного поглощения волновой функции при численном решении нестационарного уравнения Шрёдингера |
| title_full | Optimization of single-hump imaginary potentials for efficient absorption of the wave function in numerical solution of the time-dependent Schrödinger equation |
| title_fullStr | Optimization of single-hump imaginary potentials for efficient absorption of the wave function in numerical solution of the time-dependent Schrödinger equation |
| title_full_unstemmed | Optimization of single-hump imaginary potentials for efficient absorption of the wave function in numerical solution of the time-dependent Schrödinger equation |
| title_short | Optimization of single-hump imaginary potentials for efficient absorption of the wave function in numerical solution of the time-dependent Schrödinger equation |
| title_sort | optimization of single-hump imaginary potentials for efficient absorption of the wave function in numerical solution of the time-dependent schrödinger equation |
| topic | Нелинейные процессы в плазменных средах |
| topic_facet | Нелинейные процессы в плазменных средах |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/112213 |
| work_keys_str_mv | AT silaevaa optimizationofsinglehumpimaginarypotentialsforefficientabsorptionofthewavefunctioninnumericalsolutionofthetimedependentschrodingerequation AT vvedenskiinv optimizationofsinglehumpimaginarypotentialsforefficientabsorptionofthewavefunctioninnumericalsolutionofthetimedependentschrodingerequation AT silaevaa optimízacíâparametrívdzvonovidnihuâvnihpotencíalívdlâefektivnogopoglinannâhvilʹovoífunkcíípričiselʹnomurozvâzannínestacíonarnogorívnânnâšrʹodíngera AT vvedenskiinv optimízacíâparametrívdzvonovidnihuâvnihpotencíalívdlâefektivnogopoglinannâhvilʹovoífunkcíípričiselʹnomurozvâzannínestacíonarnogorívnânnâšrʹodíngera AT silaevaa optimizaciâparametrovkolokoloobraznyhmnimyhpotencialovdlâéffektivnogopogloŝeniâvolnovoifunkciipričislennomrešeniinestacionarnogouravneniâšredingera AT vvedenskiinv optimizaciâparametrovkolokoloobraznyhmnimyhpotencialovdlâéffektivnogopogloŝeniâvolnovoifunkciipričislennomrešeniinestacionarnogouravneniâšredingera |