The ideal gas of hydrogen-like atoms response to the perturbation by the external electromagnetic field
The response of the ideal gas consisting of hydrogen-like atoms to the perturbation by the external electromagnetic field in low temperature region is studied. Consideration is based on using the Green functions formalism [1] and the second quantization method in the presence of bound states of part...
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| Cite this: | The ideal gas of hydrogen-like atoms response to the perturbation by the external electromagnetic field / Yu.V. Slyusarenko, A.G. Sotnikov // Вопросы атомной науки и техники. — 2007. — № 3. — С. 390-393. — Бібліогр.: 6 назв. — англ. |
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Slyusarenko, Yu.V. Sotnikov, A.G. 2017-01-07T19:07:49Z 2017-01-07T19:07:49Z 2007 The ideal gas of hydrogen-like atoms response to the perturbation by the external electromagnetic field / Yu.V. Slyusarenko, A.G. Sotnikov // Вопросы атомной науки и техники. — 2007. — № 3. — С. 390-393. — Бібліогр.: 6 назв. — англ. 1562-6016 PACS: 05.30.d, 05.30.Jp, 03.75.Mn, 03.75.Hh https://nasplib.isofts.kiev.ua/handle/123456789/111048 The response of the ideal gas consisting of hydrogen-like atoms to the perturbation by the external electromagnetic field in low temperature region is studied. Consideration is based on using the Green functions formalism [1] and the second quantization method in the presence of bound states of particles [2]. As the most interesting phenomenon, the perturbation of system in Bose-condensation state is studied. The dispersion characteristics of such system at frequencies close to the energy interval between alkali atoms energy levels are investigated. Вивчено відгук ідеального газу, що складається із водневоподібних атомів, на збурення зовнішнім електромагнітним полем у низькотемпературній області. Дослідження базуються на використанні формалізму функцій Гріна [1] та методу вторинного квантування, що враховує можливість утворення зв’язаних станів частинок [2]. Як найбільш цікаве явище вивчено збурення системи, яка знаходиться у стані бозе-конденсації. Досліджені дисперсійні характеристики бозе-конденсату в області частот, близьких до інтервалів енергій між рівнями атомів лужних металів. Изучен отклик идеального газа, состоящего из водородоподобных атомов, на возмущение внешним электромагнитным полем в низкотемпературной области. Исследования основаны на формализме функций Грина [1] и методе вторичного квантования в присутствии связанных состояний частиц [2]. В качестве наиболее интересного явления изучено возмущение системы, находящейся в состоянии бозе-конденсации. Исследованы дисперсионные характеристики такой системы в области частот, близких к интервалам энергий между уровнями атомов щелочных металлов. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Phase transformations in condensed matter The ideal gas of hydrogen-like atoms response to the perturbation by the external electromagnetic field Відгук ідеального газу водневоподібних атомів на збурення зовнішнім електромагнітним полем Отклик идеального газа водородоподобных атомов на возмущение внешним электромагнитным полем Article published earlier |
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The ideal gas of hydrogen-like atoms response to the perturbation by the external electromagnetic field |
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The ideal gas of hydrogen-like atoms response to the perturbation by the external electromagnetic field Slyusarenko, Yu.V. Sotnikov, A.G. Phase transformations in condensed matter |
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The ideal gas of hydrogen-like atoms response to the perturbation by the external electromagnetic field |
| title_full |
The ideal gas of hydrogen-like atoms response to the perturbation by the external electromagnetic field |
| title_fullStr |
The ideal gas of hydrogen-like atoms response to the perturbation by the external electromagnetic field |
| title_full_unstemmed |
The ideal gas of hydrogen-like atoms response to the perturbation by the external electromagnetic field |
| title_sort |
ideal gas of hydrogen-like atoms response to the perturbation by the external electromagnetic field |
| author |
Slyusarenko, Yu.V. Sotnikov, A.G. |
| author_facet |
Slyusarenko, Yu.V. Sotnikov, A.G. |
| topic |
Phase transformations in condensed matter |
| topic_facet |
Phase transformations in condensed matter |
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2007 |
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English |
| container_title |
Вопросы атомной науки и техники |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Article |
| title_alt |
Відгук ідеального газу водневоподібних атомів на збурення зовнішнім електромагнітним полем Отклик идеального газа водородоподобных атомов на возмущение внешним электромагнитным полем |
| description |
The response of the ideal gas consisting of hydrogen-like atoms to the perturbation by the external electromagnetic field in low temperature region is studied. Consideration is based on using the Green functions formalism [1] and the second quantization method in the presence of bound states of particles [2]. As the most interesting phenomenon, the perturbation of system in Bose-condensation state is studied. The dispersion characteristics of such system at frequencies close to the energy interval between alkali atoms energy levels are investigated.
Вивчено відгук ідеального газу, що складається із водневоподібних атомів, на збурення зовнішнім електромагнітним полем у низькотемпературній області. Дослідження базуються на використанні формалізму функцій Гріна [1] та методу вторинного квантування, що враховує можливість утворення зв’язаних станів частинок [2]. Як найбільш цікаве явище вивчено збурення системи, яка знаходиться у стані бозе-конденсації. Досліджені дисперсійні характеристики бозе-конденсату в області частот, близьких до інтервалів енергій між рівнями атомів лужних металів.
Изучен отклик идеального газа, состоящего из водородоподобных атомов, на возмущение внешним электромагнитным полем в низкотемпературной области. Исследования основаны на формализме функций Грина [1] и методе вторичного квантования в присутствии связанных состояний частиц [2]. В качестве наиболее интересного явления изучено возмущение системы, находящейся в состоянии бозе-конденсации. Исследованы дисперсионные характеристики такой системы в области частот, близких к интервалам энергий между уровнями атомов щелочных металлов.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/111048 |
| citation_txt |
The ideal gas of hydrogen-like atoms response to the perturbation by the external electromagnetic field / Yu.V. Slyusarenko, A.G. Sotnikov // Вопросы атомной науки и техники. — 2007. — № 3. — С. 390-393. — Бібліогр.: 6 назв. — англ. |
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2025-11-25T04:21:28Z |
| last_indexed |
2025-11-25T04:21:28Z |
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1850506122535895040 |
| fulltext |
THE IDEAL GAS OF HYDROGEN-LIKE ATOMS RESPONSE TO THE
PERTURBATION BY THE EXTERNAL ELECTROMAGNETIC FIELD
Yu.V. Slyusarenko and A.G. Sotnikov
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine;
e-mail: emvg@ukr.net
The response of the ideal gas consisting of hydrogen-like atoms to the perturbation by the external electromag-
netic field in low temperature region is studied. Consideration is based on using the Green functions formalism [1]
and the second quantization method in the presence of bound states of particles [2]. As the most interesting phe-
nomenon, the perturbation of system in Bose-condensation state is studied. The dispersion characteristics of such
system at frequencies close to the energy interval between alkali atoms energy levels are investigated.
PACS: 05.30.d, 05.30.Jp, 03.75.Mn, 03.75.Hh
1. INTRODUCTION
In the process of describing a behavior of many-
particle systems a class of problems appear, that are
concerned with the system response to the perturbing
action of the external, in particular, electromagnetic
field. Widespread approach to solving such kind of
problems is based on using the Green functions formal-
ism (see in that case e.g. Ref. [1]).
It is well known that the most convenient method of
describing physical processes in quantum many-particle
theory is the second quantization method. Thus, within
the framework of the second quantization it is the sim-
plest way to formulate an approach to a description of
the system response to the perturbation by the external
field that is based on Green functions. However, if we
try to realize such an approach, we can come across an
essential difficulty, connected with the possible occur-
rence of the particle bound states.
As it has been shown in [2] for a system that con-
sists of two types of fermions (e.g. ions and electrons)
and bound states (atoms or molecules) in low kinetic
energies region we can use the approximate formulation
of the second quantization method. It makes the mathe-
matical description of such kind of systems rather sim-
ple, but preserves the required information concerning
internal degrees of freedom for the bound states.
If the creation and annihilation operators for the dif-
ferent kind of particles are constructed, it is not difficult
to broaden this theory on the external fields’ existence
situation. In this case the physical quantities operators
(such as charge and current density operators) and the
Maxwell-Lorentz system of equations that includes con-
tribution of neutral bound states can be constructed (see
Ref. [3]).
In order to simplify the following mathematical de-
scription it is convenient to consider such system (ideal
low-temperature hydrogen-like plasma) in the equilib-
rium state (including its photon component). In this case
(as it has been shown in Ref. [4]) we can neglect of free
fermions contribution in the different physical processes
that are observed in the system at extremely low tem-
peratures.
2. BASIC OPERATORS
As we consider the system in low temperature re-
gion, the creation and annihilation operators for the hy-
drogen-like atoms (as the bound states of two different
particles, see Ref. [2]) can be written as
1 1 2 2
1 2
1 2
1 2
1 2
*
x x
ˆ ˆ(x , x ) (x) (X),
ˆ(x , x ) (x) (X),
x x x , X .m m
m m
α α
α
α α
α
ϕ ϕ η
ϕ ϕ η
+ +
+
+
=
=
= − =
∑
∑ ˆ (1)
Here m1, x1 and m2, x2 are the atomic core and the outer
electron masses and coordinates, respectively, α is the
set of quantum numbers of the atom in the certain state,
characterized by the wave function , is
the annihilation operator of boson in α state.
(x)αϕ ˆ (X)αη
The expression for the Hamiltonian of non-
interacting atoms can be written as:
1
p
ˆ ˆ(p) (p) (p)p V α α α
α
ε η η− += ∑∑H , (2)
where , is the energy of an
atom in quantum state ( ), M is the bound state
mass, is the system volume. Using Eqs. (1)-(2) one
can find the expressions for the charge and current den-
sity operators (accordingly to Ref. [1]), which in the
Heisenberg representation take the form:
2 2(p) p / Mα αε ε= +
α αε
V
αε
0<
( )
1
1
2
− −− − −
+
− −− − −
+ +
=
× −
=
× − + −
∑∑
∑∑
iti
iti
M
V e e
V e e
( (p) (p '))x(p p ')
p,p ' ,
( (p) (p '))x(p p ')
p,p ' ,
(p p')
ˆ (x)
ˆ ˆ (p p') (p) (p'),
ĵ(x)
ˆ ˆ(p p') I (p p') (p) (p').
α β
α β
ε ε
α β
αβ α β
ε ε
α β
αβ αβ α β
σ
σ η η
σ η η
(3)
Here for the compactness we have introduced the
charge density σαβ(k) and the current density Iαβ(k) ma-
trix elements, which can be represented in terms of the
atomic wave functions by the following formulas (see
also Ref. [2]):
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2007, N3 (2), p. 390-393. 390
( )2 1
2 1
1 2
1 1
m m
M M
m m
M M
e d
i i
e d
i i
m m
αβ α β
β α
αβ α β
σ ϕ ϕ
ϕ ϕ
ϕ ϕ
=
× − −
∂ ∂
= +
∂ ∂
× + −
∫
∫
*
*
*
(k) y (y) (y)
exp ky exp( ky) ,
(y) (y)
I (k) y (y) (y)
y y
exp ky exp( ky) ,
(4)
where e corresponds to the absolute value of electron’s
charge.
3. MAXWELL EQUATIONS AND GREEN
FUNCTIONS
We have introduced the operators of charge
and current density of ideal gas that consists of
neutral atoms, and then it is not difficult to build
the Maxwell-Lorentz system of equations that can
describe the system response to the perturbation
by the external electromagnetic field. These equa-
tions for the mean values of operators have the
following form:
( )
( )
rot , rot 4 ( ),
div 0, div 4 ( ),
H EE H J J
H E
e
e
c c
t t
π
π σ σ
∂ ∂
= − = − +
∂ ∂
= = +
(5)
where values σ(e) and J(e) are the external charge
and current densities. If to assume that the pertur-
bation of the system by the external field source
is rather small, and the Hamiltonian of interaction
is linear in respect to the external electromagnetic
field, the mean values for the charge and current
densities in Eq. (5) can be represented in terms of
the Green functions (see Ref. [1], [3])
3 ( )
( ) ( ) ( )
3 ( )
( ) ( ) ( )
( , ) ' ' ( ', ')
1 ( ', ') ( ', ') ( ', ') ,
( , ) ' ' ( ', ')
1 ( ', ') ( ', ') ( ', ') .
i
e
i
k kl
e e
l k
t dt d x G t t
A t G t t t
c
J t dt d x G t t
A t G t t t
c
σ
ϕ
ϕ
∞
+
−∞
+
∞
+
−∞
+
= − − −
× + − −
= − − −
× + − −
∫ ∫
∫ ∫
x x x
x x x x
x x x
x x x x
e
] }T −
(6)
Here are retarded scalar, vector
and tensor Green functions respectively. Introduc-
ing the equilibrium state distribution functions for
the ideal gas of hydrogen-like (alkali) atoms
( ) ( ) ( ), ,k klG G G+ + +
[{ 1
( ) exp ( ( ) ) / 1p pfα α αε µ
−
= − , (7)
where is the atomic chemical potential, is
the temperature in the energy units, the Fourier
transforms of these Green functions, according to
Ref. [3], can be written as follows:
αµ T
( ) 1
,
( ) 1
,
( ) 1
,
( , ) ( ) ( )
( ) ( )
,
( ) ( ) 0
(2 )( , ) ( )
2
( ) ( ) ( )
( ) ,
( ) ( ) 0
(2 )( , ) ( )
2
( )
l
l
lj
G V
f f
i
G V
M
f f
i
G V
M
αβ βα
α β
α β
α β
αβ
α β
βα α β
αβ
α β
αβ
α β
αβ
ω σ σ
ε ε ω
ω σ
σ
ε ε ω
ω σ
+ −
+ −
+ −
= −
− −
×
− − + +
−=
− − −+ − − + +
−=
+
∑∑
∑∑
∑∑
p
p
p
k k
p k p
p p k
p kk k
k p k p
I k
p p k
p kk k
I k
k
(2 ) ( ) ( )
2
( ) ( )
.
( ) ( ) 0
l
jM
f f
i
βα βα
α β
α β
σ
ε ε ω
− − + −
− −
×
− − + +
p k k I k
p k p
p p k
(8)
4. THE IDEAL GAS IN THE BEC STATE
As the hydrogen-like atoms obey the Bose-statistics,
our system can also exhibit different properties peculiar
only to Bose-gases. The most interesting feature of such
systems (for our opinion) is a capability to produce the
Bose-Einstein condensate (BEC) at extremely low tem-
peratures (see more in that case in Ref. [4]). Moreover,
analyzing most of experiments with dilute gases of al-
kali atoms in BEC state (see e.g. Refs. [5], [6]) it is
clear that interaction of such system with an external
field plays one of the main roles in the investigated
phenomena.
4.1. MACROSCOPIC PARAMETERS
To find the macroscopic parameters of the system in
BEC state ( T ) we shall consider the case of zero
temperature that is equivalent to the assumption when
we can neglect the over-condensate particles contribu-
tion. Therefore, one can state that the bound states dis-
tribution functions are proportional to the Dirac
delta-function. According to (8) after integration over
momentum p the scalar Green function of the ideal gas
in BEC state will take the form:
cT<<
( )pfα
( ) 3( )
,
( , ) 2 ( ) ( )
,
0 0k k
G
nn
i i
αβ βα
α β
βα
αβ αβ
ω π σ σ
ε ε ω ε ε ω
−+ = −
× −
∆ − + + ∆ + + +
∑k k k
(9)
where is the density of condensed atoms in the cer-
tain state, is the energy interval and the
quantity (we shall neglect it below). Note,
that analogously to Eq. (9) the vector and tensor Green
functions (see Eq. (8)) for the system in BEC state can
be found.
nα
ε
αβ α βε ε ε∆ = −
2 2/k k M=
If the Green functions are known, it is not difficult
to find out the macroscopic parameters (see, for exam-
ple Ref. [1]), characterizing the system response to the
perturbation by the external electromagnetic field (e.g.
the laser radiation). The expressions for the permittivity
391
and magnetic permeability in terms of the Green func-
tions can be written as follows:
1 ( )
2
1
2 2
4( , ) 1 ( , ),
4( , ) 1 ( ),
k k
k l t
G
k
ic k
πω ω
πωµ ω σ σ
− +
−
= +
= + −
ε
(10) n
where and are longitudinal and transversal con-
ductivity coefficients, respectively:
lσ tσ
( )
2 ( )
2 2 2 ( )
2 2 2 ( )
( ) ( )
2
( , )( , )
4 ( , )
( , )( , ) ,
( ) 4 ( ,
1( , ) ( , ),
2
kk ,
k
kk
k
k k
l
t
i j
ij ij
i G
k G
k c
i k c
k k
G
k
ω ωσ ω
π ω
ω ωσ ω
ω ω π ω
ω δ
+
+
+
+
+ +
=
+
−
=
− +
= −
G
G
G
)
ω
where is the Kronekker symbol. ijδ
4.2. LIGHT DELAY PHENOMENON
As it is known from the experiments (see Ref. [6]),
if to set the frequency of the laser (as an external field
source) close to some of the atomic levels, the group
velocity of such signal can be slow down to the ex-
tremely little values. We assume that the developed the-
ory also can describe such kind of peculiarities.
Let us demonstrate it on the system that is close by
the energy structure to alkali atom’s levels (see Fig. 1).
l=0
2
1
3
l=1
Fig. 1. Three-level system. Atoms are occupying
mostly the first and the second states (hyperfine struc-
ture levels); the third state (non-occupied) corresponds
to the dipole-exited state. The laser frequency (arrow
vertical line) is dephased relatively to the energy inter-
vals and 13ε∆ 23ε∆
In that case (when we can neglect the contribution of
the other states of atoms), using Eqs. (9)-(10), one can
find the following expression for the permittivity:
1
2 2
2
13 1 23 2
13 13 23 23
2( , ) 1
(2 )
( ) ( )
,
k
n n
i i
ω
π
σ σ
ω ε γ ω ε γ
− = +
× +
− ∆ + − ∆ +
k
k k
ε
2
ω
(11)
where the quantities and are linewidths related
to the probability of the spontaneous transition from the
dipole-excited state to the lower and upper states of
hyperfine structure levels, respectively. As it is easy to
see from Eq. (11), the permittivity in general
case is the complex quantity
.
13γ 23γ
( , )k ωε
( , ) '( , ) ''( , )k k kiω ω= +ε ε ε
2 2
2 2
1
2
1
2
' ''
' '' '
n
ς
= +
= +
ε ε ε
ε ε ε
1.002
1.002
Using well-known relations for the refractive index
and for the dissipation factor ς
',+
−
one can find the refractive index and the intensity of
passed light dependencies (see Fig. 2,3)
1.015
1.010
1.005
1.000
0.095
0.090
0.085
0.0980.096 1.000
n
Fig. 2. Refractive index behavior for the three-level
system. . Left and right steep slopes
( ) correspond to the frequen-
cies and , respectively
0 13 12 2/ω ε ε= ∆ + ∆
0 0 099 1 001/ , ; ,ω ω= =
13ε= ∆ 23ω ε= ∆
x
ω
0.0980.096 1.000
0.1
0.2
0.3
0.4
0.5
Ip
Fig. 3. Relative intensity of the passed light de-
pendence on the relative frequency. At the frequencies
and the dissipation is large and
light doesn’t propagate. At frequencies that is dephased
relatively to the energy intervals, the transparency win-
dow can exist
13ω ε= ∆ 23ω ε= ∆
In the regions where the dissipation is not large one
can use the expression for the group velocity:
( / )g
cv
n dn dω
=
+ ω
. (12)
From the Fig. 2, Fig. 3 and Eq. (12) it is clear that the
group velocity depends greatly from the steepness of
the central slope. If the levels marked by 1 and 2 (levels
of hyperfine structure for alkali atoms) are situated suf-
392
ficiently close to each other, the light can be slow down
to the extremely little values (see Ref. [6]). Note, that
such phenomenon cannot be observed for the frequen-
cies and because of large dissipa-
tion (see Fig. 3).
13ω ε= ∆ 23ω ε= ∆
Finally, one can conclude that if the frequency of the
illuminating laser is dephased relatively to the energy
intervals, and ground state levels (marked here by num-
bers 1 and 2) are situated close to each other, the light
delay phenomenon can be observed.
REFERENCES
1. A.I. Akhiezer and S.V. Peletminskii. Methods of
Statistical Physics. Pergamon, Oxford, 1981.
2. S.V. Peletminskii, Yu.V. Slyusarenko. Second quan-
tization method in the presence of bound states of
particles //J. Math. Phys. 2005, v. 46, 022301.
3. Yu.V. Slyusarenko and A.G. Sotnikov. On the re-
sponse of a system with bound states of particles to
the perturbation by the external electromagnetic
field //Cond. Mat. Phys. 2006, v. 9, №3(47),
p. 459-471.
4. Yu.V. Slyusarenko and A.G. Sotnikov. On the in-
fluence of internal atomic structure on Bose–
Einstein condensation in an ideal gas of hydrogen-
like atoms //Fiz. Nizk. Temp. 2007, v. 33, №1, p. 41-
50 (in Russian).
5. C.J. Myatt, E.A. Burt, R.W. Ghrist, E.A. Cornell, and
C.E. Wieman //Phys. Rev. Lett. 1997, v. 78, p. 586.
6. L.V. Hau, S.E. Harris, Z. Datton, and C.H. Beh-
voozi //Nature. 1999, v. 397, p. 594.
ОТКЛИК ИДЕАЛЬНОГО ГАЗА ВОДОРОДОПОДОБНЫХ АТОМОВ НА ВОЗМУЩЕНИЕ
ВНЕШНИМ ЭЛЕКТРОМАГНИТНЫМ ПОЛЕМ
Ю.В. Слюсаренко, А.Г. Сотников
Изучен отклик идеального газа, состоящего из водородоподобных атомов, на возмущение внешним элек-
тромагнитным полем в низкотемпературной области. Исследования основаны на формализме функций
Грина [1] и методе вторичного квантования в присутствии связанных состояний частиц [2]. В качестве наи-
более интересного явления изучено возмущение системы, находящейся в состоянии бозе-конденсации. Ис-
следованы дисперсионные характеристики такой системы в области частот, близких к интервалам энергий
между уровнями атомов щелочных металлов.
ВІДГУК ІДЕАЛЬНОГО ГАЗУ ВОДНЕВОПОДІБНИХ АТОМІВ НА ЗБУРЕННЯ ЗОВНІШНІМ
ЕЛЕКТРОМАГНІТНИМ ПОЛЕМ
Ю.В. Слюсаренко, А.Г. Сотніков
Вивчено відгук ідеального газу, що складається із водневоподібних атомів, на збурення зовнішнім елек-
тромагнітним полем у низькотемпературній області. Дослідження базуються на використанні формалізму
функцій Гріна [1] та методу вторинного квантування, що враховує можливість утворення зв’язаних станів
частинок [2]. Як найбільш цікаве явище вивчено збурення системи, яка знаходиться у стані бозе-
конденсації. Досліджені дисперсійні характеристики бозе-конденсату в області частот, близьких до
інтервалів енергій між рівнями атомів лужних металів.
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