Excitation spectrum and electrical properties of the condensate of Bose atoms
It is shown that the condensate of a degenerated Bose gas consisting of neutral atoms possesses electrical properties which differ from a trivial polarization of the atoms in the electric field. A notion of an isotropic quadrupole moment (IQM) of a neutral atom is introduced. A distribution of I...
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| Zitieren: | Excitation spectrum and electrical properties of the condensate of Bose atoms / A.M. Kosevich // Condensed Matter Physics. — 2005. — Т. 8, № 4(44). — С. 773–778. — Бібліогр.: 6 назв. — англ. |
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irk-123456789-1210512017-06-14T03:04:43Z Excitation spectrum and electrical properties of the condensate of Bose atoms Kosevich, A.M. It is shown that the condensate of a degenerated Bose gas consisting of neutral atoms possesses electrical properties which differ from a trivial polarization of the atoms in the electric field. A notion of an isotropic quadrupole moment (IQM) of a neutral atom is introduced. A distribution of IQM reflects a specific spatial ordering in the condensate and produces a distribution of the electric potential. Small vibrations of the Bose gas are considered and a correction to the Bogoliubov spectrum of elementary excitations in the degenerated Bose gas is obtained. An additional term in the Gross-Pitaevskii equation which is responsible for such a correction is found and a new type of the nonlinear Schrodinger equation (NSE) is constructed. Since the Bose condensate is akin to the superfluid component in He II, a manifestation of its electrical activity could have a relation to the electrical activity of the superfluid liquid observed experimentally. Показано, що конденсат виродженого Бозе-газу нейтральних атомів володіє електричними властивостями, які відрізняються від тривіальної поляризації атомів у електричному полі. Вводиться поняття ізотропного квадрупольного моменту (ІКМ) нейтрального атома. Розподіл ІКМ відображає особливе просторове впорядкування в конденсаті і генерує розподіл електричного потенціалу. Розглядаються малі коливання Бозе-газу і отримується поправка до Боголюбівського спектру елементарних збуджень у виродженому Бозе-газі. Знайдено додатковий член у рівнянні Гроса-Пітаєвского, який відповідає за таку поправку та побудовано новий тип нелінійного рівняння Шредіннера. Оскільки Бозе-конденсат є споріднений з надплинною компонентою в He II, то прояв його електричної поведінки може мати відношення до експериментально спостережуваної електричної поведінки надплинної рідини 2005 Article Excitation spectrum and electrical properties of the condensate of Bose atoms / A.M. Kosevich // Condensed Matter Physics. — 2005. — Т. 8, № 4(44). — С. 773–778. — Бібліогр.: 6 назв. — англ. 1607-324X PACS: 47.27.Eq, 67.40.Db DOI:10.5488/CMP.8.4.773 http://dspace.nbuv.gov.ua/handle/123456789/121051 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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English |
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It is shown that the condensate of a degenerated Bose gas consisting
of neutral atoms possesses electrical properties which differ from a trivial
polarization of the atoms in the electric field. A notion of an isotropic
quadrupole moment (IQM) of a neutral atom is introduced. A distribution
of IQM reflects a specific spatial ordering in the condensate and produces
a distribution of the electric potential. Small vibrations of the Bose gas are
considered and a correction to the Bogoliubov spectrum of elementary excitations
in the degenerated Bose gas is obtained. An additional term in
the Gross-Pitaevskii equation which is responsible for such a correction is
found and a new type of the nonlinear Schrodinger equation (NSE) is constructed. Since the Bose condensate is akin to the superfluid component
in He II, a manifestation of its electrical activity could have a relation to the
electrical activity of the superfluid liquid observed experimentally. |
| format |
Article |
| author |
Kosevich, A.M. |
| spellingShingle |
Kosevich, A.M. Excitation spectrum and electrical properties of the condensate of Bose atoms Condensed Matter Physics |
| author_facet |
Kosevich, A.M. |
| author_sort |
Kosevich, A.M. |
| title |
Excitation spectrum and electrical properties of the condensate of Bose atoms |
| title_short |
Excitation spectrum and electrical properties of the condensate of Bose atoms |
| title_full |
Excitation spectrum and electrical properties of the condensate of Bose atoms |
| title_fullStr |
Excitation spectrum and electrical properties of the condensate of Bose atoms |
| title_full_unstemmed |
Excitation spectrum and electrical properties of the condensate of Bose atoms |
| title_sort |
excitation spectrum and electrical properties of the condensate of bose atoms |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2005 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/121051 |
| citation_txt |
Excitation spectrum and electrical properties of the condensate of Bose atoms / A.M. Kosevich // Condensed Matter Physics. — 2005. — Т. 8, № 4(44). — С. 773–778. — Бібліогр.: 6 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT kosevicham excitationspectrumandelectricalpropertiesofthecondensateofboseatoms |
| first_indexed |
2025-07-08T19:06:28Z |
| last_indexed |
2025-07-08T19:06:28Z |
| _version_ |
1837106824124301312 |
| fulltext |
Condensed Matter Physics, 2005, Vol. 8, No. 4(44), pp. 773–778
Excitation spectrum and electrical
properties of the condensate of Bose
atoms
A.M.Kosevich
B.Verkin Institute for Low Temperature Physics and Engineering
of National Academy of Sciences of Ukraine,
47, Lenin Avenue, Kharkiv, 61103 Ukraine
Received July 15, 2005, in final form October 24, 2005
It is shown that the condensate of a degenerated Bose gas consisting
of neutral atoms possesses electrical properties which differ from a triv-
ial polarization of the atoms in the electric field. A notion of an isotropic
quadrupole moment (IQM) of a neutral atom is introduced. A distribution
of IQM reflects a specific spatial ordering in the condensate and produces
a distribution of the electric potential. Small vibrations of the Bose gas are
considered and a correction to the Bogoliubov spectrum of elementary ex-
citations in the degenerated Bose gas is obtained. An additional term in
the Gross-Pitaevskii equation which is responsible for such a correction is
found and a new type of the nonlinear Schrödinger equation (NSE) is con-
structed. Since the Bose condensate is akin to the superfluid component
in He II, a manifestation of its electrical activity could have a relation to the
electrical activity of the superfluid liquid observed experimentally.
Key words: Bose condensate, Bogoliubov spectrum, electrical
polarization of He II
PACS: 47.27.Eq, 67.40.Db
1. Electrical activity of neutral atoms
The neutral He atom in the 1S0 ground state does not have an intrinsic ( in the
absence of electrical field ) dipole moment but it does have an important microscopic
electrical characteristic [1] – an isotropic quadrupole moment (IQM). The IQM is
defined by the following formula
qik =
∑
exixk, i, k = 1, 2, 3,
where the summation is over all electric charges in the system. In the case of the
He4 we have qik = q0δik and q0 = (1/3)qll = −2|e|a2 (e is the charge of the electron
and a is the Bohr radius). From the standpoint of macroscopic physics this is a
c© A.M.Kosevich 773
A.M.Kosevich
“latent” atomic characteristic, since in macroscopic interactions of electrical systems
the quantity qik − (1/3)q0δik goes to zero in the given case.
The IQM produces an electric potential inside the atom. To calculate the poten-
tial we use the simplest classical model of an atom, supposing all the electrons distri-
buted homogeneously over the surface of a sphere with radius a. If the total charge
of the atomic nucleus is Z|e|, the electrostatic potential averaged by volume inside
such a sphere-atom having the volume V0 = 4πa3/3 is equal to ϕ0 = 3Z|e|/((2a).
According to the model proposed, the atomic IQM is equal to q0 = −Z|e|a2.
Since the potential is concentrated inside the atoms, its spatial localized distri-
bution in a gas may be described in the long wave approximation by means of the
expression
ϕ(x) = ϕ0V0
∑
α
δ(x − xα) = −2πq0
∑
α
δ(x − xα), (1)
where xα is a coordinate of α-atom and the summation is extended over all the atoms
in the system. The long wave formula (1) was derived strongly in the paper [2].
Let us introduce a microscopic density of atoms
n(x) =
∑
α
δ(x − xα). (2)
As a result, we obtain a connection of the electric potential of a neutral gas ϕ(x)
with the density of the atomic IQMs Q:
ϕ(x) = −2πQ, Q = q0n(x). (3)
Equations (2), (3) make it possible to calculate the mean electric potential in
the condensate. Consider the Bose condensate either in the ground state or in a
slightly excited state. The atoms in the coherent condensate state form some ordered
structure with the wave function
Ψ(xα) = Ψ(x1,x2,x3, . . .). (4)
In order to describe the slightly excited states of a nearly ideal Bose gas we
suggest to use the approximation of a mean field type. Suppose all the atoms are
found in the same single-particle quantum state ψ0(xα):
Ψ(xα) =
∏
α
ψ0(xα). (5)
Then the mean density of the gas has got a very simple form
〈n(x)〉 =
∑
α
〈Ψ|δ(x − xα)|Ψ〉 =
∑
α
∫
|Ψ|2δ(x − xα)
∏
β
dVβ
=
∑
α
|ψ0(x)|2 = N0|ψ0(x)|2,
where N0 is the number of atoms in the condensate. Of course the function ψ0(x) is
normalized to unity.
774
Excitation spectrum of the condensate of Bose atoms
The problem of describing the electrical properties of the condensate comes to a
problem of searching the function ψ0(x). However in the mean field approximation,
the function ψ0(x) can be taken in the form of a solution of the Gross-Pitaevskii
equation [3]:
ih̄
∂ψ0
∂t
= − h̄2
2m
∆ψ0 + U0
(
|ψ0|2 − n0
)
ψ0 , (6)
where n0 = N0/V is the equilibrium density of the condensate, and a solution of
equation (6) should be normalized to the number of atoms N0.
Slightly excited states of the condensate are equivalent to small vibrations and
the wave function of such vibrations can be written as follows:
ψ(x, t) =
√
n
0
(1 + θ(x, t)) , |θ(x, t)| � 1 . (7)
Linearization of equation (6) with respect to θ(x, t) leads to the following equation
ih̄
∂θ
∂t
= − h̄2
2m
∆θ + U0n0(θ + θ∗). (8)
Equation (8) comes to two partial differential equations with the time derivatives of
the second order with respect to the sum θ + θ∗ and the difference θ− θ∗ . We take
the solution
θ + θ∗ = θ0 cos(kx − ωt), (9)
where ω and k are the frequency and wave vector connected by means of the formula
for Bogoliubov’s spectrum
ω2 = k2
U0n0
m
+
(
h̄k
2m
)2
. (10)
Equation (9) makes it possible to calculate the small oscillations if the gas den-
sity (6):
δn = n0θ0 cos(kx − ωt). (11)
The oscillations of the density produce oscillations of the electric potential (3) in
the neutral Bose gas. This effect can have a relation to the oscillations of the electric
potential produced by the vibrations of the second sound wave in a superfluid liquid
observed by Rybalko [4]. Really, eigen solutions of equation (8) describe elementary
excitations in the condensate and δn can be treated as a density of elementary
excitations above the ground state. In the theory of superfluidity [3] the vibrations
of the density of elementary excitations are associated with temperature oscillations.
In its turn a nondissipative wave of the temperature oscillations is an analog of the
second sound wave in the helium II.
2. Correction to Bogoliubov’s spectrum
The electrical activity of the Bose condensate should be taken into consideration
in solving the problem of its small vibrations. We restrict ourselves to the mean field
approximation and long wave Gross-Pitaevskii equation.
775
A.M.Kosevich
In the long wave approximation, the interaction of an atom in the point x1 with
an external field ϕ(x) is as follows:
δUint =
1
2
q∆ϕ(x1), (12)
where q is the atomic IQM. If the field ϕ(x) is created by an atom in the point x2,
we can use the expression (1) for the potential produced by the second atom. Then
in the long wave approximation the energy of the pair interaction of the atoms under
consideration can be written as
U(x1 − x2) = −2πq2∆δ(x1 − x2). (13)
The Fourier component of this interaction energy is equal to
Uk = πq2k2. (14)
However, equation (13) for the pair interaction energy does not include all pos-
sible contact interactions of two atoms. In particular, it does not take into consid-
eration two important facts. First of all, a finite size of the atomic radius a should
be taken into account [5]. And secondly, a deformation of the electron distributions
inside the atoms during their contact was excluded from the calculation of equati-
on (12). These facts give an additional contribution to the Fourier component (14)
and can change both the magnitude and the sign of Uk. Consequently, the total pair
energy of the atomic interaction in the long wave approximation should be written
as follows:
Uk = U0 + U1(ak)
2 , (15)
where the parameter U1 has an order of the magnitude of U0.
A new form of the second term in equation (15) changes the interaction energy
in the coordinate representation
δU(x1 − x2) = −U1∆δ(x1 − x2). (16)
The additional term in equation (15) which is proportional to k2 also leads to
the change of Bogoliubov ’s spectrum. Now small vibrations of the condensate have
the following frequencies squared
(h̄ω)2 = ε2(p) + (h̄Ω(p))2 , (17)
where ε(p) is given by equation (10) and the frequency Ω plays the role of an ion
plasma frequency:
Ω2(p) =
U1n0
m
(
p
h̄
)4
≡ U1n0
m
k4. (18)
Taking into account the electrical activity of the Bose condensate, one does not
change the low frequency spectrum of the sound vibrations. We see only a slight
renormalization of the mass of elementary excitations at high frequencies.
776
Excitation spectrum of the condensate of Bose atoms
In its turn the expression (16) enables us to determine an additional term in the
Gross-Pitaevskii equation associated with the electrical activity of the condensate.
Considering (6) as an equation derived in the mean field approximation, we can
include the term (16) averaged over the state ψ0 into the effective Hamiltonian.
Since
〈Ψ0(x1)|δU(x − x1)|Ψ0(x1)〉 = −U1∆|Ψ0(x)|2, (19)
we proposed [6] a little bit changed form of equation (6):
ih̄
∂Ψ0
∂t
= − h̄2
2m
∆Ψ0 + U0
(
|Ψ0|2 − n0
)
Ψ0 − U1Ψ0∆|Ψ0|2. (20)
Thus, taking the electrical properties of the Bose condensate into account we come
to a new type of the nonlinear Schröeding equation (NSE) describing the dynamics
of the Bose condensate.
3. Remarks
Return to equations (1)–(3):
ϕ(x) = −2πq n(x), n(x) =
∑
α
δ(x − xα). (21)
This expression has a relation for any gas and for any condensed matter.
Using statistical-thermodynamical averaging of equation (21) one can write
〈〈ϕ〉〉 = −2π〈〈q n〉〉. (22)
In the case of a gas the parameter q is a characteristic of a single atom and does
not depend on the states of other atoms of the gas: 〈〈q n〉〉 = q〈〈n〉〉. In the case
of a condensed matter, a distribution of the electron charge in any atom depends
on the neighbouring atoms and their states, and a calculation of the parameter q
is a special problem of the quantum mechanics of many particles. As a result the
mean electric potential (22) depends not only on the atom radius a but on the mean
interatomic distance in the matter l as well.
777
A.M.Kosevich
References
1. Kosevich A.M., On the description of electrical effects in the two-fluid model of super-
fluidity. FNT, 2005, 31, No. 1, 50–54 (in Russian); Low Temp. Phys, 2005, 31, No. 1,
37–40.
2. Kosevich A.M., Dynamic electrostricshion of Bose-condensate and a system of neutral
atoms. FNT, 2005, 31, No. 10, 1100–1102 (in Russian).
3. Lifshits E.M., Pitaevskii L.P. Statistical Physics, part 2. Nauka, Moscow, 1978, 620 p.
(in Russian).
4. Rybalko A.S., Observation of electrical induction in a wave of the second sound in
He II. FNT, 2004, 30, No. 11, 1321–1325 (in Russian); Low Temp. Phys, 2004, 30,
No. 11, 994–998.
5. Brueckner K.A., Savada K., Bose-Einstein gas with repulsive interation: hard spheres
at high density. Phys. Rev., 1957, 106, No. 6, 1128–1135.
6. Kosevich A.M., Electrical properties of a condensate of Bose-atoms. Ukr. Fiz. Zhurnal,
2005, 50, No. 8A, A130–A134 (in Russian).
Спектр збуджень та електричні властивості
конденсату Бозе-атомiв
А.М.Косевіч
Фiзико-технiчний iнститут низьких температур iм.Б.Веркiна,
просп. Ленiна, 47, Харкiв 61103
Отримано 15 липня 2005 р., в остаточному вигляді –
24 жовтня 2005 р.
Показано, що конденсат виродженого Бозе-газу нейтральних ато-
мів володіє електричними властивостями, які відрізняються від
тривіальної поляризації атомів у електричному полі. Вводиться
поняття ізотропного квадрупольного моменту (ІКМ) нейтрального
атома. Розподіл ІКМ відображає особливе просторове впорядкуван-
ня в конденсаті і генерує розподіл електричного потенціалу. Роз-
глядаються малі коливання Бозе-газу і отримується поправка до
Боголюбівського спектру елементарних збуджень у виродженому
Бозе-газі. Знайдено додатковий член у рівнянні Гроса-Пітаєвского,
який відповідає за таку поправку та побудовано новий тип неліній-
ного рівняння Шредіннера. Оскільки Бозе-конденсат є споріднений
з надплинною компонентою в He II, то прояв його електричної
поведінки може мати відношення до експериментально спостере-
жуваної електричної поведінки надплинної рідини
Ключові слова: Бозе-конденсат, Боголюбівський спектр,
електрична поляризацiя He II
PACS: 47.27.Eq, 67.40.Db
778
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