Green’s function method to the ground state properties of a two-component Bose–Einstein condensate
The elementary excitation spectrum of a two-component Bose–Einstein condensate is obtained by Green’s function method. It is found to have two branches. In the long-wave limit, the two branches of the excitation spectrum are reduced to one phonon excitation and one single-particle excitation. With t...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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nasplib_isofts_kiev_ua-123456789-1186182025-02-23T17:57:51Z Green’s function method to the ground state properties of a two-component Bose–Einstein condensate Liang, C. Wei, K. Ye, B.J. Wen, H.M. Zhou, X.Y. Han, R.D. Бозе-эйнштейновская конденсация The elementary excitation spectrum of a two-component Bose–Einstein condensate is obtained by Green’s function method. It is found to have two branches. In the long-wave limit, the two branches of the excitation spectrum are reduced to one phonon excitation and one single-particle excitation. With the obtained excitation spectrum and the Green’s functions, the depletion of the condensate and the ground state energy have also been calculated in this paper. This work has been supported by the NSF-China under grants Nos. 10974189, 10675114 and 10675115. 2011 Article Green’s function method to the ground state properties of a two-component Bose–Einstein condensate / C. Liang, K. Wei, B.J. Ye, H.M. Wen, X.Y. Zhou, R.D. Han // Физика низких температур. — 2011. — Т. 37, № 7. — С. 708–714. — Бібліогр.: 20 назв. — англ. 0132-6414 PACS: 03.75.Nt https://nasplib.isofts.kiev.ua/handle/123456789/118618 en Физика низких температур application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Бозе-эйнштейновская конденсация Бозе-эйнштейновская конденсация |
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Бозе-эйнштейновская конденсация Бозе-эйнштейновская конденсация Liang, C. Wei, K. Ye, B.J. Wen, H.M. Zhou, X.Y. Han, R.D. Green’s function method to the ground state properties of a two-component Bose–Einstein condensate Физика низких температур |
| description |
The elementary excitation spectrum of a two-component Bose–Einstein condensate is obtained by Green’s function method. It is found to have two branches. In the long-wave limit, the two branches of the excitation spectrum are reduced to one phonon excitation and one single-particle excitation. With the obtained excitation spectrum and the Green’s functions, the depletion of the condensate and the ground state energy have also been calculated in this paper. |
| format |
Article |
| author |
Liang, C. Wei, K. Ye, B.J. Wen, H.M. Zhou, X.Y. Han, R.D. |
| author_facet |
Liang, C. Wei, K. Ye, B.J. Wen, H.M. Zhou, X.Y. Han, R.D. |
| author_sort |
Liang, C. |
| title |
Green’s function method to the ground state properties of a two-component Bose–Einstein condensate |
| title_short |
Green’s function method to the ground state properties of a two-component Bose–Einstein condensate |
| title_full |
Green’s function method to the ground state properties of a two-component Bose–Einstein condensate |
| title_fullStr |
Green’s function method to the ground state properties of a two-component Bose–Einstein condensate |
| title_full_unstemmed |
Green’s function method to the ground state properties of a two-component Bose–Einstein condensate |
| title_sort |
green’s function method to the ground state properties of a two-component bose–einstein condensate |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2011 |
| topic_facet |
Бозе-эйнштейновская конденсация |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/118618 |
| citation_txt |
Green’s function method to the ground state properties of a two-component Bose–Einstein condensate / C. Liang, K. Wei, B.J. Ye, H.M. Wen, X.Y. Zhou, R.D. Han // Физика низких температур. — 2011. — Т. 37, № 7. — С. 708–714. — Бібліогр.: 20 назв. — англ. |
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Физика низких температур |
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2025-11-24T04:20:04Z |
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| fulltext |
© Chen Liang, Kong Wei, B.J. Ye, H.M. Wen, X.Y. Zhou, and R.D. Han, 2011
Fizika Nizkikh Temperatur, 2011, v. 37, No. 7, p. 708–714
Green’s function method to the ground state properties of
a two-component Bose–Einstein condensate
Chen Liang, Kong Wei, B.J. Ye, H.M. Wen, X.Y. Zhou, and R.D. Han
Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China
E-mail: phychl@mail.ustc.edu.cn
Received November 25, 2010
The elementary excitation spectrum of a two-component Bose–Einstein condensate is obtained by Green’s
function method. It is found to have two branches. In the long-wave limit, the two branches of the excitation
spectrum are reduced to one phonon excitation and one single-particle excitation. With the obtained excitation
spectrum and the Green’s functions, the depletion of the condensate and the ground state energy have also been
calculated in this paper.
PACS: 03.75.Nt Other Bose–Einstein condensation phenomena.
Keywords: Green’s function, elementary excitation, Bose–Einstein condensate.
1. Introduction
The realization of Bose–Einstein condensation (BEC)
[1,2] has attracted much interest in the past years, because
it provides the unique opportunities for exploring quantum
phenomena on a macroscopic scale. The Bose–Einstein
condensation for noninteracting particles is characterized
as the macroscopic occupation number for one of the sin-
gle-particle energy levels. For interacting systems, the cri-
terion for BEC is generalized by Penrose and Onsager and
by Yang as off-diagonal long range order [3,4]. The experi-
mentally realization of BEC is in dilute atomic gases, in
which mean field theory is well applied in the nearly zero
temperature [5,6]. The condition for diluteness is 3 1,sna <<
where n is the density of the gas and sa is the s-wave scat-
tering length. The interaction between atoms is characte-
rized by the s-wave scattering length ,sa which can be
manipulated by the use of lasers and magnetic fields. The
transition between repulsive and attractive interaction can
be controlled by a Feshbach resonance. The properties of a
gas in a trap are usually studied by Thomas–Fermi approx-
imation. When the length scale of the trap is much greater
than the coherence length ,ξ the gas is assumed to be ho-
mogeneous.
The subject of two-component Bose–Einstein conden-
sate has attracted many experimental and theoretical stu-
dies [7–15]. Much work has been devoted to the case of
double-well trapping [9–13], which is similar to the Jo-
sephson junctions of superconductors. The recent experi-
ments have directly observed the phenomena of plasmon
oscillation and macroscopic quantum self-trapping
(MQST) [12,13], which have been theoretically predicted
earlier [9–11]. Another possible case is the mixtures of the
same isotope, but in different internal spin states, such as
87Rb [7,8]. In this case, atoms can undergo transitions be-
tween hyperfine states by an external field, which corres-
ponds to the tunneling effects in the double-well case. The
two-body interaction between the atoms of the same state
may be different from the interaction of the different
hyperfine states. The dynamics of this case is similar to the
double-well condensate [11]. The thermal effects can act as
the damping term and the system under damping will
evolve into a stationary state of two equivalent components
[10]. The π phase difference of the two components corres-
ponds to the energy minimum in the mean field theory in
the zero temperature [14].
In this paper we will study the ground state properties
of a two-component BEC. The concept of elementary exci-
tations is important for the ground state BEC, and it can be
studied by several ways. The excitation spectrum can be
achieved by linearizing the hydrodynamic equations de-
rived from the Gross–Pitaevskii equation [6,15]. However
for a quantum Bose gas, the excitation spectrum was first
obtained by Bogoliubov by a special transformation [16],
which has been well extended for many quantum theories.
It is well known that the method of Green’s function can
be applied to find the elementary excitations in many fields
of condensed matter physics [17]. In this paper, we will
extend the method developed by Beliaev [18] to the Bose–
Einstein condensate of two equivalent components. In the
mean field approximation, the elementary excitation spec-
trum is found to have two branches. In the long-wave limit,
the two branches of the excitation spectrum are reduced to
Green’s function method to the ground state properties of a two-component Bose–Einstein condensate
Fizika Nizkikh Temperatur, 2011, v. 37, No. 7 709
one phonon excitation and one single-particle excitation.
By use of the obtained excitation spectrum and the Green’s
functions, we have also calculated the depletion and the
ground state energy of the condensate.
In Sec. 2, we introduce the Green’s method for a homo-
geneous Bose–Einstein condensate in the mean field ap-
proximation. In Sec. 3, we apply the method to find the
elementary excitation spectrum for a two-component BEC.
The depletion of the condensate and the ground state ener-
gy have also been calculated in this section. In Sec. 4, we
make a conclusion of the paper.
2. Green’s function method to Bose–Einstein condensate
The ground-canonical Hamiltonian for a homogeneous
Bose gas in a volume V is often written as [17]
1 21 2
1 2
2
0 ,
2 2p p k kk q k q
p k k q
UpK a a a a a a
m V
+ + +
+ −
⎛ ⎞
= −μ +⎜ ⎟⎜ ⎟
⎝ ⎠
∑ ∑ (1)
where 0U characterize the two-body interaction,
0 0U nμ = is the chemical potential used to keep the con-
servation of the total number of particles, and pa+ and pa
are the Bose creation and annihilation operators in the
momentum representation. In zero temperature the conden-
sate is well described by a field. For simplicity, the phase
of the condensate can be assumed to be zero [17], and we
can get the useful average 0 0 0 .a a N+< > = < > =
In order to get the excitation spectrum, we introduce two
single-particle Green’s functions in zero temperature [17]:
( , ) ( ) ( ) ,p pG p t t i Ta t a t+′ ′− = − < > (2)
( , ) ( ) ( ) .p pF p t t i Ta t a t+ +
−′ ′− = − < > (3)
Where the T denotes the chronological product and the
operators are in the Heisenberg picture. The first one is the
normal Green’s function and the second is abnormal. We
will try to derive their dynamical equations. For example:
( , ) ( ) [ , ] ( )p pi G p t t t t i T a K a t
t
+∂ ′ ′ ′− = δ − − < > =
∂
2
( ) ( ) ( )
2 p p
pt t i T a t a t
m
+⎛ ⎞
′ ′= δ − − < −μ > −⎜ ⎟⎜ ⎟
⎝ ⎠
0 ( ) ( ) ( ) ( ) .k q k p q p
kq
U
i Ta t a t a t a t
V
+ +
+ + ′− < >∑ (4)
The brackets of four operators in the interaction term
must be reduced to products of pair operators by Wick’s
theorem [17]:
____________________________________________________
0 0( ) ( ) ( ) ( ) 2 ( ) ( ) ( ) ( )k q k p q p k k p p
kq k
U U
i Ta t a t a t a t i a t a t Ta t a t
V
+ + + +
+ + ′ ′− < >= − < >< > −
Ω ∑ ∑
0 0
0 10 10( ) ( ) ( ) ( ) 2 ( , ) ( ) ( ) ( , )p q p q p p
q
U U
i a t a t Ta t a t nU G p t t a t a t F p t t
V V
+ + + +
− − + − ′ ′ ′− < >< > ≈ − + < > − =∑
0 02 ( , ) ( , ).nU G p t t nU F p t t′ ′= − + − (5)
_______________________________________________
To deduce (5), we have made mean field approximation
and the condensate density 0n is replaced by n. If we make
Fourier transformation of (5) into energy representation,
we can get
2
0 0( , ) 1 2 ( , ) ( , ).
2
pG p nU G p nU F p
m
⎡ ⎤⎛ ⎞
ω ω = + −μ + ω + ω⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
(6)
Similarly, we can get another equation for the abnormal
Green’s function as
2
0 0( , ) 2 ( , ) ( , ).
2
pF p nU F p nU G p
m
⎡ ⎤⎛ ⎞
ω ω = − −μ + ω − ω⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
(7)
From (6) and (7), we can get the two Green’s functions as
2
0
2 2
2( , ) ,
0
p nU
mG p
i
ω+ +
ω =
ω − ε +
(8)
0
2 2( , ) ,
0
nU
F p
i
ω = −
ω − ε +
(9)
where
2 2
0( ) 2 .
2 2
p pp nU
m m
⎛ ⎞
ε = +⎜ ⎟⎜ ⎟
⎝ ⎠
(10)
The result (10) is the well-known elementary excitation
spectrum. In the long wave-length limit 0,p → the excita-
tion (10) can be reduced to the phonon form ,spε = where
0 /s nU m= is the sound speed.
From the obtained Green’s function (8), we can get the
density of the noncondensate atoms as [17,19]
3
ex 30
lim e ( , )
2(2 )
i t
t
d p in d G p− ω
→−
= ω ω =
ππ∫ ∫
3/2
02 3
1 ( ) .
3
mnU=
π
(11)
Chen Liang, Kong Wei, B.J. Ye, H.M. Wen, X.Y. Zhou, and R.D. Han
710 Fizika Nizkikh Temperatur, 2011, v. 37, No. 7
And the energy density of the condensate can be calcu-
lated as
2
0 0 2
0
1 1 11
2 p
E U n mU
V V p≠
⎛ ⎞
⎜ ⎟= + +
⎜ ⎟
⎝ ⎠
∑
3
30
lim e ( ) ( , )
2(2 )
i t
t
d p id p G p− ω
→−
+ ω ε ω =
ππ∫ ∫
2 3 2
5 20
02 3
8 ( ) .
2 15
U n m U n= +
π
(12)
To deduce the result (12), we have replace the bare coupl-
ing constant
2
0
4 aU
m
π
= by
2 2
0 2
0
4 4 11
p
a aU
m V p≠
⎛ ⎞π π⎜ ⎟= +
⎜ ⎟
⎝ ⎠
∑
[17,20], where a is the scattering length. The first part of
(12) is the interaction energy of the condensate, and the
second part comes from the contributions of the noncon-
densed atoms. The result (12) was first obtained by Lee
and Yang [20].
The Green’s function method used here was first intro-
duced by Beliaev [18], which is base on the mean-field
approximation. The excitation spectrum can also be ob-
tained by the method of Bogoliubov transformation. For a
homogeneous BEC, the Hamiltonian should be simplified
before the transformation [16,17]:
2 2
0
2 2 p p
p
N U pH a a
V m
+= + +∑
0
0
( 2 ).
2 p p p p p p
p
NU
a a a a a a
V
+ + +
− −
≠
+ + +∑ (13)
The above Hamiltonian (13) is deduced by mean field ap-
proximation and the fact that the fluctuation in the particle
number is small. This was first done by Bogoliubov [16],
and the method of Bogoliubov transformation is to con-
struct new Bose operators to make the simplified Hamilto-
nian (13) into diagonal form. However, if we begin from
the above simplified Hamiltonian, we will be easier to get
the dynamical equations of the Green’s functions. The ex-
citation spectrums obtained by the two methods are com-
pletely same in the form [17]. The Bogoliubov transforma-
tion method and the Green’s function method in this paper
are all based on mean field approximation, which is the
first order approximation to the many-body theory [17,19].
3. Excitation spectrum and the ground state of a two-
component BEC
The Hamiltonian for a homogeneous two-component
Bose gas in a volumeV can be written in the form [14]
2
1 2 2 1
,
( )
2 ip ip p p p p
i p p
pK a a a a a a
m
+ + +⎛ ⎞
= −μ + η + +⎜ ⎟⎜ ⎟
⎝ ⎠
∑ ∑
1 21 2
1 2,
1
2 s ik ikik q ik q
i k k q
U a a a a
V
+ +
+ −+ +∑
1 21 2
1 2
1 21 2
1 ,x k kk q k q
k k q
U a a a a
V
+ +
+ −+ ∑ (14)
where η is a coupling parameter which shows the transi-
tion between the two kinds of particles, sU and xU cha-
racterize the two-body interaction of the same kinds of
particles and two different kinds of particles, μ is the
chemical potential used to keep the conservation of the
total number of particles, and ipa+ and ipa are the Bose
creation and annihilation operators in the momentum re-
presentation.
Bose–Einstein condensation occurs in a state of zero
momentum and the condensate wave function is
0 0
1 e ,ii
i i ia n
V
θΨ = < > =
where iθ is the phase of the condensate and the angle
brackets denote averaging with respect to the ground state.
When the depletion is small in nearly zero temperature, the
condensate density 0in can be replaced by the density .in
As in Ref. 14, the phase difference of π corresponds to the
minimum of the ground state energy of the system. In this
paper, we will consider the case of the phase difference
being π . In zero temperature, the ground state energy den-
sity for the homogeneous two-component condensate with
volumeV is given as
20 1 1( ) .
4 2s x
E
U U n n
V
= + − η (15)
In accordance with [14], the chemical potential can be
found as
( ).
2 2 s x
n U Uη
μ = − + + (16)
For simplicity, we assume that the phase of the first kind of
condensate is zero and the second kind is .π On this as-
sumption, we can get
10 20 ,
2
na a< > = − 10 20 .
2
na a+< > = − (17)
The Hamiltonian (14) is more complicated than the con-
densate of only one component. In order to get the excita-
tion spectrum, we introduce four single-particle Green’s
functions in zero temperature:
1 1( , ) ( ) ( ) ,p pG p t t i Ta t a t+′ ′− = − < > (18)
1 1( , ) ( ) ( ) ,p pF p t t i Ta t a t+ +
−′ ′− = − < > (19)
Green’s function method to the ground state properties of a two-component Bose–Einstein condensate
Fizika Nizkikh Temperatur, 2011, v. 37, No. 7 711
2 1( , ) ( ) ( ) ,p pH p t t i Ta t a t+′ ′− = − < > (20)
2 1( , ) ( ) ( ) ,p pL p t t i Ta t a t+ +
−′ ′− = − < > (21)
where theT denotes the chronological product and the op-
erators are in the Heisenberg representation. As the proce-
dure in last section, we will try to derive their dynamical
equations. For the normal Green’s function, we can get
1 1( , ) ( ) [ , ] ( )p pi G p t t t t i T a K a t
t
+∂ ′ ′ ′− = δ − − < > =
∂
2
1 2( ) ( ) ( )
2 2p p
pt t i T a t a t
m
⎧⎛ ⎞ η⎪′= δ − − < −μ + +⎜ ⎟⎨⎜ ⎟⎪⎝ ⎠⎩
1 1 1
1 ( ) ( ) ( )s k q k p q
kq
V a t a t a t
V
+
+ ++ +∑
2 2 1 1
1 ( ) ( ) ( ) ( ) .x k q k p q p
kq
V a t a t a t a t
V
+ +
+ +
⎫⎪ ′+ >⎬
⎪⎭
∑ (22)
The brackets of four operators in the interaction terms must
be reduced to products of pair operators by Wick’s theo-
rem [17,18]. Similar to the result (5), we will try to reduce
the following interaction term:
2 2 1 1
1 ( ) ( ) ( ) ( )x k q k p q p
kq
i U Ta t a t a t a t
V
+ +
+ + ′− < > =∑
2 2 1 1
1 ( ) ( ) ( ) ( )x k k p p
k
i U a t a t Ta t a t
V
+ + ′= − < >< > −∑
2 1 2 1
1 ( ) ( ) ( ) ( )x p q p q p p
q
i U a t a t Ta t a t
V
+ +
+ + ′− < >< > −∑
2 1 2 1
1 ( ) ( ) ( ) ( )x p q p q p p
q
i U a t a t Ta t a t
V
+ +
− − + − ′− < >< > ≈∑
2 20 10
1( , ) ( , )x xn U G p t t U a a H p t t
V
+′ ′≈ − + < > − +
20 10
1 ( , )xU a a L p t t
V
′+ < > − =
( , ) [ ( , ) ( , )].
2 2x x
n nU G p t t U H p t t L p t t′ ′ ′= − − − + − (23)
To deduce (23), we have made mean field approximation
and the condensate density 0in is replaced by .in We have
also made use of the assumption (17) and 1 2 / 2.n n n= =
All the interaction terms in (22) can be dealt with similarly.
If we make Fourier transformation of (22) into energy re-
presentation, we can get
2
( , ) 1 ( , )
2 2s x
p nG p nU U G p
m
⎡ ⎤⎛ ⎞
ω ω = + −μ + + ω +⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
( , ) ( , ) ( , ).
2 2 2 2s x x
n n nU F p U H p U L pη⎛ ⎞+ ω + − ω − ω⎜ ⎟
⎝ ⎠
(24)
As the same procedure, we can get three other equations:
2
( , ) ( , )
2 2s x
p nF p nU U F p
m
⎡ ⎤⎛ ⎞
ω ω = − −μ + + ω −⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
( , ) ( , ) ( , ),
2 2 2 2s x x
n n nU G p U H p U L pη⎛ ⎞− ω + ω − − ω⎜ ⎟
⎝ ⎠
(25)
2
( , ) ( , )
2 2s x
p nH p nU U H p
m
⎡ ⎤⎛ ⎞
ω ω = −μ + + ω +⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
( , ) ( , ) ( , ),
2 2 2 2s x x
n n nU L p U F p U G pη⎛ ⎞+ ω − ω + − ω⎜ ⎟
⎝ ⎠
(26)
2
( , ) ( , )
2 2s x
p nL p nU U L p
m
⎡ ⎤⎛ ⎞
ω ω = − −μ + + ω −⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
( , ) ( , ) ( , ).
2 2 2 2s x x
n n nU H p U G p U F pη⎛ ⎞− ω + ω − − ω⎜ ⎟
⎝ ⎠
(27)
By the use of the chemical potential (16), we can solve
the four algebra equations (24), (25), (26) and (27) with the
aid of Mathematica. And the solutions are given as
____________________________________________________
2 2 2 22
2 1 2 1 2
2 2 2 2
1 2
2 2 2 2 2 2 2
( , ) ,
( )( )
s x
p n nU U
m
G p
⎛ ⎞⎛ ⎞ ε + ε ε − εη η⎛ ⎞ω+ + + ω − − −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠⎝ ⎠ω =
ω − ε ω − ε
(28)
2 2 2 2
2 1 2 1 2
2 2 2 2
1 2
2 2 2 2
( , ) ,
( )( )
s x
n nU U
F p
⎛ ⎞ε + ε ε − ε
− ω − −⎜ ⎟⎜ ⎟
⎝ ⎠ω =
ω − ε ω − ε
(29)
Chen Liang, Kong Wei, B.J. Ye, H.M. Wen, X.Y. Zhou, and R.D. Han
712 Fizika Nizkikh Temperatur, 2011, v. 37, No. 7
Fig. 1. Excitation energy as a function of the wave number k. The
solid line shows the excitation energy 1ε , and the dash line re-
sembles the branch 2ε with the parameters: n = 4⋅1013 atoms/cm3,
0120 ,sa a= 080xa a= and ( ) / 2s xn U Uη = − .
1 2 3 4
0
10
20
30
40
50
60
70
80
k, nm
–1
�
,
n
K
2 2 2 22
2 1 2 1 2
2 2 2 2
1 2
2 2 2 2 2 2 2
( , ) ,
( )( )
x s
n p nU U
m
H p
⎛ ⎞ ⎛ ⎞ε + ε ε − εη η⎛ ⎞− ω − − ω+ + +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ω =
ω − ε ω − ε
(30)
2 2 2 2
2 1 2 1 2
2 2 2 2
1 2
2 2 2 2
( , ) ,
( )( )
x s
n nU U
L p
⎛ ⎞ε + ε ε − ε
ω − +⎜ ⎟⎜ ⎟
⎝ ⎠ω =
ω − ε ω − ε
(31)
_______________________________________________
where
2 2
1( ) ( ) ,
2 2 s x
p pp n U U
m m
⎡ ⎤
ε = + +⎢ ⎥
⎢ ⎥⎣ ⎦
(32)
2 2
2 ( ) ( ) .
2 2 s x
p pp n U U
m m
⎛ ⎞ ⎡ ⎤
ε = + η +η+ −⎜ ⎟ ⎢ ⎥⎜ ⎟ ⎢ ⎥⎝ ⎠ ⎣ ⎦
(33)
From the form of the denominator of the Green’s func-
tions, it is clear that the energy spectrum of the elementary
excitations has two branches 1( )pε and 2 ( ).pε It is possi-
ble for 2 ( )pε to be imaginary, and this situation corres-
ponds to the instabilities as indicated in Refs. 14, 15.
In the experiments of the cold atomic gas 87Rb [5], the
scattering length is often in the range 0 Rb 085 140 ,a a a< <
where 0 0,5292Åa = is the Bohr radius. The density is
12 1410 –10n ≈ atoms/cm3, which can meet the condition of
diluteness 3 0.na << And the sound speed 0 /s nU m=
is always in the order of 1 mm/s. In Fig. 1 we display
the two branches of the excitation spectrum (32) and
(33) with the parameters: the density of the 87Rb gas
n = 4⋅1013 atoms/cm3, the two scattering length
0120sa a= and 080xa a= ( 0a is the Bohr radius), and
( ) / 2.s xn U Uη = − From the selected parameters, we can
obtain the sound speed ( ) / 2s xs n U U m= + of the first
branch 1( )pε to be 1.5 mm/s. It is clear that the two
branches of excitation spectrum are of a single-particle
excitation and a phonon one.
The normal Green’s function (28) can be rewritten as
2 2
2 2 2 2
1 2
( ) ( )
2 2 2 2( , ) .
2( ) 0 2( ) 0
s x s x
p n p nU U U U
m mG p
i i
ω+ + + ω+ +η+ −
ω = +
ω − ε + ω − ε +
(34)
Based on the results of excitation spectrum (32) and
(33), we can make some discussions about the ground state
properties of the system. In the long-wavelength limit and
0 , ( ),s xn U U<< η − (32) and (33) can be reduced as
1
( )
( ) ,
2
s xn U U
p p
m
+
ε = (35)
2
2 *
( ) [ ( )] ,
2
s x
pp n U U
m
ε = η η+ − + (36)
where the effective mass *m is given by
* 2 [ ( )]
2 ( )
s x
s x
n U U
m m
n U U
η η+ −
=
η+ −
. (37)
It is clear that 1( )pε is the phonon excitations with the
sound speed ( ) / 2s xn U U m+ and 2 ( )pε corresponds to
the single-particle excitations with an effective mass and a
shift in the energy.
In the following, we will make some discussions for
two special cases:
1. If s xU U= , the excitation spectrum will be simp-
lified as
2 2
1( ) 2 ,
2 2 s
p pp nU
m m
⎛ ⎞
ε = +⎜ ⎟⎜ ⎟
⎝ ⎠
(38)
2
2 ( )
2
pp
m
ε = + η , (39)
and the Green’s function (34) can be reduced to
Green’s function method to the ground state properties of a two-component Bose–Einstein condensate
Fizika Nizkikh Temperatur, 2011, v. 37, No. 7 713
2
2 2
21
12( , )
2( ) 02( ) 0
s
p nU
mG p
ii
ω+ +
ω = +
ω− ε +ω − ε +
. (40)
From the form of (39) and (40), we can see that the second
spectrum 2 ( )pε acts only as the transition state such as 1pa
to 2 ,pa+ and there is no particles really occupying on this
spectrum. From the Green’s function (40), we can get the
depletion from the condensate of the first kind of particles:
3
1ex 30
lim e ( , )
2(2 )
i t
t
d p in d G p− ω
→−
= ω ω =
ππ∫ ∫
3/2
2 3
1 1 ( ) .
2 3
smnU=
π
(41)
The two components are equivalent and the results can be
applied to the second kind of particles. Similar to the result
(12), we can get the energy of the gases:
2 3 2
5 2
2 3
1 8 ( ) .
2 2 15
s
s
U nE mn U n
V
= − η +
π
(42)
The results (41) and (42) are completely same to (11) and
(12) as the case of one-component condensate. The phase
difference makes no effect in this case. It is well known
that the mean field theory is well applied for the cold dilute
atomic gases [5,6]. In most experiments, the depletion of
the ground state is of the order of one percent. And the
energy from the contribution of the noncondensate atoms
is only a small part of the whole energy. However, the
mean field theory is not suitable to liquid 4He for its strong
interaction and high density [6].
2. If 0,η = the excitation spectrum will be changed as
2 2
1 ( ) ,
2 2 s x
p p n U U
m m
⎡ ⎤
ε = + +⎢ ⎥
⎢ ⎥⎣ ⎦
(43)
2 2
2 ( ) .
2 2 s x
p p n U U
m m
⎡ ⎤
ε = + −⎢ ⎥
⎢ ⎥⎣ ⎦
(44)
In this case the spectrum can be reduced to two phonon
branches as 1 1s pε = and 2 2 ,s pε = where the two sound
speeds are 1 ( ) / 2s xs n U U m= + and
2 ( ) / 2 ,s xs n U U m= − respectively. The two phonon
speeds 1s and 2s can be different much from each other.
The normal Green’s function (34) can be reduced to
2 2
2 2 2 2
1 2
( ) ( )
2 2 2 2( , ) .
2( ) 0 2( ) 0
s x s x
p n p nU U U U
m mG p
i i
ω+ + + ω+ + −
ω = +
ω − ε + ω − ε +
(45)
Similarly we can get the depletion from the condensate of
the first kind of particles:
3 3
2 2
1ex 2 3 2 3
1 1 1 1 .
2 2 2 23 3
s x s xU U U U
n mn mn
+ −⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠π π
(46)
The energy of the condensate can be calculated as
3 2
2
2 3
1 8( )
4 15s x
E mU U n
V
= + + ×
π
5 2 5 2
.
2 2
s x s xU U U U
n n
⎡ ⎤+ −⎛ ⎞ ⎛ ⎞⎢ ⎥× +⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦
(47)
Comparing the results (41) and (46), we can see that the
density of excitation 1exn is reduced for the different inte-
ractions. The condition 0xU = corresponds to the mini-
mum of the excitation density (46), which is 2 / 2 times
the one of the case .s xU U= Comparing (47) with (42), the
part of the ground state energy from the contribution of the
noncondensed atoms is reduced correspondingly.
4. Conclusion
In this paper, we extend the Green’s function method to
the equivalent two-component Bose–Einstein condensate.
The elementary excitation spectrum is found to have two
branches. On the condition of strong coupling, the two
branches of the excitation spectrum are reduced to one
phonon excitation and one single-particle excitation in the
long wave-length limit. When the two different kinds of
interaction are equal, there is no particle really occupying
the branch of the single-particle excitation spectrum, which
acts only as a transition state between two different atoms.
The depletion of the condensate is same to the one of one-
component case. When the transition between the two dif-
ferent particles is forbidden, the excitation spectrum is re-
duced to two phonon forms in the long wave-length limit.
In this case the depletion of the condensate is reduced for
the two different kinds of interaction. With the obtained
excitation spectrum and the Green’s function, we have also
calculated the ground state energy in this paper.
Acknowledgments
This work has been supported by the NSF-China under
grants Nos. 10974189, 10675114 and 10675115.
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