Abstract. Based on the Cornwall-Jackiw-Tomboulis effective action approach, a
theoretical formalism is established to study the hydrodynamic instabilities in
two-component condensates of Bose gases. The effective potential is found in the
Hartree-Fock approximation and this quantity is then used to derive the expression for the
pressure, which depends on particle densities. Our numerical results show that instabilities
in our model are strongly influenced by the chemical potential.

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JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2015-0041
Mathematical and Physical Sci., 2015, Vol. 60, No. 7, pp. 121-128
This paper is available online at
HYDRODYNAMIC INSTABILITIES OF TWO-COMPONENT
BOSE-EINSTEIN CONDENSATES
Le Viet Hoa1, Nguyen Tuan Anh2, Nguyen Chinh Cuong1 and Dang Thi Minh Hue3
1Faculty of Physics, Hanoi National University of Education
2Faculty of Energy Technology, Electric Power University
3Water Resources University
Abstract. Based on the Cornwall-Jackiw-Tomboulis effective action approach, a
theoretical formalism is established to study the hydrodynamic instabilities in
two-component condensates of Bose gases. The effective potential is found in the
Hartree-Fock approximation and this quantity is then used to derive the expression for the
pressure, which depends on particle densities. Our numerical results show that instabilities
in our model are strongly influenced by the chemical potential.
Keywords: Hydrodynamic instabilities, two-component condensates, Bose gases.
1. Introduction
Theoretical studies of Bose-Einstein Condensates (BECs) [1, 2] and experimental
realizations of such systems [3, 4] have allowed us to explore many interesting physical
properties of BECs, including the superfluid dynamics of two-component BECs. In recent years
one study focused on considerations of hydrodynamic instabilities of two BECs, looking at
the Kelvin-Helmholtz instability, the Rayleigh-Taylor instability and the Richtmayer-Meshkov
instability [5, 6].
The multicomponent BEC is not a simple extension of the single component BEC. There
arise many novel phenomena such as the quantum tunneling of spin domain, vortex configuration,
phase segregation of the BEC mixture and so on [7, 8]. Moreover, it should be mentioned that
the in all experiments realizing BEC in dilute Bose gases, almost every parameter of the system
can be controlled. In connection with experimental efforts there has been theoretical progress in
describing different observed phenomena of multicomponent systems as well as testing various
models and methods, all of which are employed to consider properties of BECs.
In the present article, a theoretical formalism for studying BEC in the global U(1) × U(1)
model is formulated by means of the Cornwall-Jackiw-Tomboulis (CJT) effective action [9].
Received October 26, 2014. Accepted November 30, 2014.
Contact Le Viet Hoa, e-mail address: hoalv@hnue.edu.vn
121
Le Viet Hoa, Nguyen Tuan Anh, Nguyen Chinh Cuong and Dang Thi Minh Hue
2. Content
2.1. Effective potential in HF Approximation
Let us begin with idealized two component Bose gases given by the Lagrangian
£ = φ∗
(
−i ∂
∂t
− ∇
2
2mφ
)
φ+ ψ∗
(
−i ∂
∂t
− ∇
2
2mψ
)
ψ
− µ1φ∗φ+ λ1
2
(φ∗φ)2 − µ2ψ∗ψ + λ2
2
(ψ∗ψ)2 +
λ
2
(φ∗φ)(ψ∗ψ) (2.1)
where µ1 (µ2) represents the chemical potential of the field φ (ψ), m1 (m2) the mass of φ atom
(ψ atom), and λ1, λ2 and λ the coupling constants.
In the tree approximation the condensate densities φ20 and ψ
2
0 correspond to the local
minimum of the potential. They fulfill
−µ1φ0 + λ1
2
φ30 +
λ
4
φ0ψ
2
0 = 0
−µ2ψ0 + λ2
2
ψ30 +
λ
4
φ20ψ0 = 0 (2.2)
yielding
φ20
2
= 2
2µ1λ2 − µ2λ
4λ1λ2 − λ2 ;
ψ20
2
= 2
2µ2λ1 − µ1λ
4λ1λ2 − λ2 . (2.3)
Now let us focus on the calculation of effective potential in the Hartree-Fock (HF)
approximation. At first the field operators φ and ψ are decomposed
φ =
1√
2
(φ0 + φ1 + iφ2), ψ =
1√
2
(ψ0 + ψ1 + iψ2). (2.4)
Inserting (2.4) into (2.1) we get, among others, the interaction Lagrangian
£int =
(
λ1
2
φ0φ1 +
λ
4
ψ0ψ1
)
(φ21 + φ
2
2) +
λ1
8
(φ21 + φ
2
2)
2
+
(
λ2
2
ψ0ψ1 +
λ
4
φ0φ1
)
(ψ21 + ψ
2
2) +
λ2
8
(ψ21 + ψ
2
2)
2
+
λ
8
(φ21 + φ
2
2)(ψ
2
1 + ψ
2
2),
and the inverse propagators in the tree approximation
D−10 (k) =
~k22mφ − µ1 + 3λ12 φ20 + λ4ψ20 −ω
ω
~k2
2mφ
− µ1 + λ12 φ20 + λ4ψ20
G−10 (k) =
~k22mψ − µ2 + 3λ22 ψ20 + λ4φ20 −ω
ω
~k2
2mψ
− µ2 + λ22 ψ20 + λ4φ20
. (2.5)
122
Hydrodynamic instabilities of two-component Bose-Einstein condensates
From (2.3) and (2.5) it follows that
Eφ = +
√√√√( ~k2
2mφ
+ λ1φ
2
0
)
~k2
2mφ
Eψ = +
√√√√( ~k2
2mψ
+ λ2ψ
2
0
)
~k2
2mψ
, (2.6)
For small momenta Eqs. (2.6) reduce to
Eφ ≈ ±k
√
λ1φ
2
0
2mφ
;Eψ ≈ ±k
√
λ2ψ
2
0
2mψ
(2.7)
associating with Goldstone bosons due to U(1) × U(1) breaking.
Assuming the ansatz
D−1 =
~k22mφ +M1 −ω
ω
~k2
2mφ
+M3
G−1 =
~k22mψ +M2 −ω
ω
~k2
2mψ
+M4
for inverse propagators D, G and following closely [10] we arrive at the CJT effective potential
V CJTβ (φ0, ψ0,D,G) at finite temperature in the HF approximation
V˜ CJTβ (φ0, ψ0,D,G) = −
µ1
2
φ20 +
λ1
8
φ40 −
µ2
2
ψ20 +
λ2
8
ψ40 +
λ
8
φ20ψ
2
0
+
1
2
∫
β
tr
{
lnD−1(k) + lnG−1(k) + [D−10 (k;φ0, ψ0)D] + [G
−1
0 (k;φ0, ψ0)G]− 21
}
+
λ1
8
[ ∫
β
D11(k)
]2
+
λ1
8
[ ∫
β
D22(k)
]2
+
3λ1
4
[ ∫
β
D11(k)
][ ∫
β
D22(k)
]
+
λ2
8
[ ∫
β
G11(k)
]2
+
λ2
8
[ ∫
β
G22(k)
]2
+
3λ2
4
[ ∫
β
G11(k)
][ ∫
β
G22(k)
]
+
λ
8
[∫
β
D11(k)
][ ∫
β
G11(k)
]
+
λ
8
[ ∫
β
D11(k)
][ ∫
β
G22(k)
]
+
λ
8
[ ∫
β
D22(k)
][ ∫
β
G11(k)
]
+
λ
8
[ ∫
β
D22(k)
][ ∫
β
G22(k)
]
. (2.8)
From (2.8) we deduce immediately the following equations:
- The Schwinger-Dyson (SD) equations
D−1 = D−10 (k) + Σ
φ; G−1 = G−10 (k) + Σ
ψ, (2.9)
123
Le Viet Hoa, Nguyen Tuan Anh, Nguyen Chinh Cuong and Dang Thi Minh Hue
in which
Σφ =
(
Σφ1 0
0 Σφ2
)
; Σψ =
(
Σψ1 0
0 Σψ2
)
,
Σφ1 =
λ1
2
∫
β
D11(k) +
3λ1
2
∫
β
D22(k) +
λ
4
∫
β
G11(k) +
λ
4
∫
β
G22(k)
Σφ2 =
3λ1
2
∫
β
D11(k) +
λ1
2
∫
β
D22(k) +
λ
4
∫
β
G11(k) +
λ
4
∫
β
G22(k)
Σψ1 =
λ2
2
∫
β
G11(k) +
3λ2
2
∫
β
G22(k) +
λ
4
∫
β
D11(k) +
λ
4
∫
β
D22(k)
Σψ2 =
3λ2
2
∫
β
G11(k) +
λ2
2
∫
β
G22(k) +
λ
4
∫
β
D11(k) +
λ
4
∫
β
D22(k) (2.10)
- The gap equations
−µ1 + λ1
2
φ20 +
λ
4
ψ20 +Σ
φ
2 = 0
−µ2 + λ
4
φ20 +
λ2
2
ψ20 +Σ
ψ
2 = 0. (2.11)
Hence
φ20 = 4
A
4λ1λ2 − λ2 ; ψ
2
0 = 4
B
4λ1λ2 − λ2 . (2.12)
with
A = 2µ¯1λ2 − µ¯2λ
B = 2µ¯2λ1 − µ¯1λ
µ¯1 = µ1 − Σφ2
µ¯2 = µ2 − Σψ2 . (2.13)
Combining (2.11) and (2.9) we get the forms for inverse propagators
D−1 =
~k22mφ +M1 −ω
ω
~k2
2mφ
; M1 = −µ1 + 3λ1
2
φ20 +
λ
4
ψ20 +Σ
φ
1 ,
G−1 =
~k22mψ +M2 −ω
ω
~k2
2mψ
; M2 = −µ2 + 3λ2
2
ψ20 +
λ
4
φ20 +Σ
ψ
1 . (2.14)
It is obvious that the dispersion relations related to (2.14) read
Eφ =
√√√√ ~k2
2mφ
(
~k2
2mφ
+M1
)
−→
√
M1
2mφ
k as k → 0
Eψ =
√√√√ ~k2
2mψ
(
~k2
2mφ
+M2
)
−→
√
M2
2mψ
k as k → 0
124
Hydrodynamic instabilities of two-component Bose-Einstein condensates
which express the Goldstone theorem. Due to the Landau criteria for superfluidity [11] the
two-component BECs turn out to be superfluid in a broken phase and speeds of sound in each
condensate are given respectively by
Cφ =
√
M1
2mφ
, Cψ =
√
M2
2mψ
. (2.15)
Ultimately the one-particle-irreducible effective potential V˜ CJTβ (φ0, ψ0) is
V˜ CJTβ (φ0, ψ0) = −
µ1
2
φ20 +
λ1
8
φ40 −
µ2
2
ψ20 +
λ2
8
ψ40 +
λ
8
φ20ψ
2
0
+
1
2
∫
β
tr
{
lnD−1(k) + lnG−1(k)
}
−λ1
8
[ ∫
β
D11(k)
]2
− λ1
8
[ ∫
β
D22(k)
]2
− 3λ1
4
[ ∫
β
D11(k)
][ ∫
β
D22(k)
]
−λ2
8
[ ∫
β
G11(k)
]2
− λ2
8
[ ∫
β
G22(k)
]2
− 3λ2
4
[ ∫
β
G11(k)
][ ∫
β
G22(k)
]
−λ
8
[ ∫
β
D11(k)
][∫
β
G11(k)
]
− λ
8
[ ∫
β
D11(k)
][ ∫
β
G22(k)
]
−λ
8
[ ∫
β
D22(k)
][ ∫
β
G11(k)
]
− λ
8
[ ∫
β
D22(k)
][ ∫
β
G22(k)
]
(2.16)
2.2. Physical properties
2.2.1. Equations of state
Let us now consider Equations of State (EOS) starting from the effective potential. To this
end, we begin with the pressure defined by
P = −V˜ CJTβ (φ0, ψ0,D,G)|at minimum (2.17)
from which the total particle densities are determined
ρi =
∂P
∂µi
, i = 1, 2.
Taking into account the fact that derivatives of V˜ CJTβ (φ0, ψ0,D,G) with respect to its arguments
vanish at minimum we get
ρ1 = −
∂V CJTβ
∂µ1
=
φ20
2
+
1
2
∫
β
D11 +
1
2
∫
β
D22
ρ2 = −
∂V CJTβ
∂µ2
=
ψ20
2
+
1
2
∫
β
G11 +
1
2
∫
β
G22. (2.18)
125
Le Viet Hoa, Nguyen Tuan Anh, Nguyen Chinh Cuong and Dang Thi Minh Hue
Hence, the gap equations (2.11) become
µ1 = λ1ρ1 +
λ
2
ρ2 + λ1
∫
β
D11
µ2 = λ2ρ2 +
λ
2
ρ1 + λ2
∫
β
G11 (2.19)
and the particle densities in condensates are
φ20
2
= ρ1 − 1
2
∫
β
D11 − 1
2
∫
β
D22
ψ20
2
= ρ2 − 1
2
∫
β
G11 − 1
2
∫
β
G22. (2.20)
Combining Eqs. (2.10), (2.17) and (2.18) together produces the following expression for the
pressure
P = −V˜ = λ1
2
ρ21 +
λ2
2
ρ22 +
λ
2
ρ1ρ2 − 1
2
∫
β
tr{lnD−1(k) + lnG−1(k)} −
− λ1
2
[ ∫
β
D11
]2
− λ2
2
[ ∫
β
G11
]2
+ λ1ρ1
∫
β
D11 + λ2ρ2
∫
β
G11. (2.21)
Eqs. (2.21) constitute the EOS governing all thermodynamic processes in the system.
2.2.2. Numerical study
In order to get some insight into the hydrodynamic instabilities of the two-component
system Bose gases, let us choose a set of model parameters, which are close to the experimental
settings [12]:
λ1 = 5.10
−12eV −2;λ2 = 0, 4.10−12eV −2;λ = 4.10−12eV −2;
mφ = mψ = 80GeV ;µ1 = 1, 4.10
−11eV.
Solving the gap and the SD Eqs. (2.11) and (2.14) we obtain the phase diagram in the
(T − µ2)-plane given in Figure 1.
In this case, 4λ1λ2 − λ2 < 0, so from (2.12) A < 0 (or B < 0) corresponds to φ0 6= 0 (or
ψ0 6= 0), and there exits condensation in corresponding sectors.
As is seen in Figure 1, in the region (A > 0, B < 0) for 0 < µ2 < 2, 8.10−11eV ,
there exits hydrodynamic stability only in the ψ sector below a finite temperature. This statement
is confirmed again in Figures 2 and 3, providing the T dependence of M1,M2 and φ0, ψ0 at
µ2 = 2, 5.10
−12eV . It is clear thatM1 is negative at every temperature, and from (2.15) the speed
Cφ is an imaginary quantity, i.e. φ0 is always zero. Whereas T 0, and Cψ is a
real quantity, i.e ψ0 presents a physical condensate.
Also from Figure 1, in the region (A < 0, B < 0) for 2, 8.10−11eV < µ2 < 5.10−11eV ,
there exits hydrodynamic stability in both the φ and ψ sectors. Figures 4 and 5 show the T
dependence of M1,M2 and φ0, ψ0 at µ2 = 3.10−12eV . In this case M1 and M2 are positive
only below the temperatures Tc1 and Tc2 respectively, i.e. there are condensates in both φ and
ψ sectors.
126
Hydrodynamic instabilities of two-component Bose-Einstein condensates
Figure 1. Phase diagram in the (T − µ2)-plane
Figure 2. T dependence of M1,M2 Figure 3. T dependence of φ0, ψ0
Figure 4. T dependence of M1,M2 Figure 5. T dependence of φ0, ψ0
127
Le Viet Hoa, Nguyen Tuan Anh, Nguyen Chinh Cuong and Dang Thi Minh Hue
3. Conclusion
Due to a growing interest in binary mixtures of Bose gases we studied a non-relativistic
model of two-component complex fields. Based on the CJT effective action approach we
established a CJT effective potential in the HF approximation. The expression for the pressure,
which depends on particle densities, was derived by means of the fact that the pressure is
determined by the effective potential at minimum. Our numerical results show that instabilities
in our model are strongly influenced by the chemical potential
In order to understand better the physical properties of the two-component BECs more
detail studies of EOS could be carried out by means of numerical computation. This is left for
future study.
Acknowledgment. The authors would like to thank the HNUE for its financial support.
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