Abstract. The Cornwall-Jackiw-Tomboulis (CJT) effective action at finite temperature is
applied to study the quantum condensation of a two-component system of atoms. We derive
the renormalized effective potential in the Hartree-Fock approximation, which preserves
the Goldstone theorem. Numerical results show that there exist quantum condensations of
any component at appropriate regions of coupling constants corresponding to its positive
effective chemistry potential, i.e. to its dynamic stability. In addition, the phase quantum
transitions of both first and second order appear when the coupling constant changes.

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JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2016-0030
Mathematical and Physical Sci., 2016, Vol. 61, No. 7, pp. 41-46
This paper is available online at
QUANTUM CONDENSATION IN A TWO-COMPONENT SYSTEM OF ATOMS
Tran Huu Phat1, Le Viet Hoa2, Nguyen Tuan Anh3 and Dang Thi Minh Hue4
1Vietnam Atomic Energy Commission
2Faculty of Physics, Hanoi National University of Education
3Faculty of Energy Technology, Electric Power University, 4Water Resources University
Abstract. The Cornwall-Jackiw-Tomboulis (CJT) effective action at finite temperature is
applied to study the quantum condensation of a two-component system of atoms.We derive
the renormalized effective potential in the Hartree-Fock approximation, which preserves
the Goldstone theorem. Numerical results show that there exist quantum condensations of
any component at appropriate regions of coupling constants corresponding to its positive
effective chemistry potential, i.e. to its dynamic stability. In addition, the phase quantum
transitions of both first and second order appear when the coupling constant changes.
Keywords: Quantum condensation, two-component system, first order, second order,
coupling constant.
1. Introduction
In recent years there have appeared many papers dealing with phase transition of systems
composed of two distinct species of atoms [1, 2]. The two - component system is not a simple
extension of the single component system, because of the many novel phenomena which can arise
such as the quantum tunneling of spin domain [3], phase segregation [4] and so on.
In the present work, a theoretical formalism for studying quantum condensation of
two-component systems of atoms in the global U(1) × U(1) model is formulated by means of
the Cornwall-Jackiw-Tomboulis (CJT) effective action in the Hartree-Fock (HF) approximation
[5]. The investigation reported in Ref. [6] has provided a general view on several properties of
possible phases in this system. In order to get a deeper insight into this issue we carried out a
systematic study of quantum condensation in a system expanding on our previous work [6]. To
begin with,we first rewrite the Lagrangian density
£ = 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) (1.1)
in which 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.
Received April 30, 2016. Accepted July 14, 2016.
Contact Le Viet Hoa, e-mail address: hoalv@hnue.edu.vn
41
Tran Huu Phat, Le Viet Hoa, Nguyen Tuan Anh and Dang Thi Minh Hue
Following closely [6] we arrive at the CJT effective potential V˜ CJTβ (φ0, ψ0,D,G) at finite
temperature in double-bubble approximation which preserves the Goldstone theorem
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] − 2.1
}
+
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)
]
, (1.2)
in which ∫
β
f(k) ≡ T
n=∞∑
n=−∞
∫
d3~k
(2π)3
f(ωn, ~k), ωn = 2πnT,
and T is a temperature.
From (1.2) we deduce immediately the following equations:
a- The gap equations
δV˜ CJTβ
δφ0
= 0,
δV˜ CJTβ
δψ0
= 0,
yielding
−µ1 + λ1
2
φ20 +
λ
4
ψ20 +Σ
φ
2 = 0
−µ2 + λ
4
φ20 +
λ2
2
ψ20 +Σ
ψ
2 = 0. (1.3)
Hence
φ20 = 4
A
4λ1λ2 − λ2 ; ψ
2
0 = 4
B
4λ1λ2 − λ2 . (1.4)
with
A = 2µ¯1λ2 − µ¯2λ
B = 2µ¯2λ1 − µ¯1λ
µ¯1 = µ1 − Σφ2
µ¯2 = µ2 − Σψ2 . (1.5)
42
Quantum condensation in a two-component system of atoms
b- The Schwinger-Dyson (SD) equations
δV˜ CJTβ
δD
= 0,
δV˜ CJTβ
δG
= 0,
giving
D−1 =
~k22mφ +M1 −ωn
ωn
~k2
2mφ
; M1 = −µ1 + 3λ1
2
φ20 +
λ
4
ψ20 +Σ
φ
1 ,
G−1 =
~k22mψ +M2 −ωn
ωn
~k2
2mψ
; M2 = −µ2 + 3λ2
2
ψ20 +
λ
4
φ20 +Σ
ψ
1 . (1.6)
Here
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) (1.7)
Eqs. (1.6) leads to the dispersion relations of Nambu-Goldstone modes at small momentum
Eφ =
√√√√ ~k2
2mφ
(
~k2
2mφ
+M1
)
−→
√
M1
2mφ
k,
Eψ =
√√√√ ~k2
2mψ
(
~k2
2mψ
+M2
)
−→
√
M2
2mψ
k,
expressing not only the Goldstone theorem, but also the superfluidity of condensates due to the
Landau criteria [7] and, consequently, the speeds of sound in each condensate read
Cφ =
√
M1
2mφ
, Cψ =
√
M2
2mψ
. (1.8)
A dynamic instability takes place for condensates j when its superfluidity is broken, that is when
Mj < 0.
Based on the CJT effective potential (1.2) we can arrive at the specific heat at constant volume:
CV = −T
(
∂2V˜ CJTβ (φ0, ψ0,D,G)
∂T 2
)
V
. (1.9)
In the following the formulae (1.4), (1.6) and (1.9) will be used in the numerical study of
quantum condensation in a system.
43
Tran Huu Phat, Le Viet Hoa, Nguyen Tuan Anh and Dang Thi Minh Hue
2. Quantum condensation
In this section we explore the phase diagrams in the (T, λ)-plane and their phase structures.
Then, making use of these phase diagrams, we will prove that:
- In diagram several phases are provided among them following phases (φ0 6= 0, ψ0 =
0), (φ0 = 0, ψ0 6= 0), (φ0 6= 0, ψ0 6= 0) and (φ0 = 0, ψ0 = 0). The last phase is considered to be
a phase without condensate.
- The values of the coupling constant λ of each component are, for example φ0, might settle
in one among two states: φ0 = 0, φ0 6= 0.
When investigating the phase diagram in the (T, λ)-planes, it was found that.
The phase diagram in the (T, λ)-plane is determined by the equations A(T, λ) =
B(T, λ) = 0 at fixed values of other parameters. Fig. 1 shows the phase diagram plotted at
λ1 = 5×10−12eV −2, λ2 = 0.4×10−12eV −2, µ1 = −1.4×10−11eV and µ2 = −5.5×10−12eV .
Figure 1. Phase diagram in the (T, λ)-plane
Three different phases and desert are distributed in all regions which are separated by the
phase lines A = B = 0 and two straight lines λ = ±λc = ±
√
4λ1λ2 =
√
8 × 10−12eV −2.
Referring to Fig. 1, some values of T can be selected, such as T = 5 nK, which leads to
quantum condensation and non-condensation scenarios for both condensates as plotted in Figs.
2a and 2b. There are three regions separated by two vertical lines ±λc, whith the first-order
quantum transitions shown in Fig. 2a. Besides these special transition points, there exist other
points shown as second-order transition at λ = 2.04 × 10−12eV −2 for φ0 component and at
λ = 3.8 × 10−12eV −2 for the ψ0 component. Based on Eq. (1.8), in Fig. 2b it is shown that
quantum condensation of one the component appears only because its effective chemistry potential
44
Quantum condensation in a two-component system of atoms
is positive, i.e. its sound speed is real. This corresponds to the dynamic stability of for that
component. At the phase transition points, in this case, the effective chemistry potentials change
their sign.
(a) (b)
Figure 2. λ-dependence of condensates and effective chemistry potentials at T = 10 nK
Figure 3. λ-dependence of the specific heat at constant volume at T = 10 nK
However, a question which immediately arises is whether or not the scenarios of
condensation and non-condensation, presented in the foregoing cases, possibly exist in nature.
To answer this, let us investigate the λ dependence of the specific heat at constant volume based
on the formula (1.9). The graph of CV (λ) plotted in Fig. 3 proves that the scenarios seen in Fig. 2
possibly exist in nature because their corresponding specific heat is positive, CV > 0. It is easily
to realize that the quantum processes only happen when λ > −λc.
45
Tran Huu Phat, Le Viet Hoa, Nguyen Tuan Anh and Dang Thi Minh Hue
3. Conclusion
In the previous sections we focused on the quantum condensation and dynamic stability in
a two-component system of atoms. Our results are presented in the following order:
- We found two distinctive properties among others of the system:
- There manifest two scenarios, condensation and non-condensation, depending upon the
coupling constant setting.
- Independent of the setting of the coupling constant we discovered that the dynamic
stability is the main factor causing condensation in the system, and in this connection, we
formulated sufficient condition for condensation scenarios to happen: the presence of any
condensation scenario is determined by the dynamic stability at a certain value of the coupling
constant.
We predicted that the quantum condensations in a two-component system of atoms might
be present in nature. In addition, phase quantum transitions of both first and second order appear
when the coupling constant changes. These findings have not been presented in any other work,
except for Ref. [6].
Acknowledgment. The authors would like to thank the HNUE for financial support.
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