ЖЭТФ, 2022, том 162, вып. 2 (8), стр. 177-180

THE CASIMIR EFFECT IN BOSE-EINSTEIN CONDENSATE

MIXTURES CONFINED BY A PARALLEL PLATE GEOMETRY

IN THE IMPROVED HARTREE-FOCK APPROXIMATION

Nguyen Van Thu^*

Department of Physics, Hanoi Pedagogical University 2

100000, Hanoi, Vietnam

Received February 4, 2022

revised version February 4, 2022.

Accepted for publication March 18, 2022

DOI: 10.31857/S0044451022080028

integrals were calculated at the lowest-order approxi-

EDN: Van Thu EFLSKB

mation, which we call the lowest-order Hartree-Fock

(LIHF) approximation. Therefore, the vanishing of the

Casimir force in the limit of the full strong segregation

The Casimir effect was first discovered [1] for the

did not change.

electromagnetic field confined between two neutral par-

In this paper, the Casimir effect in the BECs at zero

allel plates at zero temperature. This effect has been

temperature is researched in the IHF approximation

studied for other fields [2-7]. In field of the Bose-Ein-

with the higher-order terms of the momentum integrals

stein condensate (BEC), the Casimir effect has been

and it is called the higher-order improved Hartree-Fock

considered in both experiment [8-12] and theory, for

(HIHF) approximation. To do so, we start from the

example, [13-18] in the grand canonical ensemble and

Lagrangian density of the BECs without external field

[17,19] in the canonical ensemble. In these papers, the

[23, 24],

Casimir effect was investigated in the one-loop approxi-

(

)

mation within the framework of perturbative theory for

∑

ℏ²

ψ^∗

-iℏ∂_t -

∇²

ψ_j - V,

(1)

a single dilute BEC. For two-component BEC (BECs),

2m_j

j=1,2

our previous work [20] pointed out that the Casimir

force is not a simple superposition of the one of two sin-

with

gle component BEC because of the mutual repulsive in-

(

)

∑

g_jj

teractions. It was also proven to be zero in some cases:

V =

-µ_j|ψ_j|² +

|ψ_j|4

+ g₁₂|ψ₁|²|ψ₂|².

(2)

(i) the inter-distance between two plates becomes large

j=1,2

enough; (ii) both the inter- and intraspecies interac-

Here ℏ is the reduced Planck constant, µ_j and m_j are

tions are zero (ideal gases); and an important case (iii)

the chemical potential and atomic mass of component

the interspecies interaction is the full strong segrega-

j, respectively. The coupling constant

tion. Even so, the case (iii) result is controversial be-

cause of the interpretation that the original Casimir

g_jj = 4πℏ²a_jj/m_j > 0

force and interspecies interactive force are of the same

order in the full strong separation. As an improvement,

represents the strength of the repulsive intraspecies in-

in Ref. [21] the Casimir effect in the BECs was studied

teraction and

in the improved Hartree-Fock (IHF) approximation,

(

)

which based on the Cornwall-Jackiw-Tomboulis (CJT)

g_jj′ = 2πℏ²

a_jj′ > 0

m_j

m_j′

effective action formalism [22]. In this approximation

the two-loop diagrams were taken into account and the

is the strength of the repulsive interspecies interaction,

Goldstone theory is obeyed. However, the momentum

a_jj′ being the s-wave scattering length between com-

ponents j and j^′. The field operator ψ_j has the ex-

^* E-mail: nvthu@live.com

pectation value ψ_j0, which plays the role of the order

177

Nguyen Van Thu

ЖЭТФ, том 162, вып. 2 (8), 2022

parameter. Recall that for g²¹² > g₁₁g₂₂ the two com-

ponents are immiscible and a phase-segregated BEC

forms [25], and vice versa. Shifting the field operators

ψ_j → ψ_j0 +

√

(ψ_j1 + iψ_j2)

one can obtain the CJT effective potential in the

Hartree-Fock approximation. Unfortunately, this CJT

effective potential was proved violating the Goldstone

theorem [26] by finding the dispersion relation from the

request for vanishing of the determinant of the inverse

propagators [27]. To solve this problem, the method

developed in [28] is invoked. After adding the extra

term in the CJT effective potential, one arrives at the

IHF approximation. Minimizing the CJT effective po-

tential with respect to the order parameter one has the

gap equations

Fig. 1. (Color online) The effective masses (top panel) and or-

der parameters (bottom panel) as a function of 1/K at L = 1.

The solid red (first component) and solid blue (second com-

-1 + φ²¹ + Kφ²² +

Σ⁽¹⁾² = 0,

g₁₁n₁₀

ponent) lines are in the HIHF approximation, the dashed red

(3)

(first component) and dashed blue (second component) are

-1 + φ²² + Kφ²¹ +

Σ⁽²⁾² = 0.

the corresponding quantities in the LIHF approximation

g₂₂n₂₀

Similarly, the Schwinger-Dyson equations can be

0.000

achieved by minimizing the CJT effective potential

with respect to the elements of the propagators

-0.005

-0.010

M²¹ = -1 + 3φ²¹ + Kφ²² +

Σ⁽¹⁾¹,

g₁₁n₁₀

(4)

-0.015

M²² = -1 + 3φ²² + Kφ²¹ +

Σ⁽²⁾¹.

g₂₂n₂₀

-0.020

In Eqs. (3) and (4), n_j0 is the bulk density of the com-

-0.025

ponent j, K = g₁₂/√g11g22, φj = ψj0/√nj0 is the di-

mensionless order parameter, and M_j is dimensionless

-0.030

effective mass. The self energies are defined as

-0.035

0.0

0.2

0.4

0.6

0.8

Σ⁽¹⁾¹ =

(g₁₁P₁₁ + 3g₁₁P₂₂ + g₁₂Q₁₁ + g₁₂Q₂₂),

Σ⁽²⁾¹ =

(g₂₂Q₁₁ + 3g₂₂Q₂₂ + g₁₂P₁₁ + g₁₂P₂₂),

Fig. 2. (Color online) The Casimir force versus 1/K at L = 1 in

(5)

the HIHF approximation (red line). The blue line corresponds

Σ⁽¹⁾² =

(3g₁₁P₁₁ + g₁₁P₂₂ + g₁₂Q₁₁ + g₁₂Q₂₂),

to the LIHF and one-loop approximations, respectively

Σ⁽²⁾² =

(3g₂₂Q₁₁ + g₂₂Q₂₂ + g₁₂P₁₁ + g₁₂P₂₂),

[29]. The periodic boundary condition is imposed at

with P_aa, Q_bb are the momentum integrals.

the plates, which can be realized in experiments by us-

We now consider a binary mixture of Bose gases

ing toroidal traps [30,31] or optical lattices [32]. Due to

confined between two parallel plates, which perpen-

the confinement, the wave vectors are quantized. Us-

dicular to the z-axis. This means that the system

ing the Euler-Maclaurin formula [33], the momentum

is confined to a parallel plate geometry with the size

integrals are calculated in the HIHF approximation at

ℓ_x, ℓ_y and distance between the two plates of ℓ = ℓ_z,

zero temperature

which satisfies condition ℓ_x, ℓ_y ≫ ℓ as was discussed in

178

ЖЭТФ, том 162, вып. 2 (8), 2022

The Casimir effect in Bose-Einstein condensate mixtures. . .

π²m₁g₁₁n₁₀ξ²¹

P₁₁ = -

sponding result in the LIHF approximation, that is the

90ℏ²ℓ³M₁

effective masses depend on both the distance ℓ and K.

The comparison of the evolution of the effective masses

π²m₂g₂₂n₂₀ξ²²

Q₁₁ = -

(top panel) and order parameters (bottom panel) in

90ℏ²ℓ³M₂

(6)

the HIHF approximation (solid lines) with those in the

m₁g₁₁n₁₀M₁

m₁g₁₁n₁₀ξ²¹π

LIHF approximation (dashed lines) is sketched in Fig. 1

P₂₂ =

12ℏ²ℓ

90ℏ²M₁ℓ³

at L = 1. A remarkable difference in regime of the

strong segregated region can be observed. In this re-

m₂g₂₂n₂₀M₂

m₂g₂₂n₂₀ξ²²π

Q₂₂ =

gion, at a given value of 1/K, the values of the effective

12ℏ²ℓ

90ℏ²M₂ℓ3

masses and order parameters in the HIHF approxima-

where ξj = ℏ/√2mj gjj nj0 is the healing length of

tion are bigger than the corresponding ones in the LIHF

jth component.

approximation. At the full strong segregation, both

In order to illustrate these calculations, numeri-

the effective masses and order parameters are different

cal computations are performed for a dual-species Bo-

from zero whereas they vanish in the LIHF approxi-

se-Einstein condensates of rubidium 87 (first compo-

mation. This fact confirms that the influence of the

nent) and cesium 133 (second component). For this sys-

higher-order terms in the momentum integrals is not

tem the parameters are in order m₁ = 86.909u, a₁₁ =

negligible. In the regime of the full strong separation,

= 100.4a₀ for rubidium 87 and m₂ = 132.905u, a₂₂ =

the HIHF approximation gives

= 280a₀ for cesium 133 [34]. Here u = 1.66 · 10^-27 kg

M_j0

and a₀ = 0.529 nm are the atomic mass unit and Bohr

M_j ≃

ℓ

radius, respectively. The dimensionless length is cho-

(7)

φ_j0

sen as L = ℓ/ξ₁ with ξ₁ = 4000 nm being the healing

φ_j ≃

ℓ²

length of rubidium 87. The numerical computations

show a significant difference compared with the corre-

in which

(

)_1/3

jjξ^jπ²

M_j0 =^jg

45ℏ²

(8)

√

5π^4/3g^2/3

m^2/3jξ^4/3jℏ^4/3 - (15π)^2/3g^4/3jjm^4/3jξ^2/3j

φ_j0 =

√

360³

3ℏ^8/3

Formally resemble the momentum integrals, the

F_C =

(

)

Casimir force energy density is found

∑

π²m_jg_jjξ^2jM_j

π²m_jg_jjξ^2j ∂M_j

(10)

120ℏ²ℓ⁴

360ℏ²ℓ³

∂ℓ

j=1,2

∑

π²m_jg_jjξ^2jM_j

E_C =

(9)

360ℏ²ℓ³

j=1,2

Owing to the ℓ-dependence of the effective masses, the

contribution of the higher-order terms in the momen-

It is clear that the Casimir energy is negative and, ne-

tum integrals is shown by second term in right-hand

glecting the ℓ-dependence of M_j , it has the same form

side of Eq. (10). The evolution of the Casimir force

as the one in the LIHF approximation and one-loop

versus 1/K is plotted in Fig. 2 at L = 1 and other

approximation. However, in the LIHF and one-loop

parameters are the same as in Fig. 1. The red line

approximations, the effective masses are independent

is drawn in the HIHF approximation whereas the blue

of the distance so that the Casimir energy density will

line corresponds to the LIHF and one-loop approxi-

be the same as the one in the one-loop approximation.

mations. This figure shows that the strength of the

The Casimir force is defined as the negative derivative

Casimir force decreases as the interspecies interaction

of the Casimir energy with respect to a change in the

increases. This fact is understandable if we note that

distance between two parallel plates. Combining with

the Casimir force is attractive whereas the interspecies

(9) one obtains the Casimir force acting on per unit

interaction is repulsive. The red line in Fig. 2 show

area of the plates

that the Casimir force is non-zero in the limit of the

179

ЖЭТФ, вып. 2 (8)

Nguyen Van Thu

ЖЭТФ, том 162, вып. 2 (8), 2022

full strong segregation within the HIHF approximation,

13.

J. Schiefele and C. Henkel, J. Phys. A 42, 045401

whereas it vanishes in the LIHF and one-loop approxi-

(2009).

mations as shown by the blue line in Fig. 2. This is an

14.

J. O. Andersen, Rev. Mod. Phys. 76, 599 (2004).

interesting result in comparison with that in [20]. This

15.

D. C. Roberts and Y. Pomeau, Phys. Rev. Lett. 95,

result gives us the conclusion that the Casimir force is

145303 (2005).

always on top of interspecies interaction and this is an

important improvement on the result in our previous

16.

S. Biswas, J. K. Bhattacharjee, D. Majumder, K. Sa-

paper [20]. Mathematically, the Casimir force in full

ha, and N. Chakravarty, J. Phys. B 43, 085305

strong separation is

(2010).

∑

(m_j g_jj π²)^4/3

17.

Nguyen Van Thu, Phys. Lett. A 382, 1078 (2018).

F_C = -

(11)

90.3^2/35^1/3ℏ^8/3ℓ

j=1,2

18.

Nguyen Van Thu and Pham The Song, Physica

A 540, 123018 (2020).

This equation confirms again that the Casimir force is

non-zero in the full strong separation limit.

19.

Nguyen Van Thu, Luong Thi Theu, and Dang Thanh

Hai, JETP 130, 321 (2020).

Acknowledgments. We are grateful to Shyamal

20.

Nguyen Van Thu and Luong Thi Theu, J. Stat. Phys

Biswas for their useful discussions.

168, 1 (2017).

Funding. This work is funded by the Vietnam Na-

21.

Nguyen Van Thu and Luong Thi Theu, Int. J. Mod.

tional Foundation for Science and Technology Develop-

Phys. B 33, 1950114 (2019).

ment (NAFOSTED) under Grant No. 103.01-2018.02.

22.

J. M. Cornwall, R. Jackiw, and E. Tomboulis, Phys.

Rev. D 10, 2428 (1974).

The full text of this paper is published in the English

version of JETP.

23.

L. Pitaevskii and S. Stringari, Bose-Einstein Con-

densation, Oxford University Press (2003).

REFERENCES

24.

C. J. Pethick and H. Smith, Bose-Einstein Conden-

sation in Dilute Gases, Cambridge University Press

1. H. B. G. Casimir, Proc. K. Ned. Akad. Wet. 51, 793

(1948).

(2008).

25.

P. Ao and S. T. Chui, Phys. Rev. A 58, 4836 (1998).

2. F. Chen, G. L. Klimchitskaya, V. M. Mostepanenko,

and U. Mohideen, Phys. Rev. B 76, 035338 (2007).

26.

T. H. Phat, L. V. Hoa, N. T. Anh, and N. V. Long,

Ann. Phys. 324, 2074 (2009).

3. G. Bimonte, Phys. Rev. A 78, 062101 (2008).

4. J. F. Babb, Adv. Atom. Mol. Opt. Phys. 59, 1 (2010).

27.

S. Floerchinger and C. Wetterich, Phys. Rev. A 79,

013601 (2009).

5. Tran Huu Phat and Nguyen Van Thu, Int. J. Mod.

Phys. A 29, 1450078 (2014).

28.

Yu. B. Ivanov, F. Riek, and J. Knoll, Phys. Rev.

D 71, 105016 (2005).

6. M. Bordag, U. Mohideen, and V. M. Mostepanenko,

Phys. Rep. 353, 1 (2001).

29.

R. Lipowsky, in Random Fluctuations and Pattern

Growth, ed. by H. Stanley and N. Ostrowsky, NATO

7. G. L. Klimchitskaya and U. Mohideen, Rev. Mod.

ASI Series E, Vol. 157, Kluwer Akad. Publ., Dor-

Phys. 81, 1827 (2009).

drecht (1988), pp. 227-245.

8. M. Fukuto, Y. F. Yano, and P. S. Pershan, Phys. Rev.

30.

P. D. Drummond, A. Eleftheriou, K. Huang, and

Lett. 94, 135702 (2005).

K. V. Kheruntsyan, Phys. Rev. A 63, 053602 (2001).

9. A. Ganshin, S. Scheidemantel, R. Garcia, and

31.

J. Brand and W. P. Reinhardt, J. Phys. B 34, L113

M. H. W. Chan, Phys. Rev. Lett. 97, 075301 (2006).

(2001).

10. D. M. Harber, J. M. Obrecht, J. M. McGuirk, and

32.

A. A. Shams and H. R. Glyde, Phys. Rev. B 79,

E. A. Cornell, Phys. Rev. 72, 033610 (2005).

214508 (2009).

11. J. M. Obrecht, R. J. Wild, M. Antezza, L. P. Pitaev-

33.

G. B. Arfken and H. J. Weber, Mathematical Methods

skii, S. Stringari, and E. A. Cornell, Phys. Rev. Lett.

for Physicists, Academic, San Diego (2005).

98, 063201 (2007).

34.

D. J. McCarron, H. W. Cho, D. L. Jenkin, M. P. Kop-

12. G. L. Klimchitskaya and V. M. Mostepanenko, J.

pinger, and S. L. Cornish, Phys. Rev. A 84, 011603

Phys. A 41, 312002 (2008).

(2011).

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