On the finite-size effects in two segregated Bose-Einstein condensates restricted by a hard wall
The finite-size effects in two segregated Bose-Einstein condensates (BECs) restricted by a hard wall is studied by means of the Gross-Pitaevskii equations in the double-parabola approximation (DPA). Starting from the consistency between the boundary conditions (BCs) imposed on condensates in confin...
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Інститут фізики конденсованих систем НАН України
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irk-123456789-1576332019-06-21T01:29:27Z On the finite-size effects in two segregated Bose-Einstein condensates restricted by a hard wall Quyet, H.V. Thu, N.V. Tam, D.T. Phat, T.H. The finite-size effects in two segregated Bose-Einstein condensates (BECs) restricted by a hard wall is studied by means of the Gross-Pitaevskii equations in the double-parabola approximation (DPA). Starting from the consistency between the boundary conditions (BCs) imposed on condensates in confined geometry and in the full space, we find all possible BCs together with the corresponding condensate profiles and interface tensions. We discover two finite-size effects: a) The ground state derived from the Neumann BC is stable whereas the ground states derived from the Robin and Dirichlet BCs are unstable. b) Thereby, there equally manifest two possible wetting phase transitions originating from two unstable states. However, the one associated with the Robin BC is more favourable because it corresponds to a smaller interface tension. Ефекти скiнченного розмiру у двох вiдокремлених конденсатах Бозе-Ейнштейна, обмежених твердою стiнкою, дослiджено з допомогою рiвнянь Гросса-Пiтаєвского в наближеннi подвiйної параболи. Виходячи з узгодженостi мiж граничними умовами, якi накладаються на конденсати в обмеженiй геометрiї i в повному просторi, ми знаходимо усi можливi граничнi умови разом iз вiдповiдними профiлями конденсату i мiжфазовi натяги. Нами вiдкрито два ефекти скiнченного розмiру: a) основний стан, отриманий з граничної умови Ньюмана, є стiйким, а основнi стани, отриманi з граничних умов Робiна i Дiрiхлє, є нестiйкими, b) отже, однаковим чином проявляються два можливих переходи змочування як результат двох нестiйких станiв. Проте, перехiд, пов’язаний з граничними умовами Робiна, виявляється бiльш сприятливим, тому що вiн вiдповiдає меншому мiжфазовому натягу. 2019 Article On the finite-size effects in two segregated Bose-Einstein condensates restricted by a hard wall / H.V. Quyet, N.V. Thu, D.T. Tam, T.H. Phat // Condensed Matter Physics. — 2019. — Т. 22, № 1. — С. 13001: 1–14. — Бібліогр.: 31назв. — англ. 1607-324X PACS: 03.75.Hh, 05.30.Jp, 68.03.Cd, 68.08.Bc DOI:10.5488/CMP.22.13001 arXiv:1903.11457 http://dspace.nbuv.gov.ua/handle/123456789/157633 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
The finite-size effects in two segregated Bose-Einstein condensates (BECs) restricted by a hard wall is studied
by means of the Gross-Pitaevskii equations in the double-parabola approximation (DPA). Starting from the consistency between the boundary conditions (BCs) imposed on condensates in confined geometry and in the full
space, we find all possible BCs together with the corresponding condensate profiles and interface tensions. We
discover two finite-size effects: a) The ground state derived from the Neumann BC is stable whereas the ground
states derived from the Robin and Dirichlet BCs are unstable. b) Thereby, there equally manifest two possible
wetting phase transitions originating from two unstable states. However, the one associated with the Robin BC
is more favourable because it corresponds to a smaller interface tension. |
| format |
Article |
| author |
Quyet, H.V. Thu, N.V. Tam, D.T. Phat, T.H. |
| spellingShingle |
Quyet, H.V. Thu, N.V. Tam, D.T. Phat, T.H. On the finite-size effects in two segregated Bose-Einstein condensates restricted by a hard wall Condensed Matter Physics |
| author_facet |
Quyet, H.V. Thu, N.V. Tam, D.T. Phat, T.H. |
| author_sort |
Quyet, H.V. |
| title |
On the finite-size effects in two segregated Bose-Einstein condensates restricted by a hard wall |
| title_short |
On the finite-size effects in two segregated Bose-Einstein condensates restricted by a hard wall |
| title_full |
On the finite-size effects in two segregated Bose-Einstein condensates restricted by a hard wall |
| title_fullStr |
On the finite-size effects in two segregated Bose-Einstein condensates restricted by a hard wall |
| title_full_unstemmed |
On the finite-size effects in two segregated Bose-Einstein condensates restricted by a hard wall |
| title_sort |
on the finite-size effects in two segregated bose-einstein condensates restricted by a hard wall |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2019 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/157633 |
| citation_txt |
On the finite-size effects in two segregated Bose-Einstein condensates restricted by a hard wall / H.V. Quyet, N.V. Thu, D.T. Tam, T.H. Phat // Condensed Matter Physics. — 2019. — Т. 22, № 1. — С. 13001: 1–14. — Бібліогр.: 31назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT quyethv onthefinitesizeeffectsintwosegregatedboseeinsteincondensatesrestrictedbyahardwall AT thunv onthefinitesizeeffectsintwosegregatedboseeinsteincondensatesrestrictedbyahardwall AT tamdt onthefinitesizeeffectsintwosegregatedboseeinsteincondensatesrestrictedbyahardwall AT phatth onthefinitesizeeffectsintwosegregatedboseeinsteincondensatesrestrictedbyahardwall |
| first_indexed |
2025-07-14T10:03:26Z |
| last_indexed |
2025-07-14T10:03:26Z |
| _version_ |
1837616235124097024 |
| fulltext |
Condensed Matter Physics, 2019, Vol. 22, No 1, 13001: 1–14
DOI: 10.5488/CMP.22.13001
http://www.icmp.lviv.ua/journal
On the finite-size effects in two segregated
Bose-Einstein condensates restricted by a hard wall
H.V. Quyet1, N.V. Thu1, D.T. Tam2, T.H. Phat3
1 Department of Physics, Hanoi Pedagogical University 2, Hanoi, Vietnam
2 Tay Bac University, Son La, Vietnam
3 Vietnam Atomic Energy Commission, 59 Ly Thuong Kiet, Hanoi, Vietnam
Received November 21, 2018, in final form February 2, 2019
The finite-size effects in two segregated Bose-Einstein condensates (BECs) restricted by a hard wall is studied
by means of the Gross-Pitaevskii equations in the double-parabola approximation (DPA). Starting from the con-
sistency between the boundary conditions (BCs) imposed on condensates in confined geometry and in the full
space, we find all possible BCs together with the corresponding condensate profiles and interface tensions. We
discover two finite-size effects: a) The ground state derived from the Neumann BC is stable whereas the ground
states derived from the Robin and Dirichlet BCs are unstable. b) Thereby, there equally manifest two possible
wetting phase transitions originating from two unstable states. However, the one associated with the Robin BC
is more favourable because it corresponds to a smaller interface tension.
Key words: finite-size effect, Bose-Einstein condensates, boundary condition, double-parabola approximation
PACS: 03.75.Hh, 05.30.Jp, 68.03.Cd, 68.08.Bc
1. Introduction
The physics of binary Bose-Einstein condensate mixture in infinite space has explored many inter-
esting instabilities. It was proved theoretically that the well-known instabilities in classical fluid also take
place in quantum fluid, including the Rayleigh-Taylor instability [1, 2], the Kelvin-Helmholtz instability
[3, 4], the Richtmyer-Meshkov instability [5], and the Rayleigh-Plateau instability [6] for immiscible
case. For the miscible case, the counter-superflow instability was discovered in [7–9] and the dynamical
instability was recently predicted to occur in both single-component [10] and two-component Bose-
Einstein condensates (BECs) [11]. This instability is associated with the superfluidity breaking of BECs
and, therefore, it is purely a quantum effect. In the semi-infinite space where the system of two segregated
BECs is restricted by a hard wall, its ground state is fully determined by the boundary conditions (BCs)
at hard wall. In [12, 13], with the Dirichlet BCs imposed for both condensates, the authors indicated
impeccably that the system of interest can undergo a wetting phase transition. In the present paper we
explore the unstable states associated with the BCs at a hard wall, and then we proceed to the study of
wetting phase transitions. Here, we prove that the wetting phase transition associated with the Robin BC
at hard wall is more favourable.
To this end, we make use of the double-parabola approximation (DPA) method which was proposed
in [14, 15] and was developed in [16–18]. The boundary problem we deal with is schematically given in
figure 1 where the hard wall (optical wall) is placed at z = −h′, the interface at z = L and the curve j
( j = 1, 2) represent the condensate profile φ j(z)which will be introduced in the next section. The interval
L Aj is the so-called penetration depth of condensate j [14] inside the condensate j ′ , j . Its expression
reads
L Aj = ξj/
√
K − 1
This work is licensed under a Creative Commons Attribution 4.0 International License . Further distribution
of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
13001-1
https://doi.org/10.5488/CMP.22.13001
http://www.icmp.lviv.ua/journal
http://creativecommons.org/licenses/by/4.0/
H.V. Quyet, N.V. Thu, D.T. Tam, T.H. Phat
Figure 1. (Colour online) The hard wall is located at z = −h′, the interface at z = L and the component 1
(2) occupies the region z > L (z < L). The curve j represents the condensate profile j and the interval
L Aj is the penetration depth of the condensate j.
with
ξj = ~/
√
2mj µj
being the healing length of condensate j. The parameters K , mj and µj will be cleared up in what follows.
Therefore, the interval A1 A2 is the coexistence region of two condensates.
Following [19, 20], H can be written in the form
H =
∫
V
dVHGP +
∫
W
dSHW , (1.1)
here, the Gross-Pitaevskii (GP) Hamiltonian HGP in the bulk and the wall Hamiltonian HW are given as
follows:
– The GP Hamiltonian
HGP =
2∑
j=1
ψ∗j
(
−
~2
2mj
∇2
)
ψj + V (ψ1, ψ2), (1.2a)
where
V (ψ1, ψ2) =
2∑
j=1
(
−µjψ
∗
jψj +
gj j
2
��ψj
��4) + g12 |ψ1 |
2 |ψ2 |
2, (1.2b)
here, ψj = ψj(®x), mj and µj ( j = 1, 2) are respectively the wave function, the atomic mass and the
chemical potential of each species j, with ®x = (®a, z), ®a = (x, y) to be denoted in what follows. For
inhomogeneous state, in which two condensates are separated by interface, we have K = g12√
g11g22
> 1 and
hereafter we will consider the case of all coupling constants being positive.
– The wall Hamiltonian. Usually, the wall Hamiltonian is chosen in the phenomenological form
[19, 20]
HW =
2∑
j=1
~2
2mjλj
ψ∗WjψWj , (1.3)
where ψWj ( j = 1, 2) is the wall field induced by ψj on the wall and λj is its extrapolation length,
introduced by de Gennes [21]. Next, let us pay special attention to the wall fields when the wall is flat.
For the first condensate, figure 1 indicates that the Dirichlet BC
ψ1(®a, z = −h′) = 0
is justified when h′ + L > ξ1√
K−1
or K >
ξ2
1
(h′+L)2
+ 1.
13001-2
On the finite-size effects in two segregated Bose-Einstein condensates restricted by a hard wall
Moreover, the Dirichlet BC is also consistent with the condition in the infinite space ψ1(®a,−∞) = 0
[16, 18]. As to the second condensate, we have two options:
either
ψ2 (®a, z = −h′) = 0,
or
ψ2 (®a, z = −h′) , 0.
It is easily seen that the consistency between the BCs in the semi-infinite space and in the infinite space
eliminates the first option. Indeed, if h′→∞ one obtains
lim
h′→∞
ψ2 (®a, z = −h′) = ψ2 (®a,−∞) = 0,
which contradicts the BC for this condensate in the infinite space [14, 22], ψ2 (®a,−∞) , 0.
Hence, in reality the wall Hamiltonian (1.3) is simplified to
HW =
~2
2m2λ2
ψ∗2 (®a, z = −h′)ψ2 (®a, z = −h′) (1.4)
for the flat wall.
The fact that the equilibrium values of the fields minimizes the total Hamiltonian given in (1.1), (1.2)
and (1.4) leads to the time-independent GP equations in the bulk(
−
~2
2m1
∆ − µ1 + g11 |ψ1 |
2 + g12 |ψ2 |
2
)
ψ1 = 0, (1.5a)(
−
~2
2m2
∆ − µ2 + g22 |ψ2 |
2 + g12 |ψ1 |
2
)
ψ2 = 0, (1.5b)
together with the BCs at the wall
®n∇ψWj =
1
λj
ψWj ,
where ®n is the unit vector normal to the wall and pointing inside the system. For a flat wall, the foregoing
condition turns out to be the Robin BC(
∂ψ2 (®a, z)
∂z
)
z=−h′
=
1
λ2
ψ2 (®a, z = −h′) . (1.6)
It is easily seen that (1.6) is reduced to the Neumann BC(
∂ψ2 (®a, z)
∂z
)
z=−h′
= 0, (1.7)
when λ2 tends to infinity and to the Dirichlet BC
ψ2 (®a, z = −h′) = 0, (1.8)
when λ2 → 0.
The present paper is organized as follows. In section 2, employing the DPA, we calculate the ana-
lytical solutions of the GP equations obeying the Robin BC at a hard wall. Section 3 is devoted to the
determination of the interface tension in the grand canonical ensemble (GCE) and the wetting phase
transition. The conclusion is given in section 4.
13001-3
H.V. Quyet, N.V. Thu, D.T. Tam, T.H. Phat
2. Ground state in DPA
For two segregated BECs, the translation along the Oz axis is spontaneously broken by the presence
of the domain wall-the interface, and one usually focuses on the condensate profiles that only depend on
z and for simplicity we do not change their symbols
ψj = ψj(z),
which are assumed to be real without loss of generality. In terms of ψj (z), the GP Hamiltonian (1.2)
becomes
HGP =
2∑
j=1
ψj
(
−
~2
2mj
d2
dz2
)
ψj + V(ψ1, ψ2) (2.1a)
with
V(ψ1, ψ2) =
2∑
j=1
(
− µjψ
2
j +
gj j
2
ψ4
j
)
+ g12ψ
2
1ψ
2
2 . (2.1b)
It is convenient to write all quantities in the dimensionless forms by introducing new quantities % = z/ξ1,
h = h′/ξ1, ` = L/ξ1 and φ j = ψj/
√nj, where the densities nj ( j = 1, 2) were determined in [16]. In the
dimensionless forms, the GP Hamiltonian (2.1) is rewritten as follows:
H̃ (φ1, φ2) =
2∑
j=1
φ j
(
−
ξ2
j
ξ1
d2
d%2
)
φ j + Ṽ (φ1, φ2) , (2.2a)
in which
Ṽ (φ1, φ2) =
2∑
j=1
(
−φ2
j +
φ4
j
2
)
+ Kφ2
1φ
2
2. (2.2b)
In the dimensionless form, equations (1.5) read
−
d2φ1
d%2 − φ1 + φ
3
1 + Kφ2
2φ1 = 0, (2.3a)
− ξ2 d2φ2
d%2 − φ2 + φ
3
2 + Kφ2
1φ2 = 0, (2.3b)
here, ξ = ξ2/ξ1.
Correspondingly, the wall Hamiltonian (1.4) is rewritten in the form
HW =
µ2ξ
2
2
λ2
n2φ
2
2 (−h) . (2.4)
Now, let us present in detail the formulation of the DPA [16]. The DPA method consists of three
constituents:
a) We temporarily introduce % = ` the interface location which separates the system into two regions
as plotted in figure 1.
b) The foregoing potential (2.2b) is linearized by the DPA potential VDPA(φ1, φ2) in appropriate half
spaces separated by the interface
ṼDPA(φ j, φ2) = 2(φ j − 1)2 + β2φ2
j′ −
1
2
,
{
% > `, ( j, j ′) = (1, 2),
% < `, ( j, j ′) = (2, 1),
(2.5)
where β =
√
K − 1.
c) The smooth variation and the continuity of both condensate profiles when passing the interface are
13001-4
On the finite-size effects in two segregated Bose-Einstein condensates restricted by a hard wall
assumed. Mathematically, this assumption is expressed by(dφ j (%)
d%
)
%=`−0
=
(dφ j (%)
d%
)
ρ=`+0
, (2.6)
φ j (% = ` − 0) = φ j (% = ` + 0) . (2.7)
It is important to note that the introduction of the interface position ` is merely a mathematical tactics
based on which one sets up the DPA potential. Indeed, the interface is defined as the intersection of two
condensates
φ1 (% = `; `,K, ξ) = φ2 (% = `; `,K, ξ) , (2.8)
which indicates that ` is a function of K and ξ
` = f (K, ξ) . (2.9)
Therefore, in reality, both condensates depend only upon % and the model parameters.
Taking into account (2.2a), (2.4) and (2.5), we arrive at the total Hamiltonian in the DPA, HDPA.
Minimizing it, we are led to the GP equations in DPA together with the BCs, namely:
– GP equations.
Equations in the right-hand side of the interface
−
d2φ1
d%2 + 2(φ1 − 1) = 0, (2.10a)
− ξ2 d2φ2
d%2 + β
2φ2 = 0. (2.10b)
Equations in the left-hand side of the interface
−
d2φ1
d%2 + β
2φ1 = 0, (2.10c)
− ξ2 d2φ2
d%2 + 2(φ2 − 1) = 0. (2.10d)
– The BCs that both condensate profiles fulfil are:
The Dirichlet BC at hard wall for the first condensate
φ1 (−h) = 0. (2.11)
The Robin BC at hard wall for the second condensate(
dφ2 (%)
d%
)
%=−h
= cφ2 (−h) (2.12)
with c = ξ1
λ2
.
Besides these, we still have the conditions for both condensates at infinity
φ1 (+∞) = 1, φ2 (+∞) = 0. (2.13)
The analytical expressions of the solutions of equations (2.10) which obey the conditions (2.11), (2.12)
and (2.13) are in order:
– In the RHS of the interface
φ1(%) = 1 + A1e−
√
2%, (2.14a)
φ2(%) = A2e−
β%
ξ . (2.14b)
13001-5
H.V. Quyet, N.V. Thu, D.T. Tam, T.H. Phat
– In the LHS of the interface
φ1(%) =B1e−β(2h+%)
[
−1 + e2β(h+%)
]
, (2.15a)
φ2(%) =1 + B2e
√
2%
ξ +
1
√
2 + cξ
e−
√
2(2h+%)
ξ
[
√
2B2 − c
(
B2 + e
√
2h
ξ
)
ξ
]
. (2.15b)
Substituting (2.14), (2.15) into (2.6), (2.8), we find the analytical expressions of Aj , Bj ( j = 1, 2) which
are given in appendix A.
Themost important requirement for any approximationmethod is its reliability. To check the reliability
of DPA, let us plot in figure 2 the condensate profiles corresponding to several values of ξ and at c = 1/
√
2,
h = 0: a) ξ = 1.0, K = 3; b) ξ = 0.5, K = 2.5; c) ξ = 1.5, K = 3.5. The solid lines denote the solutions
(2.14), (2.15) of the DPA equations (2.10) and the dashed lines denote the numerical solutions of the GP
equations (2.3), with the same BCs (2.11), (2.12) and (2.13).
Figure 2 tells us that:
– The graphs obtained in DPA are close to those found in the GP theory in the whole variation region
of %. This implies that the DPA is a reliable method.
– The condensates calculated respectively in DPA and in GP equations provide the same interface
position `.
Note that the foregoing results remain valid for other values of c, including c = 0 (Neumann BC) and
c = ∞ (Dirichlet BC).
– Finally, let us calculate the value c appearing in the Robin BC (2.12). At first, let us look for φ2 (−h)
0 5 10 15 20 25 30
0.0
0.2
0.4
0.6
0.8
1.0
j
(a) c = 1/
√
2, h = 0, ξ = 0.5, K = 2.5.
0 5 10 15 20 25 30
0.0
0.2
0.4
0.6
0.8
1.0
j
(b) c = 1/
√
2, h = 0, ξ = 1.0, K = 3.
0 5 10 15 20 25 30
0.0
0.2
0.4
0.6
0.8
1.0
j
(c) c = 1/
√
2, h = 0, ξ = 1.5, K = 3.5.
Figure 2. (Colour online) The condensate profiles calculated in DPA (solid lines) and in GP equations
(dashed lines). The red and blue lines correspond to the first and the second component.
13001-6
On the finite-size effects in two segregated Bose-Einstein condensates restricted by a hard wall
from (2.15)
φ2L(% = −h) =
1 + 4e−
√
2(3h+2`)
ξ (M1 + M2 + M3)(√
2 + cξ
) {√
2 cosh
[√
2(h+`)
ξ
]
+ cξ sinh
[√
2(h+`)
ξ
]}
M4
. (2.16)
The expressions of Mj , 1 6 j 6 4 are given in appendix B.
When h tends to infinity, (2.16) becomes
lim
h→∞
φ2 (−h) = φ2 (−∞) =
1
1 + cξ/2
, (2.17)
which must coincide with the BC for φ2 (%) in the infinite space [14, 16], φ2 (−∞) = 1. Hence, we get
c = 0 and then the Robin BC (2.12) turns out to be the Neumann BC.
3. Interface tension in grand canonical ensemble
The interface tension was determined in grand canonical ensemble (GCE) and canonical ensemble
(CE) by [23, 24] and [14], accordingly. The system can be viewed as having a direct contact with the bulk
reservoirs of condensates and we have µj = gj jnj . The interface tension defined in GCE are equal to 4
times their counterparts in CE [13] in the infinite space and different from 4 times in the semi-infinite
space [17]. To begin with, let us follow closely [23, 24] to start from the grand potential of the interface
in the dimensionless form calculated at the bulk coexistence of two condensates
Ω = 2P0 A
+∞∫
−h
d%
[ ∑
j=1,2
(
−φ j∂
2
%φ j
)
+ Ṽ
]
, (3.1)
where P1 = P2 = P0 = gj jn2
j/2, A is the interface area and Ṽ is given by (2.2b).
Replacing Ṽ by the DPA potential (2.5) in equation (3.1), we arrive at the grand potential in DPA
Ω = 2P0ξ1 A
∞∫
−h
d%
(
− φ∗1∂
2
%φ1 − ξ
2φ∗2∂
2
%φ2 + ṼDPA
)
. (3.2)
Then, taking into account (2.5), (2.10) and (3.2), the interface tension is determined as the excess grand
potential per unit area,
γ12 =
Ω −Ωb
A
= P0ξ1
[ ∫̀
−h
4(1 − φ2)d% +
∞∫
`
4(1 − φ1)d%
]
, (3.3)
with Ωb = −PV being the bulk grand potential.
Substituting (2.14), (2.15) into (3.3), leads to
γ̃12 =
γ12
P0ξ1
= −2
√
2A1e−
√
2` −
1
√
2 + cξ
X, (3.4)
in which
X = 2e−
√
2(2h+`)
ξ
[
−1 + e
√
2(h+`)
ξ
]
ξ − ce
√
2h
ξ ξ + B2
[
√
2 − cξ + e
√
2(h+`)
ξ
(√
2 + cξ
) ]
.
(3.4) is the interface tension with Robin BC.
13001-7
H.V. Quyet, N.V. Thu, D.T. Tam, T.H. Phat
0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
j
Figure 3. (Colour online) The profiles of condensate 2 are plotted in the interval 0 6 % 6 ` at K = 3 and
ξ = 1 and several values of c, c = 0 (blue line), c = 1 (green line) and c = ∞ (red line).
When c tends to infinity, equation (3.4) turns out to be
γ̃12 =
2
√
2β
β +
√
2 tanh [(h + `) β]
+
4
[
−1 + e
√
2(h+`)
ξ
]
√
2 − β + e
2
√
2(h+`)
ξ
(√
2 + β
) [
1 −
√
2β + e
√
2(h+`)
ξ
(
1 +
√
2β
) ]
ξ. (3.5)
Expression (3.5) totally coincides with the interface tension calculated imposing the Dirichlet condition
at a hard wall [17]. When c tends to zero, equation (3.4) reduces to the interface tension with Neumann
BC
γ̃12 =
4ξβ
2 +
√
2β coth
[√
2(h+`)
ξ
] + 2
√
2β
β +
√
2 tanh [(h + `) β]
. (3.6)
In order to better understand the physical significance of the Neumann BC it is convenient to let h = 0
and rewrite the interface tension (3.3) in the following form
γ̃12 =
γ12
P0ξ1
= 4
∫̀
0
(1 − φ2)d% + 4
∞∫
`
(1 − φ1)d% = b − 4
∫̀
0
φ2d%, (3.7)
where
b =
2
√
2β
β +
√
2 tanh(β`)
+ 4` > 0.
The expression
∫`
0 φ2d% is exactly the area of the shaded domain limited by the condensate 2 and the
interval 0 6 % 6 ` in the % axis, as is shown in figure 3. It is easily realized that the area monotonously
decreases for c changing from Neumann BC (blue line), c = 0, to Robin BC (green line), c = 1, for
example, and then to Dirichlet BC (red line), c → ∞. This implies that the interface tension (3.7) obeys
the inequality
γ12 (Neumann) < γ12 (Robin) < γ12 (Dirichlet) . (3.8)
The above inequality is illustrated in figure 4 which displays the K dependence of interface tensions
(3.4), (3.5) and (3.6), corresponding, respectively, to the Robin, Dirichlet and Neumann BCs at ξ = 1.
For other values of ξ, the above inequality (3.8) remains unchanged (illustrated in figure 5).
13001-8
On the finite-size effects in two segregated Bose-Einstein condensates restricted by a hard wall
0.0 0.2 0.4 0.6 0.8 1.0
0
2
4
6
8
1/K
1
2
P
0
1
Figure 4. (Colour online) The evolution of the GCE interface tension versus K at ξ = 1. The blue, green
and red lines correspond, respectively, to Neumann, Robin and Dirichlet BCs.
0.0 0.5 1.0 1.5 2.0
0
2
4
6
8
10
1
2
P
0
1
Figure 5. (Colour online) The evolution of the GCE interface tension versus ξ at K = 3. The blue, green
and red lines correspond, respectively, to Neumann, Robin and Dirichlet BCs.
Combining the inequality (3.8) and the figure 4, we arrive at the important statement:
a) The state derived from the Neumann BC is stable while those which are derived from the other
BCs are unstable.
b)
lim
K→1
∂γ̃12
∂K
= ∞. (3.9)
13001-9
H.V. Quyet, N.V. Thu, D.T. Tam, T.H. Phat
Figure 6. (Colour online) The evolution of ` versus K at ξ = 1.
Next, let us consider whether or not the systemwould undergo a transition from immiscible tomiscible
states when K tends to 1. To this end, we numerically compute the evolution of the function ` = f (K, ξ)
given in (2.8) versus K and the result is visualized in figure 6, which gives
lim
K→1
` (K, ξ = 1) = ∞. (3.10)
For other values of ξ, the preceding results remain unchanged. Based on (3.10), we immediately have
lim
K→1
γ̃12 = lim
K→1
2
√
2
1 +
√
2 [h + ` (K, ξ)]
= 0, (3.11)
which is valid for all interface tensions (3.4), (3.5) and (3.6).
Combining (3.9) and (3.11) proves that as K tends to 1, there occurs a first-order phase transition
from immiscible to miscible states.
To proceed to the wetting phase transition, let us remark that this phase transition could be realized
in the system which was initially in the unstable state. In our case, the wetting phase transition could
equally depart from one of the two unstable states as follows:
a) Ground state derived from the Dirichlet BC. We consider the behaviour of the corresponding
interface tension given in (3.5) in strong segregation
lim
K→∞
γ12 = 2
√
2P0ξ1 + 4
√
2P0ξ2 tanh
(
h + `
√
2ξ
)
. (3.12)
On the other hand, for strong segregation, the behaviour γ12 is expressed through the sum of two wall
tensions γWj ( j = 1, 2)
lim
K→∞
γ12 = γW1 + γW2 . (3.13)
Comparing (3.12) and (3.13), immediately gives
γW1 = 2
√
2P0ξ1 , (3.14a)
γW2 = 4
√
2P0ξ2 tanh
(
h + `
√
2ξ
)
. (3.14b)
13001-10
On the finite-size effects in two segregated Bose-Einstein condensates restricted by a hard wall
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
1
K
Figure 7. (Colour online) Wetting phase transition at h = 0. The red line (blue line) corresponds to the
Robin (Dirichlet) BC.
The wetting phase transition takes place when the Antonov rule is realized [13, 17]
γW1 = γW2 + γ12. (3.15)
The phase diagram of (3.15) is plotted in figure 7 with blue line.
b) Ground state derived from the Robin BC. Analogously, the strong segregation behaviour of the
interface tension derived from the Robin BC (3.4) reads
lim
K→∞
γ12 = 2
√
2P0ξ1
+ 4P0ξ2
2ce
2
√
2`
ξ ξ(
√
2 + cξ) + 2
√
2ce
√
2`
ξ ξ(−2 + c2ξ2) − (−1 +
√
2cξ)(−2 + c2ξ2) − Y
(
√
2 + cξ)
[
√
2 − cξ + e
2
√
2`
ξ (
√
2 + cξ)
]2 (3.16)
with
Y = −2ce
3
√
2`
ξ ξ
[
2
√
2 + cξ(4 +
√
2cξ)
]
+ e
4
√
2`
ξ
{
2 + cξ
[
4
√
2 + cξ(5 +
√
2cξ)
]}
.
Combining (3.13) and (3.16), yields
γw1 = 2
√
2P0ξ1 , (3.17a)
γw2 = 4P0ξ2
2ce
2
√
2`
ξ ξ(
√
2 + cξ) + 2
√
2ce
√
2`
ξ ξ(−2 + c2ξ2) − (−1 +
√
2cξ)(−2 + c2ξ2) − Y
(
√
2 + cξ)
[
√
2 − cξ + e
2
√
2`
ξ (
√
2 + cξ)
]2 . (3.17b)
The phase diagram of the wetting phase transition determined by the Antonov rule (3.15) is depicted
in figure 7 by red line. There is easily seen a distinction between two diagrams corresponding to two
different phase transitions. Although both phase transitions are possible, but the one associated with the
Robin BC is more favourable because it corresponds to the smaller interface tension (3.8).
13001-11
H.V. Quyet, N.V. Thu, D.T. Tam, T.H. Phat
4. Conclusion and outlook
Based on the GP equations in the DPA, we investigated the system of two segregated Bose-Einstein
condensates restricted by a hard wall. Our main results are in order:
a) Making use of the consistency between the the BCs imposed on the condensates in confined
geometry and in infinite space, we determined the BCs for both condensates at a hard wall, namely, the
Neumann BC for the second condensate and Dirichlet BC for the first condensate.
b) The analytical expressions for both condensate profiles, calculated in different BCs, are very close
to those derived from the GP equations by numerical computations. This proves the reliability of the
DPA.
c) Based on the analytical expressions of interface tension, calculated in different BCs, we showed
that
γ12 (Neumann) < γ12 (Robin) < γ12 (Dirichlet) ,
which proves that the ground state corresponds to the Neumann BC and is stable while other two BCs
lead to unstable states.
d) The above mentioned result leads to an important consequence: In our system there exist two
scenarios of wetting phase transitions which correspond to two different unstable states, but the one
associated with the Robin unstable state is more favourable because it corresponds to a smaller interface
tension.
e) As K tends to 1, the system undergoes a first-order phase transition from immiscible to miscible
states.
The results mentioned in items a), c) and d) are our major successes, they are very meaningful
because the system of binary BECs mixture is also useful for many other systems of great interest, such
as multi-component superconductors, Mott-insulators and so on [25]. Especially, it is very interesting to
extend the method presented in this work, in a straightforward manner, to two-gap superconductors [26]
whose Hamiltonians have the same structure of our system. It is expected that the application of the DPA
to these systems could provide new interesting effects.
Last but not least, it is worth to remark that our previous results were derived within the framework
of the Gross-Pitaevskii theory based on the mean field approximation, whereby the thermal and quantum
fluctuations were completely neglected [27, 28]. Although a great number of the effects related to the
Bose-Einstein condensations can be successfully explained by the Gross-Pitaevskii theory, in recent years
a more complete theory is most welcome in order to comprehend various experiments associated with
quantum fluctuations, solitons, vortices and so on [29–31].
Acknowledgements
This paper is supported by the Vietnam Ministry of Education and Training in the framework of
the Scientific Research Project under Grant B 2015-25-33. The discussion with Pham The Song is
acknowledged with thanks.
A. The analytical expressions of A j , B j ( j = 1, 2)
A1 = −
e
√
2`β
β +
√
2 tanh [(h + `) β]
, B1 =
√
2e(2h+`)β
−
√
2 + β + e2(h+`)β (√2 + β
) ,
A2 =
e
`β
ξ
[
−1 + e
√
2(h+`)
ξ
]
A21(√
2 + cξ
) (
−2 +
√
2β + A22
) , B2 =
B21 + e
√
2(2h+`)
ξ β
(
−2 + c2ξ2) − B22 − B23
B24
[
−2 +
√
2β + c
(√
2 − β
)
ξ + B25
] ,
A21 =
√
2
(
2 − c2ξ2) + e
√
2(h+`)
ξ
[
2
√
2 + cξ
(
4 +
√
2cξ
) ]
,
13001-12
On the finite-size effects in two segregated Bose-Einstein condensates restricted by a hard wall
A22 = c
(√
2 − β
)
ξ + e
2
√
2(h+`)
ξ
[
2 +
√
2β + c
(√
2 + β
)
ξ
]
,
B21 = ce
√
2h
ξ ξ
[
−2 +
√
2β + c
(√
2 − β
)
ξ
]
,
B22 = ce
√
2(3h+2`)
ξ ξ
[
2 +
√
2cξ − β
(√
2 + cξ
) ]
, B23 = e
√
2(4h+3`)
ξ β
[
2 + cξ
(
2
√
2 + cξ
) ]
,
B24 =
√
2 − cξ + e
2
√
2(h+`)
ξ
(√
2 + cξ
)
, B25 = e
2
√
2(h+`)
ξ
[
2 +
√
2β + c
(√
2 + β
)
ξ
]
.
B. The analytical expressions ofM j , 1 6 j 6 4
M1 = 2ce
√
2(3h+2`)
ξ ξ
(√
2 + cξ
)
(−2 + cβξ) + 2
√
2e
√
2(2h+`)
ξ β
(
−2 + c2ξ2),
M2 = −ce
√
2(5h+4`)
ξ ξ
[
2
(√
2 + β
)
+ 2c
(
2 +
√
2β
)
ξ + c2 (√2 + β
)
ξ2],
M3 = ce
√
2h
ξ
(√
2 − β
)
ξ
(
−2 + c2ξ2) − 2e
√
2(4h+3`)
ξ β
[
2
√
2 + cξ
(
4 +
√
2cξ
) ]
,
M4 =
√
2 (β + cξ) cosh
[√
2 (h + `)
ξ
]
+ (2 + cβξ) sinh
[√
2 (h + `)
ξ
]
.
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Ефекти скiнченного розмiру у двох вiдокремлених
конденсатах Бозе-Ейнштейна, обмежених твердою стiнкою
Х.В. Квует1, Н.В. Тху1, Д.Т. Там2, Т.Х. Пхат3
1 Фiзичний факультет, Ханойський педагогiчний унiверситет 2, Ханой, В’єтнам
2 Унiверситет Тау Бак,Шон Ла, В’єтнам
3 Комiсiя атомної енергетики В’єтнаму, Ханой, В’єтнам
Ефекти скiнченного розмiру у двох вiдокремлених конденсатах Бозе-Ейнштейна, обмежених твердою
стiнкою, дослiджено з допомогою рiвнянь Гросса-Пiтаєвского в наближеннi подвiйної параболи. Вихо-
дячи з узгодженостi мiж граничними умовами, якi накладаються на конденсати в обмеженiй геометрiї i
в повному просторi, ми знаходимо усi можливi граничнi умови разом iз вiдповiдними профiлями кон-
денсату i мiжфазовi натяги. Нами вiдкрито два ефекти скiнченного розмiру: a) основний стан, отриманий
з граничної умови Ньюмана, є стiйким, а основнi стани, отриманi з граничних умов Робiна i Дiрiхлє, є
нестiйкими, b) отже, однаковим чином проявляються два можливих переходи змочування як резуль-
тат двох нестiйких станiв. Проте, перехiд, пов’язаний з граничними умовами Робiна, виявляється бiльш
сприятливим, тому що вiн вiдповiдає меншому мiжфазовому натягу.
Ключовi слова: ефекти скiнченного розмiру, конденсати Бозе-Ейнштейна, гранична умова, наближення
подвiйної параболи
13001-14
https://doi.org/10.1103/PhysRevLett.89.067001
https://doi.org/10.1103/PhysRevA.81.013631
https://doi.org/10.1103/PhysRevA.94.063633
Introduction
Ground state in DPA
Interface tension in grand canonical ensemble
Conclusion and outlook
The analytical expressions of Aj, Bj (j =1, 2)
The analytical expressions of Mj, 1 j 4
|