Study of finite-Size effects in two segregated bose-einstein condensates with spatial restriction

MINISTRY OF SCIENCE AND TECHNOLOGY MINISTRY OF EDUCATION AND TRANING VIETNAM ATOMIC ENERGY INSTITUTE ? ? ? ? ? Pham The Song STUDY OF FINITE-SIZE EFFECTS IN TWO SEGREGATED BOSE-EINSTEIN CONDENSATES WITH SPATIAL RESTRICTION Major: Theoretic and mathematic physics Code: 62.44.01.03 DRAFT SUMMARY OF THE DOCTOR THESIS Hanoi, 2017 Abbreviations Signs English Vietnamese BEC Bose-Einstein conden-sate ngưng tụ Bose-Einstein BECs two segregated Bose-Einstein condensates ngưng tụ Bose-Einstein hai thành phần phân tách CE Canonical ensemble tập hợp chính tắc GCE Grand canonical ensem-ble tập hợp chính tắc lớn DPA Double-parabola approx-imation gần đúng parabol kép MDPA Modified double-parabola approximation gần đúng parabol kép mở rộng GP Gross-Pitaevskii Gross-Pitaevskii GPE(s) Gross-Pitaevskii equa-tion(s) (hệ) phương trình Gross- Pitaevskii TIGPEs Time-independent Gross-Pitaevskii equa- tions hệ phương trình Gross- Pitaevskii không phụ thuộc thời gian TPA Tripple-parabola approx-imation gần đúng ba parabol MFA Mean-field approxima-tion gần đúng trường trung bình 2INTRODUCTION 1. The research subject By applying the DPA method, J. O. Indekeu and his partner stud- ied surface tension and wetting phase transition in the infinity system of BECs and have gained lots of important results (Phys. Rev. A 91, 033615, (2015)). However, the influence of spatial restriction on physics properties of this system had not mentioned yet, while study the con- fined space effects of quantum systems meaningful to modern technology. That is the reasons why we choose the subject Study of finite-size ef- fects in two segregated Bose-Einstein condensates with spatial restriction. 2. History of problem The Nobel prize 2001 in physics was awarded to E. A. Conell and W. Ketterle as a result of the first successful experiment of BEC (1995), which attracted attention of many scientists in the world. Based on the MFA, the GPE (s) are found by E. P. Gross and L. P. Pitaevskii, which is the most important development in theory of BEC. Using order parameters linearization method of P. Ao and S. T. Chui, J. O. Indekeu and his partner has found DPA method. Accord- ingly, they investigated analytical solution to GPEs and compared with its numerical solution in GP theory. From here, the authors calculated in detail about surface tension and wetting phase transition of infinity BECs system. In this thesis, the influence of spatial restriction on the static prop- erties of BECs system limited by hard walls is studied by means of the MDPA applied to GP theory. 3. The aims of this thesis Study the influence of spatial restriction on the static properties of BECs in ground state. 4. The object, problems, scope of the thesis • Studying object: The system of BECs confined by one (or two) hard wall(s). • Studying problems: – Find the wave functions of condensates under Dirichlet and Robin boundary conditions at hard walls; – Investigate the surface tensions; – Study wetting phase transition and the influence of space limitation on the static properties of BECs system. 3– Propose the experimental verification the finite-size effects in BECs system. • Studying scope: System of time-independent BECs at ultra-low temperature, in GCE and CE. 5. The methods MFA method, MDPA method, numerical method. 6. The contributions of the thesis The thesis has many contributions in physics of BECs. 7. The structure of the thesis Besides introduction and conclusion, this thesis consists of 3 chap- ters. Chapter 1. Overview of Bose-Einstein condensate. Theory of BECs. Chapter 2. Interfacial tension and wetting phase transition in the system of BECs limited by one hard wall. Chapter 3. Finite-size effects of interfacial tension in the system of BECs limited by two hard walls. 4Chapter 1. Overview of Bose-Einstein condensate. Theory of BECs. 1.1. Overview of Bose-Einstein condensate 1.1.1. Bose-Einstein condensate state BEC is a state of a boson gas cooled to critical temperature Tc ap- proximately 0(K). Under such condition, a large ratio of bosons occupy the lowest energy level. Particle number at condensated state: N(ε = 0) = N −N(ε > 0) = N [ 1− ( T Tc )3/2] , here N is total particle number of system. 1.2. The Gross-Pitaevskii equation and hydrodynamic equa- tions of condensated wave function a. The Gross-Pitaevskii equation + The time-dependent GPE i~∂tψ = − ~ 2 2m ∇2ψ + U(~x)ψ +G|ψ|2ψ. + The time-independent GPE − ~ 2 2m ∇2ψ(~x) + U(~x)ψ(~x) +G|ψ(~x)|2ψ(~x) = µψ(~x). b. Healing length of condensated wave function: ξ = ~√ 2mGn . c. The hydrodynamic equations of condensated wave function + The continuity equation of condensate: ∂tn+∇(n~v) = 0, in which n = |ψ|2, ~v = i~2mn ( ψ∇ψ∗ − ψ∗∇ψ ) is velocity of condensate, ~j = i~2 ( ψ∇ψ∗ − ψ∗∇ψ ) is density of momentum. + The equations of motion: ∂t|ψ0|2 = − ~ m ∇(|ψ0|2∇φ) and ∂tφ = −1~ δE δn . 51.2. The Gross-Pitaevskii theory for infinity system of BECs + Density of Hamiltonian and interacting potential: Hˆb = Hb 2P = 2∑ j=1 φj ( − ξ 2 j ξ21 ∂2z ) φj + Vˆ(φ1, φ2), Vˆ(φ1, φ2) = V(Ψ1,Ψ2) 2P = 2∑ j=1 [ − |φj |2 + 1 2 |φj |4 ] +K|φ1|2|φ2|2. + TIGPEs (1.26): −∂2zφ1 − φ1 + |φ1|3 +K|φ2|2φ1 = 0, −ξ2∂2zφ2 − φ2 + |φ2|3 +K|φ1|2φ2 = 0. + Component 1 (component 2) occupies the region z > 0 (z < 0), wave functions of condensates satisfy the boundary condition (1.28) φ1(z → +∞) = φ2(z → −∞) = 1, φ2(z → +∞) = φ1(z → −∞) = 0. + Constant of the motion (The first integral) (1.33): (∂zφ1) 2 + ξ2(∂zφ2) 2 + 2∑ j=1 [ |φj |2 − 1 2 |φj |4 ] −K|φ2|2|φ1|2 = 1 2 . 1.3. DPA method for infinity system of BECs We assume that in the haft space z > z0 (z 6 z0) profile of conden- sates behave like |φj | = 1 + εj , |φj′ | = δj′ , { z > z0, (j, j′) = (1, 2), z 6 z0, (j, j′) = (2, 1), here z0 = 0, the dimensionless real quantities εj , δj′ are treated as small perturbations, (εj , δj′) 1. + TIGPEs in DPA: In the right hand side of interface (z > 0) −∂2zφ1 + 2(φ1 − 1) = 0, −ξ2∂2zφ2 + ηφ2 = 0; 6Fig. 1.1: Profile of condensates in the each side of interface in DPA. In the left hand side of interfac (z 6 0) −∂2zφ1 + ηφ1 = 0, −ξ2∂2zφ2 + 2(φ2 − 1) = 0, with η = K − 1. + Hamiltionian density in DPA (1.37): HˆbDPA = Hb 2P = 2∑ j=1 φj ( − ξ 2 j ξ21 ∂2z ) φj + VˆDPA(φ1, φ2), VˆDPA(φ1, φ2) = 2(|φj | − 1)2 + η|φj′ |2 − 1 2 , { z > z0, (j, j′) = (1, 2), z 6 z0, (j, j′) = (2, 1). 1.4. MDPA method for system of BECs restricted by hard walls + Density of surface Hamiltonian: HˆA = HA 2P = 2∑ j=1 ξ2j Λ˜j φAj ∗ φAj , HˆWi = HWi 2P = 2∑ j=1 ξ2j λ˜Wij φWij ∗ φWij . + Density of Hamiltonian in MDPA: HˆMDPA = ∫ V HˆbDPAdV + ∫ A HˆAdS + ∑ i ∫ Wi HˆWidS. 7+ Dimensionless form of TIGPEs in MDPA for system of BECs restricted by hard walls: In the right hand side of interface (z > z0) (1.43) −∂2zφ1 + 2(φ1 − 1) = 0, −ξ2∂2zφ2 + ηφ2 = 0; In the left hand side of interface (z 6 z0) (1.44) −∂2zφ1 + ηφ1 = 0, −ξ2∂2zφ2 + 2(φ2 − 1) = 0. + The wave functions φj(j = 1, 2) satisfy the boundary conditions (1.45) as follow: Robin condition ∂zφj |z=z0−0 = 1 Λj φj(z = z0) = ∂zφj |z=z0+0 and continuity condition of wave functions at interface φj(z = z0 − 0) = φj(z0) = φj(z = z0 + 0); Robin boundary condition ∂zφj |z=zWi = 1 λWij φj(z = zWi) or Dirichlet condition at hard walls φj(z = zWi) = 0 when the wall order parameters vanish, where Λj = Λ˜j/ξ1, λWij = λ˜ Wi j /ξ1. 1.5. Excess energy on the interface + In GCE: ∆Ω = 2APξ1 ∫ dz { − φ∗1∂2zφ1 − ξ2φ∗2∂2zφ2 + Vˆ(φ1, φ2) + 1 2 } , here A is interface area. + In CE: ∆E = PAξ1 ∫ dz(−φ∗1∂2zφ1 − ξ2φ∗2∂2zφ2). 8Summarization of chapter 1 • BEC state in the system of none-interacting bose gas was de- scribed by using the Bose–Einstein statistics distribution; • Based on the GPE(s) in MFA, the hydrodynamic equations of condensated wave functions were found; • DPA method was modified to MDPA method, which employed in system of BECs limited by hard walls; • The formulae of interacting energy between two-components of BECs, which called excess energy, were found in both GCE and CE. 9Chapter 2. Interfacial tension and wetting phase transition in the system of BECs limited by one hard wall. 2.1. Ground state Fig. 2.1: The system of BECs limited by one hard wall at z˜ = −h˜, the interface between two-components at z˜ = z˜0. 2.1.1. Ground state under Dirichlet boundary condition at hard wall + Boundary condition: φ1(z = −h) = φ2(z = −h) = 0, φ1(z → +∞) = 1, φ2(z → +∞) = 0. + The wave functions: φ1 = 1−A1e− √ 2z, φ2 = B1e − √ ηz ξ , in the region z > z0, φ1 = A2e √ η(−2h−z)(e2 √ η(h+z) − 1), φ2 = e − √ 2(2h+z) ξ (e √ 2(h+z) ξ − 1)(B2e √ 2(h+z) ξ +B2 + e √ 2h ξ ), in the region z 6 z0. 2.2. Ground state under Robin boundary condition at hard wall 10 Fig. 2.2b: The wave functions of condensates under Dirichlet boundary condition (φj(−h) = 0), K = 1.01, ξ = 1, h = 50(b). Red line and blue line correspond to MDPA and GP. K = 3, ξ = 3, h = 20∂zϕ2 z=-h = 0ϕ2 ϕ1 -20 -15 -10 -5 0 5 100.0 0.2 0.4 0.6 0.8 1.0 z Fig. 2.4b: The wave functions of condensates under Robin boundary condition (λW2  1), K = 3, h = 20; ξ = 3. Solid line and dashed line correspond to GP and MDPA. + Boundary condition: φ1(z = −h) = 0, ∂zφ2|z=−h = 0, φ1(z → +∞) = 1, φ2(z → +∞) = 0. 11 + The wave functions: φ1 = 1−A1e− √ 2z, φ2 = B1e − √ ηz ξ , in the region z > z0, φ1 = A2e √ η(−2h−z)(e2 √ η(h+z) − 1), φ2 = 1−B2(e− √ 2(2h+z) ξ + e √ 2z ξ ), in the region z 6 z0. Profile of condensates in MDPA is close to profile of condensates in GP theory. The interface position depends strongly on the hard wall position. 2.2. Interfacial tension and wetting phase transition in GCE 2.2.1. Under Dirichlet boundary condition at hard wall γ˜12 = 4 +∞∫ −h dz{−φ∗1∂2zφ1 − ξ2φ∗2∂2zφ2} = 4 +∞∫ −h dz{(∂zφ1)2 + ξ2(∂zφ2)2}. + Wall surface tension: γ˜1W = 2 √ 2− 4 √ 2(h+ z0)( e √ 2(h+z0) ξ + 1 )2 − 4 √ 2(h+ z0) e √ 2(h+z0) ξ + 1 , γ˜2W = ( 2 √ 2− 4 e √ 2(h+z0) ξ + 1 ) ξ. + Antonov rule: γ˜1W = γ˜2W + γ˜12. Figs. 2.5 and 2.9 shown that under Dirichlet boundary condition the influence of hard wall position on the interfacial tension and wetting phase diagram is insignificant. 2.2.2. Under Robin boundary condition at hard wall γ˜12 = 2(I1 + ξ 2I2 + C1 + C2), 12 Fig. 2.5: (GCE) The evolution of interfacial tension versus 1/K at h = 0(red line) and h→ +∞(blue line). Fig. 2.9: (GCE) Wetting phase diagram at h = 0 (red line) and h→ +∞(blue line). where I1 = +∞∫ −h (−φ∗1∂2zφ1)dz,C1 = z0∫ −h η|φ1|2dz + +∞∫ z0 2(|φ1| − 1)2dz, I2 = +∞∫ −h (−φ∗2∂2zφ2)dz,C2 = z0∫ −h 2(|φ2| − 1)2dz + +∞∫ z0 η|φ2|2dz. 13 ∂zϕ2 -h = 0 = 0, ξ = 1 h→+∞ h = 20 0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 4 5 6 1 K γ 12 Pξ 1 Fig. 2.11: (GCE) Spatial restriction effect of interfacial tension at ξ = 1. ∂zϕ2 z=-h = 0 partial wetting region complete wetting region ζ→+∞ζ = 20 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 ξ Fig. 2.12: (GCE) Wetting phase diagram at ζ = (h+ z0) = 20 and ζ = (h+ z0)→ +∞. + Wall surface tension: γ˜1W = 4(h+ z0)√ 2(h+ z0) + 1 , γ˜2W = 2 √ 2 tanh (√2(h+ z0) ξ ) ξ. The vanishing of the influence of hard wall position on the interface tension and wetting phase transition shows in figs. 2.11 and 2.12. 14 (a) ϕ j(-h) = 0, ξ = 5, n21 = 1ζ = 100ζ = 20 0.0 0.2 0.4 0.6 0.8 1.0 0 10 20 30 40 1/K Γ 12 N 1 σ 10 (b) ∂zϕ2 z=-h= 0, ξ = 10, n21 = 1ζ = 100ζ = 20 0.0 0.2 0.4 0.6 0.8 1.0 0 5 10 15 1/K Γ 12 N 1 σ 10 Fig. 2.14: (CE) The evolution of interfacial tension versus 1/K at n21 = 1, ζ = (h+ z0) = 20, 100; φj(−h) = 0, ξ = 5(a); ∂zφ2|z=−h = 0, ξ = 10(b). 15 (a) ϕ j(-h)= 0, ζ = 20, n21=1ξ = 1ξ = 6ξ = 11 0.0 0.2 0.4 0.6 0.8 1.0 0 10 20 30 40 1/K Γ 12 N 1 σ 10 (b) ∂zϕ2 z=-h= 0, ζ = 20, n21=1ξ = 5ξ = 10ξ = 15 0.0 0.2 0.4 0.6 0.8 1.0 0 5 10 15 20 1/K Γ 12 N 1 σ 10 Fig. 2.15: (CE) The evolution of interfacial tension versus 1/K at n21 = 1, ζ = (h+ z0) = 20; φj(−h) = 0, ξ = 1, 6, 11(a); ∂zφ2|z=−h = 0, ξ = 5, 10, 15(b). 16 (a) ϕ j(-h)= 0, ζ = 20, ξ=1 n21 = 0.5 n21 = 1.0 n21 = 2.0 0.0 0.2 0.4 0.6 0.8 1.0 0 5 10 15 20 1/K Γ 12 N 1 σ 10 (b) ∂zϕ2 z=-h= 0, ζ = 20, ξ=1 n21 = 0.5 n21 = 1.0 n21 = 2.0 0.0 0.2 0.4 0.6 0.8 1.0 0 5 10 15 1/K Γ 12 N 1 σ 10 Fig 2.16: The evolution of interfacial tension versus 1/K at ξ = 1, ζ = (h+ z0) = 20 với n21 = 0.5, 1.0, 2.0; φj(−h) = 0(a), ∂zφ2|z=−h = 0(b). 17 2.3. Interfacial tension in CE Γ˜12 = ∆E APξ1 = (I1 + ξ 2I2), or Γ˜12 = 1 N1 [ I1 + n 3/2 21 ξI2 ] , in which σ10 = g11n102 , n21 = n20 n10 ,N1 = +∞∫ −h φ21dz. The evolution of interfacial tension versus 1/K at each other position of hard wall with different values of ξ and n21 plotted in figs. 2.14, 2.15, 2.16, which shown that spatial restriction effect of interfacial tension is also weakly in CE. Summarization of chapter 2 • Solution to TIGPEs in MDPA is very close to its numerical solu- tion in GP theory. • In the system of BECs limited by one hard wall, the interface position depends strongly on the position of hard wall. • Under Dirichlet boundary condition, the h-dependence of interfa- cial tension is insignificant, which vanishes under Robin boundary condition. • Wetting phase diagram is also weakly depending on hard wall position. 18 Chapter 3. Finite-size effects of interfacial tension in the system of BECs limited by two hard walls. 3.1. Ground state Fig. 3.1: The system of BECs limited by two hard walls at z˜ = ±h˜, interface between two-components at z˜ = z˜0. 3.1.1. Under Dirichlet boundary condition at hard walls Fig. 3.2b: The profile of condensates under Dirichlet boundary condition (φj(±h) = 0) in MDPA (solid line) and in GP theory (dashed line), at K = 3 and ξ = 1. 19 + Boundary condition (3.1): φj(z = ±h) = 0 với j = (1, 2). + The wave functions of condensates: φ1 = e −√2z(e √ 2z − e √ 2h)(A1(e √ 2h + e √ 2z) + 1), φ2 = B1(e √ ηz ξ − e √ η(2h−z) ξ ), in the right hand side of interface (z > z0), φ1 = A2e √ η(−(2h+z))(e2 √ η(h+z) − 1), φ2 = e − √ 2(2h+z) ξ (e √ 2(h+z) ξ − 1)(B2e √ 2(h+z) ξ +B2 + e √ 2h ξ ), in the left hand side of interface (z 6 z0). 3.1.2. Under Robin boundary condition at hard walls K = 1.2, ξ = 1, h = 10 ∂zϕ2 z=-h = 0 ϕ2 ϕ1 -10 -5 0 5 100.0 0.2 0.4 0.6 0.8 1.0 z Fig. 3.3b: The profile of condensates under Robin boundary condition (λW11 , λ W2 2  1) in MDPA (dashed line) and in GP theory (solid line). + Boundary condition (3.4): φ1(z = −h) = 0, ∂zφ2|z=−h = 0, ∂zφ1|z=+h = 0, φ2(z = +h) = 0. 20 + The wave functions of condensates: φ1 = 1−A1(e √ 2z + e √ 2(2h−z)), φ2 = B1e − √ ηz ξ (e 2 √ ηz ξ − e 2 √ ηh ξ ), in the right hand side of interface (z > z0), φ1 = A2e √ η(−2h−z)(e2 √ η(h+z) − 1), φ2 = 1−B2(e √ 2z ξ + e− √ 2(2h+z) ξ ), in the left hand side of interface (z 6 z0). 3.2. Finite-size effects of interfacial tension under Dirichlet boundary condition at hard walls. Casimir-like force 3.2.1. In GCE Fig. 3.4b: (GCE) Finite-size effect of interfacial tension at ξ = 3(b) with different values of K = 1(solid line), 1.1 (dashed line), 3 (dotted line). γ˜12 = 4(I1 + ξ 2I2) + 2[−(ap + am − 4)h+ (ap − am)z0], in which 21 I1 = +h∫ −h (∂zφ1) 2dz,I2 = +h∫ −h (∂zφ2) 2dz, ap = [(∂zφ1) 2 + ξ2(∂zφ2) 2] ∣∣∣ z=+h = 2(2A1e √ 2h + 1)2 + 4ηB21e 2h √ η/ξ, am = [(∂zφ1) 2 + ξ2(∂zφ2) 2] ∣∣∣ z=−h = 4A22ηe −2h√η + 2(2B2e− √ 2h/ξ + 1)2. In fig. 3.4, interfacial tension depends strongly on the distance be- tween two hard walls when h 6 ξ, this dependence decrease as h > ξ, interfacial tension tends to constant when h ξ. 3.2.2. In CE Fig. 3.5b: (CE) The h-dependence of interfacial tension at ξ = 3 and K = 3. Γ12 = ∆E A = Pξ1 +h∫ −h dz(−φ1∂2zφ1 − ξ2φ2∂2zφ2) = Pξ1 +h∫ −h dz[(∂zφ1) 2 + ξ2(∂zφ2) 2] = Pξ1(I1 + ξ 2I2). 22 In the case m1 = m2 = m, g11 = g22 = g, σ10 = σ20 = σ0: Γ˜12 = 1 N 31 [ I1 + (n20 n10 )3/2 I2 ] , where Γ˜12 = Γ12/σ0N31 . Fig. 3.6b: The evolution of interfacial tension versus 1/K at ξ = 3 with h→ +∞(solid line), h = 12 (dotted line), h = 8(dashed line). Blue and red correspond to GCE and CE, respectively. Fig. 3.5 shown that the inter-species interaction leads to the growth of pressure when the volume of system is reduced. Inversely, when the volume expands the decrease of pressure leads to the decrease of inter- facial tension. + The formula (3.24): γ12 Γ12 = 4 + 2[(ap − am)z0 − (ap + am − 4)h] I1 + ξ2I2 . The ratio of γ12Γ12 not exactly equal to 4, this result is different from which was found in semi-infinity system (chapter 2) and infinity system (Phys. Rev. A 91, 013626 (2015)). This phenomenon shown clear in fig. 3.6. In addition, the interfacial tension do not vanishing at de-mixing limit. In the limit of h→∞, γ12Γ12 → 4. 23 Fig. 3.7b: (GCE) The h-dependence of Casimir-like force acts on a unit area of the hard walls at ξ = 3 with K = 1(solid line), K = 1.1 (dashed line), K = 3(dotted line). 3.2.3. Casimir-like force Casimir-like force acts on a unit of the hard walls F˜GCE = −1 2 ∂hγ˜12, F˜CE = −1 2 ∂hΓ˜12. 3.3.Finite-size effects of interfacial tension under Robin bound- ary condition at hard walls 3.3.1. In GCE γ˜12 = 4 +h∫ −h [(∂zφ1) 2 + ξ2(∂zφ2) 2]dz = 4(I1 + ξ 2I2). The interfacial tension only vanishing at de-mixing limit when h ξ. 3.3.2. In CE + The ratio of γ˜12/Γ˜12 = 4. 24 ξ = 1, ϕ1(-h) = ϕ2(h)= 0∂zϕ1 z=h = ∂zϕ2 z=-h = 0 K=3.0 K=1.1 K=1.0 0 2 4 6 8 10 12 0.0 0.5 1.0 1.5 2.0 2.5 3.0 h γ 12 Pξ 1 Fig. 3.8: (GCE) Finite-size effect of interfacial tension at ξ = 1, with different values of K = 1.0 (solid line), 1.1 (dashed line), 3 (dot-dashed line). ξ=1 h→+∞ h=10 h = 7 h=4 ϕ1(h) = ϕ2(h)= 0 zϕ1 z
h = zϕ2 z
h = 0 0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 4 5 6 1 K γ 1 2 P ξ 1 0.95 0.96 0.97 0.98 0.99 1.00 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Fig. 3.9: (GCE) The evolution of interfacial tension versus 1/K at ξ = 1 with h = 4, 7, 10 and h→∞. 25 n21 = 1, ξ = 1∂zϕ1 z=h = ∂zϕ2 z=-h = 0 K=3 K=2 K=1.1 0 2 4 6 8 10 12 0.00 0.05 0.10 0.15 0.20 0.25 h Γ 12 N 13 σ 10 Fig 3.11: (CE) The h-dependence of interfacial tension at n21 = 1, ξ = 1 with K = 1.1, 2.0, 3.0. Summarization of chapter 3 • The ground state of BECs system limited by two hard walls un- der both Dirichlet and Robin boundary conditions were found by MDPA method, which very close to ground state in GP theory. • The appearance of hard walls leads to the interface position de- pends strongly on the healing length of condensated wave func- tions (ξ). The interface at z = 0 when condensated state of two- components is symmetric or h→∞. • Finite-size effect of interfacial tension is only significant when h ∼ ξ. • We found a new type of forces acting on two hard walls, which behave like Casimir force, so called Casimir-like force. 26 CONCLUSION In this thesis, we have investigated systematically the static proper- ties in the system of BECs limited by hard walls. Among many results obtained, some important results listed as follow: 1. The MDPA method was proposed, which employed in many spa- tial profiles, while the DPA method only employed in infinity or haft-infinity system of BECs. 2. In the system of BECs limited by one hard wall, the interfacial tension and the wetting phase diagram depend weakly on the spa- tial restriction. In the system of BECs limited by two hard walls, the interfacial tensions and their finite-size effects were found to- gether with a new type of longrange forces acting on two walls in both GCE and CE. Some points of view: 1. The influence of temperature on the static properties of condensed interface and wetting phase transition. 2. The fluctuations of interface, the deformations of condensed sur- face. 27 LIST OF PAPERS RELATE WITH THE THESIS • Le Viet Hoa, To Manh Kien and Pham The Song (2010), Study the phase transition in binary mixtures, Journal of science of HNUE Natural.Sci. Vol. 55, 6, p.3-13. • N. V. Thu, T. H. Phat, P. T. Song (2016),Wetting phase transition of two segregated Bose-Einstein condensates restricted by one hard wall, Phys. Lett. A 380, 1487. • N. V. Thu, T. H. Phat, P. T. Song (2017), Finite-size effects of surface tension in two segregated BECs confined by two hard walls, J. Low Temp. Phys. Vol. 186, 127.