Hydrodynamic instabilities of two-component Bose-Einstein condensates

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. REFERENCES [1] E. Timmermans, 1998. Phys. Rev. Lett. 81, 5718. [2] P. Ao and S.T.Chui, 1998. Phys. Rev. A58, 4836. [3] C. J. Myatt, E. A. Burt, R. W. Ghrist, E. A. Cornell and C. E. 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