Hydrodynamic instabilities of two component bose Einstein condensates

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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.

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Nội dung Text: Hydrodynamic instabilities of two component bose Einstein condensates

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 http://stdb.hnue.edu.vn HYDRODYNAMIC INSTABILITIES OF TWO-COMPONENT BOSE-EINSTEIN CONDENSATES Le Viet Hoa1 , Nguyen Tuan Anh2 , Nguyen Chinh Cuong1 and Dang Thi Minh Hue3 1 Faculty of Physics, Hanoi National University of Education 2 Faculty of Energy Technology, Electric Power University 3 Water 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 ∗ ∂ ∇2 ∗ ∂ ∇2 £ = φ −i − φ + ψ −i − ψ ∂t 2mφ ∂t 2mψ λ1 λ2 λ − µ1 φ∗ φ + (φ∗ φ)2 − µ2 ψ ∗ ψ + (ψ ∗ ψ)2 + (φ∗ φ)(ψ ∗ ψ) (2.1) 2 2 2 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 ψ02 correspond to the local minimum of the potential. They fulfill λ1 3 λ −µ1 φ0 + φ + φ0 ψ02 = 0 2 0 4 λ2 λ −µ2 ψ0 + ψ03 + φ20 ψ0 = 0 (2.2) 2 4 yielding φ20 2µ1 λ2 − µ2 λ ψ02 2µ2 λ1 − µ1 λ =2 ; =2 . (2.3) 2 4λ1 λ2 − λ2 2 4λ1 λ2 − λ2 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 1 1 φ = √ (φ0 + φ1 + iφ2 ), ψ = √ (ψ0 + ψ1 + iψ2 ). (2.4) 2 2 Inserting (2.4) into (2.1) we get, among others, the interaction Lagrangian λ1 λ λ1 £int = φ0 φ1 + ψ0 ψ1 (φ21 + φ22 ) + (φ21 + φ22 )2 2 4 8 λ2 λ λ2 + ψ0 ψ1 + φ0 φ1 (ψ12 + ψ22 ) + (ψ12 + ψ22 )2 2 4 8 λ 2 + (φ + φ22 )(ψ12 + ψ22 ), 8 1 and the inverse propagators in the tree approximation   ~k 2 3λ1 2 λ 2 −1  2mφ − µ1 + 2 φ0 + 4 ψ0 −ω  D0 (k) = ~k 2 λ1 2 ω 2mφ − µ1 + 2 φ0 + λ4 ψ02   ~k 2 −1 2m − µ2 + 3λ2 2 ψ02 + λ4 φ20 −ω G0 (k) =  ψ ~k 2 λ2 2 . (2.5) ω 2mψ − µ2 + 2 ψ0 + λ4 φ20 122
Hydrodynamic instabilities of two-component Bose-Einstein condensates From (2.3) and (2.5) it follows that v ! u u ~k2 ~k2 Eφ = +t + λ1 φ20 2mφ 2mφ v ! u u ~k2 ~k2 Eψ = +t + λ2 ψ02 , (2.6) 2mψ 2mψ For small momenta Eqs. (2.6) reduce to s s λ1 φ20 λ2 ψ02 Eφ ≈ ±k ; Eψ ≈ ±k (2.7) 2mφ 2mψ associating with Goldstone bosons due to U (1) × U (1) breaking. Assuming the ansatz   ~k 2 2m + M1 −ω D −1 =  φ ~k 2  ω 2mφ + M3   ~k 2 −1 2m + M2 −ω G =  ψ ~k 2  ω 2mψ + M4 for inverse propagators D, G and following closely [10] we arrive at the CJT effective potential VβCJT (φ0 , ψ0 , D, G) at finite temperature in the HF approximation µ1 λ1 µ2 λ2 λ V˜βCJT (φ0 , ψ0 , D, G) = − φ20 + φ40 − ψ02 + ψ04 + φ20 ψ02 Z 2 8 2 8 8 1 −1 −1 −1 −1 + tr ln D (k) + ln G (k) + [D0 (k; φ0 , ψ0 )D] + [G0 (k; φ0 , ψ0 )G] − 211 2 β Z 2 Z 2 Z Z λ1 λ1 3λ1 + D11 (k) + D22 (k) + D11 (k) D22 (k) 8 β 8 β 4 β β Z 2 Z 2 Z Z λ2 λ2 3λ2 + G11 (k) + G22 (k) + G11 (k) G22 (k) 8 β 8 β 4 β β Z Z Z Z λ λ + D11 (k) G11 (k) + D11 (k) G22 (k) 8 β β 8 β β Z Z Z Z λ λ + D22 (k) G11 (k) + D22 (k) G22 (k) . (2.8) 8 β β 8 β β From (2.8) we deduce immediately the following equations: - The Schwinger-Dyson (SD) equations D −1 = D0−1 (k) + Σφ ; G−1 = G−1 ψ 0 (k) + Σ , (2.9) 123
Le Viet Hoa, Nguyen Tuan Anh, Nguyen Chinh Cuong and Dang Thi Minh Hue in which ! ! Σφ1 0 Σ ψ 0 Σφ = ; Σψ = 1 , 0 Σφ2 0 Σψ 2 Z Z Z Z λ1 3λ1 λ λ Σφ1 = D11 (k) + D22 (k) + G11 (k) + G22 (k) 2 β 2 β 4 β 4 β Z Z Z Z 3λ1 λ1 λ λ Σφ2 = D11 (k) + D22 (k) + G11 (k) + G22 (k) 2 β 2 β 4 β 4 β Z Z Z Z λ2 3λ2 λ λ Σψ 1 = G11 (k) + G22 (k) + D11 (k) + D22 (k) 2 β 2 β 4 β 4 β Z Z Z Z 3λ2 λ2 λ λ Σψ 2 = G11 (k) + G22 (k) + D11 (k) + D22 (k) (2.10) 2 β 2 β 4 β 4 β - The gap equations λ1 2 λ 2 −µ1 + φ + ψ + Σφ2 = 0 2 0 4 0 λ λ2 −µ2 + φ20 + ψ02 + Σψ 2 = 0. (2.11) 4 2 Hence A B φ20 = 4 2 ; ψ02 = 4 . (2.12) 4λ1 λ2 − λ 4λ1 λ2 − λ2 with µ 1 λ2 − µ A = 2¯ ¯2 λ µ 2 λ1 − µ B = 2¯ ¯1 λ ¯1 = µ1 − Σφ2 µ ¯ 2 = µ2 − Σ ψ µ 2. (2.13) Combining (2.11) and (2.9) we get the forms for inverse propagators   ~k 2 + M −ω D −1 =  2mφ 1 ~k 2  ; M1 = −µ1 + 3λ1 φ20 + λ ψ02 + Σφ1 , ω 2 4 2mφ   ~k 2 2mψ + M2 −ω G −1 =  ~k 2  ; M2 = −µ2 + 3λ2 ψ02 + λ φ20 + Σψ1. (2.14) ω 2 4 2mψ It is obvious that the dispersion relations related to (2.14) read v ! s u u ~k2 ~k2 M1 Eφ = t + M1 −→ k as k → 0 2mφ 2mφ 2mφ v ! s u u ~k2 ~k2 M2 Eψ = t + M2 −→ k as k → 0 2mψ 2mφ 2mψ 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 s s M1 M2 Cφ = , Cψ = . (2.15) 2mφ 2mψ Ultimately the one-particle-irreducible effective potential V˜βCJT (φ0 , ψ0 ) is µ1 λ1 µ2 λ2 λ V˜βCJT (φ0 , ψ0 ) = − φ20 + φ40 − ψ02 + ψ04 + φ20 ψ02 Z 2 8 2 8 8 1 + tr ln D −1 (k) + ln G−1 (k) 2 β Z 2 Z 2 Z Z λ1 λ1 3λ1 − D11 (k) − D22 (k) − D11 (k) D22 (k) 8 β 8 β 4 β β Z 2 Z 2 Z Z λ2 λ2 3λ2 − G11 (k) − G22 (k) − G11 (k) G22 (k) 8 β 8 β 4 β β Z Z Z Z λ λ − D11 (k) G11 (k) − D11 (k) G22 (k) 8 β β 8 β β Z Z Z Z λ λ − D22 (k) G11 (k) − D22 (k) G22 (k) (2.16) 8 β β 8 β β 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˜βCJT (φ0 , ψ0 , D, G)|at minimum (2.17) from which the total particle densities are determined ∂P ρi = , i = 1, 2. ∂µi Taking into account the fact that derivatives of V˜βCJT (φ0 , ψ0 , D, G) with respect to its arguments vanish at minimum we get Z Z ∂VβCJT φ20 1 1 ρ1 = − = + D11 + D22 ∂µ1 2 2 β 2 β Z Z ∂VβCJT ψ2 1 1 ρ2 = − = 0 + G11 + G22 . (2.18) ∂µ2 2 2 β 2 β 125
Le Viet Hoa, Nguyen Tuan Anh, Nguyen Chinh Cuong and Dang Thi Minh Hue Hence, the gap equations (2.11) become Z λ µ1 = λ1 ρ1 + ρ2 + λ1 D11 2 β Z λ µ2 = λ2 ρ2 + ρ1 + λ2 G11 (2.19) 2 β and the particle densities in condensates are Z Z φ20 1 1 = ρ1 − D11 − D22 2 2 β 2 β Z Z ψ02 1 1 = ρ2 − G11 − G22 . (2.20) 2 2 β 2 β Combining Eqs. (2.10), (2.17) and (2.18) together produces the following expression for the pressure Z ˜ λ1 2 λ2 2 λ 1 P = −V = ρ + ρ2 + ρ1 ρ2 − tr{ln D −1 (k) + ln G−1 (k)} − 2 1 2 2 2 β Z 2 Z 2 Z Z λ1 λ2 − D11 − G11 + λ1 ρ1 D11 + λ2 ρ2 G11 . (2.21) 2 β 2 β β β 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 = 5.10−12 eV −2 ; λ2 = 0, 4.10−12 eV −2 ; λ = 4.10−12 eV −2 ; mφ = mψ = 80GeV ; µ1 = 1, 4.10−11 eV. 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, 4λ1 λ2 − 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, 8.10−11 eV , 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, 5.10−12 eV . It is clear that M1 is negative at every temperature, and from (2.15) the speed Cφ is an imaginary quantity, i.e. φ0 is always zero. Whereas T < Tc2 , then M2 > 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, 8.10−11 eV < µ2 < 5.10−11 eV , 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 = 3.10−12 eV . 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. Wieman, 1997. Phys. Rev. Lett. 78, 586. [4] D. M. Stamper-Kurn, M. R. Andrews, A. P. Chikkatur, S. Inouye, H.-J. Miesner, J., 1998. Stenger and W.Ketterle, Phys. Rev. Lett. 80, 2027. [5] H. Takeuchi, N. Suzuki, K. Kasamatsu, H. Saito and M. Tsubota, 2010. Phys. Rev. B81, 094517. [6] K. Sasaki, N. Suzuki and H. Saito, 2011. Phys.Rev. A83, 053606. [7] H. -J. Miesner, D. M. Stemper-Kurn, J. Stenger, S. Inouye, A. P. Chikkatur and W. Ketterle, 1999. Phys. Rev. Lett. 82, 2228. [8] M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman and E. A. Cornell, 1999. Phys. Rev. Lett. 83, 2498. [9] J. M. Cornwall, R. Jackiw and E. Tomboulis, 1974. Phys. Rev. D10, 2428. G. Amelino-Camelia and S. Y. Pi, 1993. Phys. Rev. D47, 2356. [10] Tran Huu Phat, Nguyen Tuan Anh, and Le Viet Hoa, 2007. Eur. Phys. J. A19, 359; Tran Huu Phat, Le Viet Hoa, Nguyen Tuan Anh, and Nguyen Van Long, 2004. Phys. Rev. D76, 125027; Tran Huu Phat, Nguyen Van Long, Nguyen Tuan Anh, and Le Viet Hoa, 2008. Kaon condensation in linear sigma model, Phys. Rev. D78, 105016. [11] L. Landau, and E. M. Lifshitz, 1969. Statistical Physics. Pergamon Press. [12] Tran Huu Phat, Le Viet Hoa, Nguyen Tuan Anh, and Nguyen Van Long, 2009. Annals of Physics 324, 2074. 128