The interface properties of two-component Bose - Einstein condensates

JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2015-0037 Mathematical and Physical Sci., 2015, Vol. 60, No. 7, pp. 88-93 This paper is available online at THE INTERFACE PROPERTIES OF TWO-COMPONENT BOSE-EINSTEIN CONDENSATES Le Viet Hoa1, Nguyen Tuan Anh2, Le Huy Son3 and Nguyen Van Hop1 1Faculty of Physics, Hanoi National University of Education 2 Faculty of Energy Technology, Electric Power University 3Faculty of Physics, Hanoi Metropolitan University Abstract. The analytical expressions for the wave functions of two-component Bose-Einstein Condensates are derived by means of the Gross-Pitaevskii equations in linear approximation. Based on the Bernoulli equation the shape of the interface and the dispersion relations for both phonon and ripplonare studied. It is shown that the numbers of Nambu-Goldstone modes in a system obey a modified counting rule which states that the number of type I plus twice the number of type II Nambu-Goldstone modes are greater than or equal to the generators of spontaneously broken symmetries. Here the type I (type II) consists of Nambu-Goldstone modes with linear (fractional) dispersion relation. Keywords: Bose-Einstein Condensates, shape of interface, Nambu-Goldstone modes. 1. Introduction The theoretical studies of two immiscible Bose-Einstein Condensates (BECs) [1, 2] and the experimental realizations of such systems [3-5] have allowed us to explore many interesting physical properties of BECs, in which the superfluid dynamics of interface between two segregated BECs has attracted special attention. Following this trend, in recent years one has focused on considerations of hydrodynamic instabilities at the interface of two BECs, such as the Kelvin-Helmholtz instability, the Rayleigh-Taylor instability and the Richtmayer-Meshkov instability [6, 7]. Combining the hydrodynamic approach and the Bogoliubov- de Gennes method these considerations confirmed that the foregoing instabilities of fluid in classic hydrodynamics are also to take place for two segregated BECs. The present paper deals with two-immiscible BECs with planar configuration, in which the first (second) component occupies the space left (right) to the z = 0 plane and their interface is represented by a surface z = η(x, y, t). We neglect the interface thickness (Figure 1). Our main aim is to investigate the capillary waves at this interface focusing on the expressions for the wave functions and the shape of the interface. Received May 13, 2015. Accepted October 5, 2015. Contact Le Viet Hoa, e-mail address: hoalv@hnue.edu.vn 88 The interface properties of two-component bose-einstein condensates The shape of interface 2. Content 2.1. Wave functions of condensates Let us start with the Lagrangian density £ = P1 + P2 − g12|ψ1|2|ψ2|2, (2.1) where Pj = i~ ψ ∗ j ∂ψj ∂t − ~ 2 2mj |∇ψj |2 − gjj 2 |ψj |4, (2.2) ψj(j = 1, 2) are wave functions, mj are atomic masses and the interaction coupling constants are defined as gjk = 2π~ 2ajk ( m−1j +m −1 k ) with ajk being the s-wave scattering length between the atoms in components j and k. In the following we assume that g212 > g11g22 implying that the two components are immiscible. From Lagrangian (2.1) the Gross-Pitaevskii (GP) equation is deduced i~ ∂ψ1 ∂t = ( − ~ 2 2m1 ∇2 + g11|ψ1|2 + g12|ψ2|2 ) ψ1, i~ ∂ψ2 ∂t = ( − ~ 2 2m2 ∇2 + g22|ψ2|2 + g12|ψ1|2 ) ψ2. (2.3) It is clear that (2.1) and (2.3) generally allow the excitation of two independent phonons corresponding to the spontaneous breaking of symmetry U(1) × U(1). For immiscible condensates the translational invariance is also spontaneously broken by the presence of a domain wall-interface. This leads to the appearance of a new gapless Nambu-Goldstone (NG) mode: the ripplon mode displaying the ripple waves propagating along the interface. 89 Le Viet Hoa, Nguyen Tuan Anh, Le Huy Son and Nguyen Van Hop To explore Eq. (2.3), let us first determine the wave functions in some approximation. We follow closely [8] to write the wave function as ψj(x, y, z, t) = √ nj(x, y, x, t)e i φj(x, y, z, t). (2.4) in which nj and φj are real functions. Next the density and phase are separated as follows nj(x, y, z, t) = nj 0 + δnj(x, y, z, t), (2.5a) φj(x, y, z, t) = −gjjnj 0 ~ t+ δφj(x, y, z, t). (2.5b) Inserting (2.4), (2.5) into (2.3) and taking only the first order of δnj , δφj we get approximately the equations −→∇−→vj = 0, (2.6a) ~ ∂ ∂t (δφj) + gjj δnj = 0, (2.6b) assuming that the relative density changes following a fluid particle are small compared to the velocity gradients. The velocity ~vj in (2.6a) is defined as −→vj = ~ mj −→∇δφj . (2.7) Eq. (2.6a) says that in the approximation under consideration the condensates are subject to incompressible flow. Next we adopt the ansatz δφj = ϕj(z)χj(σ), σ = k1x+ k2y − ωt, (2.8) which combines with (2.6a) and (2.7) providing d2 dz2 ϕj(z)− k2ϕj(z) = 0, (2.9) ( ∂2 ∂x2 + ∂2 ∂y2 + k2 ) χj(σ) = 0, (2.10) k2 = k21 + k 2 2 Imposing the conditions ϕ1(z)→ 0 as z → −∞ ϕ1(0) < +∞, 90 The interface properties of two-component bose-einstein condensates and ϕ2(z)→ 0 as z → +∞ ϕ2(0) < +∞, we get from (2.9) ϕj(z) = exp[−(−1)jkz], (2.11) and Eq (2.10) yields the general solutions χj(σ) = Aj cos σ +Bj sinσ. (2.12) Combining (2.11) and (2.12) leads to δφj = (Aj cos σ +Bj sinσ) exp[−(−1)jkz], (2.13) in which Aj and Bj are small parameters. For simplicity we restrict ourselves to the cases δφj = Aj exp[−(−1)jkz] cos σ. (2.14) Substituting (2.14) into (2.6b) we find gjj δnj = −Aj~ ω exp[−(−1)jkz] sin σ. (2.15) Taking into account (2.4), (2.5), (2.14) and (2.15) we are led to the expression for the wave functions in the first-order approximation ψ˜j = √ nj0 − Aj~ ω gjj exp[−(−1)jkz] sin σ.ei ( − gjjnjo ~ t+Aj exp[−(−1)jkz]. cos σ ) . (2.16) Although the coupling constant of the inter-species interaction g12 is not present in the preceding formulas it is not possible to consider the two BEC condensates as being independent because the inter-species interaction is implicitly included in their wave functions which have common parameters. The wave functions derived from foregoing equations describe the incompressible fluids. Any modification of these functions will leads to compressible fluid. 2.2. The shape of the interface and the dispersion relations In order to determine the shape of interface z = η(x, y, t) let us remember that the Lagrangian L = ∫ d~r£ is approximated by L = ∫ dx dy (∫ η −∞ dz P1 + ∫ +∞ η dz P2 ) − α S, (2.17) 91 Le Viet Hoa, Nguyen Tuan Anh, Le Huy Son and Nguyen Van Hop where α is the interface tension and S is its area given by S = ∫ dx dy [ 1 + ( ∂η ∂x )2 + ( ∂η ∂y )2]1/2 . (2.18) The variation in Eqs (2.17) and (2.19) with respect to η gives the equation P1(x, y, z = η, t)− P2(x, y, z = η, t) + α ( ∂2 ∂x2 + ∂2 ∂y2 ) η = 0, (2.19) which corresponds to the Bernoulli equation in fluid mechanics [9]. Now we go on to determine the shape of the interface and the dispersion relations for both the phonon and ripplon. To do this, we first assume that the interface is expressed by η(x, y, t) = η(σ), then accept the boundary condition ∂η ∂t = ~ m1 ( ∂δφ1 ∂z ) z=0 = ~ m2 ( ∂δφ2 ∂z ) z=0 . (2.20) in which δφj given by (2.14). The solution to Eq. (2.20) is derived immediately as η(σ) = η0 sinσ (2.21) with |η0| ≪ 1 and ω satisfying the phonon dispersion relation ω = − ~A1 η0m1 k = ~A2 η0m2 k. (2.22) Note that the boundary condition (2.20) is justified in the low energy limit, |kη0| ≪ 1. Substituting (2.14), (2.15), (2.21) and (2.22) into (2.19) we get the ripplon dispersion relation ω2 = α k3 ρ1 + ρ2 , (2.23) in which ρ1 = m1n1(η − 0), ρ2 = m2n2(η + 0) Thus the interface η = η0 sin(k1x+ k2y − ωt) is a solution of the Bernoulli equation (2.19) if the frequency fulfills the dispersion relation (2.22) and (2.23). Moreover, (2.22) and (2.23) show that in a system of two segregated Bose-Einstein condensates (BECs) there exist only two NG modes, a phonon and a ripplon, even though the number of broken symmetries is 3. Our results are then in agreement with that of [10] which is that the number of Nambu-Goldstone (NG) modes in a system obey a modified counting rule which states that the number of type I plus twice the number of type II NG modes is greater than or equal to the generators of spontaneously broken symmetries, here the type I (type II) consisting of NG modes with a linear (fractional) dispersion relation [11]. 92 The interface properties of two-component bose-einstein condensates 2.3. Conclusion In the preceding section the properties of capillary waves at the planar interface of two separate BECs at rest were investigated by means of approximated GP equations. Our main results are in order. - Based on the GP equation the wave functions of condensates are derived. - Based on the Bernoulli equation the shape of the interface and the dispersion relations for both the phonon and ripplonare are studied. 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