Ripplon modes of two segregated bose-einstein condensates in confined geometry

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The ripplon modes of two segregated Bose-Einstein condensates (BECs) confined by one and two hard walls are respectively studied by means of the hydrodynamic approach within the Gross-Pitaevskii (GP) theory. For the system at rest we find that due to the spatial restriction the dispersion relations are of the form ω ∼ k 2 in low momentum limit for both cases, while for the system in motion parallel to the interface the dispersion relations for both cases are ω ∼ k at low momentum limit and, furthermore, the system becomes unstable.

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Nội dung Text: Ripplon modes of two segregated bose-einstein condensates in confined geometry

Communications in Physics, Vol. 26, No. 1 (2016), pp. 11-18 DOI:10.15625/0868-3166/26/1/7790 RIPPLON MODES OF TWO SEGREGATED BOSE-EINSTEIN CONDENSATES IN CONFINED GEOMETRY TRAN HUU PHAT Vietnam Atomic Energy Commission, 59 Ly Thuong Kiet, Hanoi, Vietnam HOANG VAN QUYET† Department of Physics, Hanoi Pedagogical University No. 2, Hanoi, Vietnam † E-mail: hoangvanquyetsp2@gmail.com Received 20 February 2016 Accepted for publication 12 April 2016 Abstract. The ripplon modes of two segregated Bose-Einstein condensates (BECs) confined by one and two hard walls are respectively studied by means of the hydrodynamic approach within the Gross-Pitaevskii (GP) theory. For the system at rest we find that due to the spatial restriction the dispersion relations are of the form ω ∼ k2 in low momentum limit for both cases, while for the system in motion parallel to the interface the dispersion relations for both cases are ω ∼ k at low momentum limit and, furthermore, the system becomes unstable. Keywords: Bose-Einstein condensates, hydrodynamic approach, ripplon modes, Kelvin-Helmholtz instability, Bernoulli equation. Classification numbers: 03.75.Fi, 05.30.Jp, 67.60.- g. I. INTRODUCTION The phase separation in binary mixtures of Bose-Einstein condensates was theoretically predicted [1, 2] and observed later in experiments [3–8]. Since then many works have been devoted to explore the statics, the dynamics and the ripplon modes of two segregated BECs [9–25]. However, most of them concerned with the systems in infinite space, while all experimental realizations have been carried out in restricted regions of space with spatial structure being more and more complicated. In this respect, the present paper deals with the problem: how ripplon modes are influenced by the spatial restriction when a system of two segregated BECs is confined by one or two hard walls and the method we use to tackle the problem is the hydrodynamic approach. To c 2016 Vietnam Academy of Science and Technology 12 TRAN HUU PHAT AND HOANG VAN QUYET begin with, let us start from the GP Lagrangian −r P + P − g |ψ |2 |ψ |2 , £ = ∫ d→ 1 2 12 1 2 Pj = i¯hψ ∗j 2 g j j 4 ∂ψj h¯ 2 ψ j , ∇ψ j − + ∂t 2m j 2 (1a) (1b) with m j being the atomic mass of the j component. The intra-s and inter-species interaction coupling constants are defined as −1 g jk = 2π h¯ 2 a jk m−1 + m , j k where a jk is the scattering length between the atoms in components j and k. In the following we assume that two BECs are separated g212 > g11 g22 . From the Lagrangian (1) the GP equations are deduced straightforwardly ∂ ψ1 h¯ 2 2 2 2 i¯h = − ∇ + g11 |ψ1 | + g12 |ψ2 | ψ1 , (2a) ∂t 2m1 ∂ ψ2 h¯ 2 2 2 2 i¯h = − ∇ + g22 |ψ2 | + g12 |ψ1 | ψ2 . (2b) ∂t 2m2 This paper is organized as follows. The main body of the work is presented in Sec. II where the Bernoulli equation is established and the ripplon modes are derived. The conclusion and discussion are given in Sec. III. II. RIPPLON MODES First of all let us establish the Bernoulli equation for the system of two BECs separated by the interface which is represented by the equation z = z0 + η (σ ) , σ = (kx x + ky y − ωt) and located at z = z0 . Assume that the component 1 (component 2) resides in the region z > z0 (z < z0 ). Then the Lagrangian (1a) is approximated by η+z +∞ 0 £ = 2π ∫ dxdy ∫ dzP2 + ∫ dzP1 − αS, (3) −∞ η+z0 where α is the interface tension and the interface area S is given by Z S= ∂η dxdy[1 + ∂x 2 ∂η + ∂y 2 1 2 ] ≈ Z 1 ∂η 2 1 ∂η 2 + ], dxdy[1 + 2 ∂x 2 ∂y for interface with negligible thickness. From (3) and (4) we arrive at the Bernoulli equation 2 ∂ ∂2 P1 (x, y, z = z0 + η,t) − P2 (x, y, z = z0 + η,t) = α + , ∂ x 2 ∂ y2 (4) (5) which corresponds to the classic Bernoulli equation in hydrodynamics [26, 27]. Next , we proceed to the GP equations in linear approximation as made in Ref. [17] writing q ψ j (x, y, z, t,) = n j (x, y, z, t)eiφ j (x,y,z,t) , (6) RIPPLON MODES OF TWO SEGREGATED BOSE-EINSTEIN CONDENSATES IN CONFINED GEOMETRY 13 in which the particle number density and phase, n j (x, y, z,t) and φ j (x, y, z,t), are decomposed as n j (x, y, z,t) = n j0 + δ n j (x, y, z,t) , (7a) g j j n j0 φ j (x, y, z,t) = − t + δ φ j (x, y, z,t) . (7b) h¯ Inserting (6) and (7) into (2) and taking only the first order of δ n j , δ φ j we are led to the approximate equations − ∇→ v = 0, (8a) j ∂ δ φj + g j j δ nj = 0, (8b) ∂t assuming that the relative density changes following fluid particles are small compared to the velocity gradient [28]. The velocity ~v j in (8a) is defined as h¯ h¯ → − v j = ~∇δ φ j , mj (9) Eq. (8a) tells that in the approximation under consideration the condensates are incompressible fluid. Based on the approximate equations (8) and (9) the pressure given in (1b) takes the form 1 (10) Pj = g j j n2j0 + g j j n j0 δ n j . 2 To solve (8a) and (9) we adopt the ansatz δ φ j (x) = ϕ j (z) χ j (σ ) , from which we get the equations d2 2 − k ϕ j (z) = 0, dz2 2 ∂ ∂2 2 + + k χ j = 0, ∂ x 2 ∂ y2 (11a) (11b) with k2 = kx2 + ky2 . The Bernoulli equation together with the solutions to Eqs. (11) are the basic ingredients for calculating the ripplon modes in the following two subsections. A- One hard-wall case Assume that the hard-wall is located at z = - h as depicted in Fig. 1. The geometrical configuration of the system suggests that the reasonable conditions to be imposed on the two condensates are ∂ ϕ2 (z) = 0, ∂z z=−h ϕ1 (z) → 0 as z → +∞, which lead to δ φ1 = (A1 cos σ + B1 sin σ ) exp (−kz) , 14 TRAN HUU PHAT AND HOANG VAN QUYET Fig. 1. The interface is located at z = z0 and the hard wall at z = −h. δ φ2 = (A2 cos σ + B2 sin σ ) cosh [k(z + h) ] , where A j , B j are small parameters. For simplicity let us restrict to the case δ φ1 = A1 exp (−kz) cos σ , (12a) δ φ2 = A2 cosh [k(z + h) ] cos σ . (12b) Note that the results to be obtained later remain unchanged if cos σ in (12) is replaced by sin σ . The kinetic condition at the interface reads ∂ η (σ ) h¯ ∂ δ φ1 h¯ ∂ δ φ2 = = . (13) ∂t m1 ∂ z z=z0 m2 ∂ z z=z0 Inserting (12) into (13) yields −ω dη (σ ) h¯ k h¯ k = −A1 exp (−kz0 ) cos σ = A2 sinh [k(z0 + h) ] cos σ . dσ m1 m2 (14) The solution to Eqs. (14) provides immediately η (σ ) = η0 sinσ , (15) and η0 m1 ω exp(kz0 ) , h¯ k η0 m2 ω A2 = − . h¯ k sinh [k(z0 + h)] A1 = (16a) (16b) Eqs. (14) are justified for |kη0 | 1. Substituting (10) , (12) , (15) , (16) into the Bernoulli equation (5) we arrive at ω2 = αk3 , ρ2 coth [k(z0 + h)] + ρ1 here ρ1 = m1 n10 (z0 + 0), ρ2 = m2 n20 (z0 − 0). As h tends to infinity Eq. (17) turns out to be r ω= α k3/2 , ρ1 + ρ2 (17) RIPPLON MODES OF TWO SEGREGATED BOSE-EINSTEIN CONDENSATES IN CONFINED GEOMETRY 15 which is the well known formula for dispersion relation of ripplon in classic hydrodynamics [26, 27]. At the low momentum limit Eq. (17) behaves like α (h + z0 ) 4 k , (18) ω2 ≈ ρ2 which is the main result of this subsection. In order to get a deeper insight into the issue let us extend to the case when condensates flow with velocity ~V j parallel to the interface. Then the corresponding stationary state is also represented in the form (7b) with g j j n j0 mj → − → φj = − t+ V j .−r ⊥ + δ φ j , h¯ h¯ where δ φ j is given by Eqs. (12). In this case the kinetic condition at the interface (13) is modified as h¯ ∂ δ φ j ∂ → − ∂ + Vj − η= . (19) ∂t mj ∂ z z=z0 ∂ r→ ⊥ Substituting (12) and (15) into (19) gives η0 m1 ω − ~V1~k exp(kz0 ) A1 = , (20a) h¯ k η0 m2 ω − ~V2~k A2 = − . (20b) h¯ k sinh [k(z0 + h)] After inserting (20) into (5) we are led to the equation 2 2 ρ1 ω − ~V1~k + ρ2 coth [k (z0 + h)] ω − ~V2~k = αk2 , yielding k ω± = Vc k ± √ ρ1 + ρ2 here Vc = r ρ1 ρ2 2 αk − V ρ1 + ρ2 r (21) ρ1V1 cos θ1 + ρ2 coth [k (z0 + h)]V2 cos θ2 , ρ1 + ρ2 coth [k (z0 + h)] \ with θ j = ~V j ,~k . It is easily seen that as h tends to infinity (21) turns out to be s αk3 ρ1 ρ2Vr2 k2 ω± = Vc k ± − , ρ1 + ρ2 (ρ1 + ρ2 )2 (22) which is a well known result. The small-k behavior of (21) reads r ρ1 ω ≈ V2 cos θ2 k ± − (h + z0 )Vr k3/2 . ρ2 Eq. (23) indicates that the Kelvin-Helmholtz instability always occurs for Vr = ~V1 − ~V2 > 0. Next let us go over to the two hard-wall case. (23)