A new counting rule of nambu-goldstone modes in system of two segregated Bose-Einstein condensates

HNUE JOURNAL OF SCIENCE DOI: 10.18173/2354-1059.2018-0027 Natural Science, 2018, Volume 63, Issue 6, pp. 39-45 This paper is available online at A NEW COUNTING RULE OF NAMBU-GOLDSTONE MODES IN SYSTEM OF TWO SEGREGATED BOSE-EINSTEIN CONDENSATES Le Viet Hoa Faculty of Physics, Hanoi National University of Education Abstract. The counting rule of Nambu-Goldstone (NG) modes in a system of two-segregated Bose-Einstein condensates (BECs) is studied by means of the hydrodynamic approach for the Gross - Pitaevskii equations. It is shown that although the numbers of NG modes in system at rest and in moving system differ from each other, but they obey the same modified counting rule. Moreover, depending on the velocities the Landau instability may occur in moving system. Keywords: Counting rule of Nambu-Goldstone (NG) modes, two-segregated Bose-Einstein condensates, the Gross - Pitaevskii equations, the Landau instability. 1. Introduction It is known that the low energy property of any system, relativistic or non-relativistic, is entirely determined by its NG modes. In relativistic physics the existence and the number of NG modes are determined by the Goldstone theorem [1, 2] which states that any relativistic system for which a continuous, global symmetry is spontaneously broken, must contain in its spectrum gapless modes called the NG modes and, moreover, the number of NG modes coincides with the number of broken continuous symmetries. However, this counting rule is violated for systems without Lorentz invariance or with spontaneously broken space-time symmetry [3, 4]. For example, in systems without Lorentz-invariance a new counting rule was formulated by Nielsen and Chadha in Ref.[3]: The number of type I plus twice the number of type II of NG modes are greater than or equal to the number of broken generators. The type I (type II) consists of NG modes which possess the dispersion relations with odd (even) power momentum in the low momentum limit. Recently, the Goldstone theorem has been re-examined in the non-relativistic systems [5, 6]. In Ref. [7] Takeuchi and Kasamatsu indicated that in system of two segregated Bose-Einstein condensates (BECs) there exist only two NG modes, a phonon and a ripplon, despite the number of broken symmetries equals 3. The recent progress of experimental realization for ultra-cold gases allows us to consider BECs as an important ground for re-considering the counting rule of NG modes. In this regards, the present paper deals with the subject on how counting rule of NG modes be modified in two segregated BECs. The hydrodynamic approach will be used since it is very powerful to provide all expected dispersion relations of NG modes. To this end, let us begin with the Lagrangian Received April 12, 2018. Revised August 10, 2018. Accepted August 17, 2018. Contact Le Viet Hoa, e-mail: hoalv@hnue.edu.vn 39 Le Viet Hoa £ = ∫ d3x ( P1 + P2 − g12|ψ1| 2|ψ2| 2 ) , (1.1) where Pj = i~ ψ ∗ j ∂ψj ∂t + ~ 2 2mj |∇ψj| 2 + µj|ψj | 2 − gjj 2 |ψj | 4 (1.2) with mj and µj being the atomic mass and chemical potential of the j-th component, respectively. The intra-and inter-component interaction parameters have the form gjk = 2π~ 2ajk(m −1 j +m −1 k ). We assume that ajk > 0 in what follows. The Gross-Pitaevskii equations are derived from (1.1) 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, (1.3) which describe the immiscible condensates for g212 > g11g22. It is clear that (1.1) and (1.3) 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 NG mode: the ripplon mode displaying the ripple waves propagating along interface. 2. Content For two immiscible BECs we assume that the first (second) component occupies the space left (right) to the z = 0 plane and and their interface is represented by a surface z = η(x, y, t). Then the Lagrangian (1.1) is approximated by £ = ∫ dx dy (∫ η −∞ dz P1 + ∫ +∞ η dz P2 ) − α S, (2.1) where α is the interface tension and the interface area S is given as S = ∫ dx dy [ 1 + ( ∂η ∂x )2 + ( ∂η ∂y )2]1/2 . (2.2) For interface with small deviation from a plane, i.e. |η| ≪ 1, from (2.1) and (2.2) we arrive at the equation P1(x, y, z = η, t)− P2(x, y, z = η, t) + α ( ∂2 ∂x2 + ∂2 ∂y2 ) η = 0, (2.3) which corresponds to the Bernoulli equation in classic fluid mechanics [8]. To explore Eq. (2.3), let us first determine the wave functions in some approximation. To do this, we follow closely [9] 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) 40 A new counting rule of nambu-goldstone modes in system of two segregated Bose-Einstein condensates 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 (1.3) 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. Now we adopt the ansatz δφj = ϕj(z)χj(σ), σ = k1x+ k2y − ωt, (2.8) which together with (2.6a) and (2.7) provides d2 dz2 ϕj(z)− k 2ϕj(z) = 0, (2.9a)( ∂2 ∂x2 + ∂2 ∂y2 + k2 ) χj(σ) = 0, (2.9b) where k2 = k21 + k 2 2 . Imposing the conditions ϕ1(z)→ 0 as z →∞, ϕ2(z)→ 0 as z → −∞, we get from (2.9a) ϕj(z) = exp[−(−1) jkz], (2.10) and Eq.(2.9b) yields the general solutions χj(σ) = Aj cos σ +Bj sinσ (2.11) 41 Le Viet Hoa with arbitrary small constants Aj , Bj . The substitution of (2.10) and (2.11) into (2.8) gives δφj = exp[−(−1) jkz](Aj cos σ +Bj sinσ). For the sake of simplicity in what follows we will restrict ourselves to the case δφj = Aj exp[−(−1) jkz] cos σ. (2.12) Inserting (2.12) into (2.6b) provides gjj δnj = −Aj~ ω exp[−(−1) jkz] sin σ. (2.13) Finally, taking into account (2.4), (2.5), (2.12) and (2.13) we are led to the analytical expression for the wave functions in the chosen approximation ψj = √ nj0 − Aj~ ω gjj exp[−(−1)jkz] sin σ.ei ( − gjjnjo ~ t + Aj exp[−(−1) jkz]. cos σ ) . (2.14) Thus, we already established successfully the analytical expressions for the wave functions in the linear approximation. Therefore from now we go over to determining the shape of the interface and the dispersion relations for both phonon and ripplon. To do this, we first accept the boundary condition ∂η ∂t = ~ m1 ( ∂δφ1 ∂z ) z=0 = ~ m2 ( ∂δφ2 ∂z ) z=0 , (2.15) which yields the differential equations for η = η(σ) for δφj given by (2.12) ω dη(σ) dσ = −A1 ~ k m1 cos σ = A2 ~ k m2 cosσ. (2.16) The solution to Eq. (2.16) which represents the interface locating at z = 0 is derived immediately η(σ) = η0 sinσ (2.17) with |η0| ≪ 1 and ω satisfying the phonon dispersion relation ω = − ~A1 η0m1 k = ~A2 η0m2 k. (2.18) Note that the boundary condition (2.15) is justified in the low energy limit, |kη0| ≪ 1. For simplicity we take η0 > 0, then the first (second) component of BECs is superfluid for A1 0). In the opposite case, the Landau instability emerges. The speed of sound in both condensate reads c = − ~A1 η0m1 = ~A2 η0m2 . (2.19) (2.18) and (2.19) prove the quantum character of phonon. 42 A new counting rule of nambu-goldstone modes in system of two segregated Bose-Einstein condensates Next, substituting (2.12), (2.13), (2.17) and (2.18) into Eq. (2.3) and neglecting second and higher orders of Aj and kη0 we find the ripplon dispersion relation ω2 ( m1n10 k + m2n20 k ) = αk2 or ω = √ α ρ1 + ρ2 k3/2, (2.20) where ρ1 = m1n10 = n1(−0); ρ2 = m2n20 = n2(+0). In order to get a deeper insight into the problem, let us extend the investigation to the case when the j-component flows with a velocity ~Vj parallel to the interface. Consequently, the corresponding stationary state is also represented in the form (2.5) with φj = − gjjnj0 ~ t+ mj ~ ~Vj .~r⊥ + δφj ; ~r⊥ = (x, y), (2.21) here δφj is given in (2.12). In this case the boundary condition (2.15) is modified as( ∂ ∂t + ~Vj ∂ ∂~r⊥ ) η(σ) = ~ mj ( ∂δφj ∂z ) z=0 . (2.22) The substitution of (2.12) and (2.17) into (2.22) yields the dispersion relations of two phonon branches attaching to two components ω1 = c1k, ω2 = c2k, (2.23) here c1, c2 are the speeds of sound in corresponding condensates c1 = V1 cos(~V1, ~k)− ~A1 η0m1 , c2 = V2 cos(~V2, ~k) + ~A2 η0m2 . (2.24) It is evident from (2.24) that the Landau instability might occurs if ~Vj fulfill, at least, one of the following inequalities V1 cos(~V1, ~k) < ~A1 η0m1 , V2 cos(~V2, ~k) < − ~A2 η0m2 . (2.25) It is similar to what we did before, the Bernoulli equation (2.3) in combination with (2.12), (2.17) and (2.22) leads to the ripplon spectrum ω± = VC ± √ α k3 ρ1 + ρ2 − ρ1ρ2V 2Rk 2 (ρ1 + ρ2)2 , (2.26) 43 Le Viet Hoa in which ~VC = ρ1~V1 + ρ2~V2 ρ1 + ρ2 ; ~VR = ~V1 − ~V2. Eq.(2.26) indicates that the system of two-component BECs is dynamically unstable for V 2R > 0, i.e ~V1 6= ~V2. This is the well-known Kelvin-Helmholtz instability in hydrodynamics. Transforming to the laboratory frame ~VC = 0, Eq. (2.26) becomes ω2 = α k3 ρ1 + ρ2 − ρ1V 2 1 ρ2 k2, which describes the ripplon mode generalized to the moving system and the ripplon mode causes dynamical instability of the system for k < ρ1(ρ1 + ρ2) αρ2 V 21 . 3. Conclusion Let us first resume what was presented in the preceding section. The NG modes of capillary waves were investigated by means of the linear approximation applied to the Gross-Pitaevskii equations. Two main ingredients of the approach are the boundary condition and the Bernoulli equation. Our main results are in order as follows: 1- In the long-wave length limit the boundary condition and the Bernoulli equation provide the dispersion relations of two NG modes for system at rest. 2- For moving system the modified boundary condition leads to the dispersion relations of two distinct phonons attaching to two condensates. The Landau instability might take place depending on the velocities of condensate flows. In addition, based on the Bernoulli equation we arrive at the ripplon with fractional dispersion relation, whose frequency becomes complex when the velocities of two components are different. This is the well-known Kelvin-Helmholtz instability in hydrodynamics. It is worth noting that the phonon mode always come out as a quantum object, whereas ripplon remains classic in character. What we found above evidently asserts that the number of NG modes in a system at rest and in a moving differs from each other. For system at rest our result is in agreement with that of [7], which, unfortunately, fails for moving system. However, both of them obey the new counting rule replacing that of Nielsen and Chadha: The number of type I plus twice the number of type II of NG modes are greater than or equal to the number of broken generators. Here the type I (type II) is defined as a collection of NG modes with linear (fractional) dispersion relation. As mentioned above, we made use of the appropriate approximation which leads to Eq. (2.6a) characterizing the incompressible fluid. Now a question which arises is that how our obtained results be modified when the compressibility is taken into account. Remember that the NG modes appears as low-energy excitations in the spectrum of energies. Taking into consideration the compressibility means that we must calculate additionally the correction of the orders of Mach number ( v/vs )2 [10], here v is typical speed of the system and vs the speed of sound. In the low 44 A new counting rule of nambu-goldstone modes in system of two segregated Bose-Einstein condensates energy limit this ratio can be treated as negligibly small, of course. Hence, the full low-energy picture of our system remains unchanged with the involvement of compressibility. Last but not least it is worth to point out that the numerical diagonalization of the Bogoliubov-de Gennes equations is very powerful to provide all excited energies, in which the lowest energies corresponding to NG modes [7] are derived from the hydrodynamic approach. 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