Many-particle Quantum Hydrodynamics of Spin-1 Bose-Einstein Condensates
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n Many-particle Quantum Hydrodynamics of Spin-1 Bose-Einstein Condensates
Mariya Iv. Trukhanova
1, 2, a) and Yuri N. Obukhov M. V. Lomonosov Moscow State University, Faculty of Physics, Leninskie Gory, Moscow,Russia Russian Academy of Sciences, Nuclear Safety Institute, B. Tulskaya 52, 115191 Moscow,Russia
We develop a novel model of the magnetized spin-1 Bose-Einstein condensate (BEC) of neutral atoms, usingthe method of many-particle quantum hydrodynamic (QHD) and propose an original derivation of the systemof continual equations. We consider bosons with a spin-spin interaction and a short range interaction in thefirst order in the interaction radius, on the of basis of the self-consistent field approximation of the QHDequations. We demonstrate that the dynamics of the fluid velocity and magnetization is determined by anontrivial modification of the Euler and Landau-Lifshitz equation, and show that a nontrivial modificationof the spin density evolution equation contains the spin torque effect that arises from the self-interactionsbetween spins of the bosons. The properties of the dispersion spectrum of collective excitations are described.We obtain the new contribution of the self-interaction of spins in the spin wave spectrum together with theinfluence of an external magnetic field and spin-spin interactions between polarized particles. The anisotropicspin wave instability is predicted.
CONTENTS
I. Introduction II. General theory
III. The Madelung decomposition
IV. Excitations in the polarized BEC k = k k e z k = k ⊥ e ⊥ V. Discussion
I. INTRODUCTION
Magnetized Bose-Einstein condensates (BEC) are inthe focus of numerous studies. The dynamics ofthe scalar Bose-Einstein condensate had been analyzedin details in the mean-field approach by Gross andPitaevskii . Recently, the attention has been growingto the study of BEC for atoms with magnetic polariza-tion, due to the experimental realization of the polarizedmagnetic BEC of Cr, and to the study of various non-trivial magnetic structures in the BEC, that can exist ina spinor Bose-Einstein condensate. a) Electronic mail: [email protected]
In a system of cold atoms with spins in the BEC state,a number of interesting exotic topological excitationssuch as vortices and skyrmions can exist. Skyrmionsare investigated in many condensed matter systems, suchas liquid crystals , bulky magnetic materials and thinfilms , as well as the quantum Hall systems . The sta-bility of skyrmions in a fictitious spin- condensate of Rb atoms had been studied in Ref. . It was foundthat skyrmions can exist in the two-component spinoronly as a metastable state, and in addition, the size andthe lifetime of the skyrmion, its spin texture and its dy-namical properties were obtained. The two-dimensionalskyrmions were investigated theoretically in a spin-1BEC and realized for the first time experimentally ina spin-2 Bose-Einstein condensate. The nonequilibriumdynamics of a rapidly quenched spin-1 Bose gas withspin-orbit coupling had been studied by solving thestochastic projected Gross-Pitaevskii equation. It wasshown that the crystallization of half-skyrmions (merons)can be occur in a spinor condensate of Rb atoms. Thethree-dimensional skyrmions, in the general case, areunstable towards shrinkage due to an energy gradient.But the new way of creation of the three-dimensionalskyrmions in a ferromagnetic spin-1 BEC by manipu-lating a multipole magnetic field and a pair of counter-propagating laser beams had been proposed recently .The hydrodynamical description of the BEC is basedon the continuum mechanics concepts such as the densityof the particles, the mass current density as well as theforce fields densities. The construction of the hydrody-namical description of a spinor condensate helps to sim-plify the consideration of the dynamics of different exotictopological excitations such as skyrmions, merons andvortices. A set of hydrodynamical equations had beenobtained recently for a spin-1 BEC, which are equiva-lent to the multi-component Gross-Pitaevskii equations,but in terms of the observable physical quantities suchas the spin density and the nematic (or quadrupolar)tensor in addition to the density and the mass current.The hydrodynamical equations of motion for a ferromag-netic BEC of arbitrary spin in the long wavelength limitwere derived by A. Lamacraft . A nontrivial modifi-cation of the Landau-Lifshitz equation was obtained, inwhich the magnetization is defined by the superfluid ve-locity, and the anisotropic spin wave instability was dis-cussed. The mean-field hydrodynamical equations for thedynamics of a spin-F spinor condensates had been alsoderived in Ref. , where the generalization to spin-1 ofthe Mermin-Ho relation was proposed and an analyticsolution for the skyrmion texture in the incompressibleregime of a spin-half condensate had been found. Thethree-fluid hydrodynamical description had been devel-oped in Ref. to treat the low frequency and long wave-length excitations of the spin-1 BEC. The hydrodynam-ical formulation of the mean-field dynamics of a system,derived in Refs. , is based on the introduction of asecond-quantized Hamiltonian. The well-known mean-field method cannot take into account inter-particle in-teractions, based on the collective dynamics of individualparticles, and it also does not take into account ther-mal fluctuations in a system of a large number of parti-cles. Although the mean-field method makes it possibleto obtain a hydrodynamical model of a Bose condensateof ultracold atoms, it cannot take into account thermalfluctuations and inter-particle interactions in an explicitform.In this article we revisit the original derivation of theequations of quantum hydrodynamics of BEC on the ba-sis of the collective dynamics of individual quantum par-ticles described by the many-particle Schr¨odinger equa-tion. Our aim is to derive the macroscopic equationsfor the evolution of physical fields, and to confirm andexpand the previous results by making use of the many-particle quantum hydrodynamics method.The method of many-particle quantum hydrodynamicsmakes it possible to pass from a description in terms ofa many-particle wave function of the Schr¨odinger equa-tion to the hydrodynamical equations formulated in theterms of macroscopic variables such as the particles den-sity, the mass current density, the spin density and thespin current density. This allows one to study multi-particle interactions and nonequilibrium processes in asystem of a large number of interacting particles. Themethod of many-particle quantum hydrodynamics wasdeveloped for various physical systems such as the quan-tum plasmas ? , the spinning plasmas , the relativis-tic quantum plasma , ultracold Bose and Fermi gaseswith the short range interaction and magnetizedBEC . Andreev established the equations of the quan-tum hydrodynamics (QHD) for the magnetized spin-1neutral BEC, where the spin-spin interaction along withthe short range interaction had been taken into account.This paper is organized as follows. In Sec. II wepresent the derivation of the set of QHD equations fora ferromagnetic Bose condensate of spin-1 with the helpof the method of many-particle quantum hydrodynam-ics. In Sec. III we demonstrate the derivation of the macroscopic equations of the hydrodynamics of a Bosecondensate, based on the explicit representation of themany-particle wave function. In Sec. IV we analyze thesmall excitations in the magnetized BEC. II. GENERAL THEORYA. Euler angles description
When the magnetization vector of the ferromagneticBEC is directed along the z -axis, the order parameteris described by ψ (0) = . The mean-field state ofthe BEC can be determined by the five variables: thedensity ρ , the phase ξ and, the ˆ U gauge transformationthat depends on the three Euler angles ϕ, ϑ, χ . A generalorder parameter ψ is given by performing a gauge trans-formation with the phase ξ and rotating to an arbitrarydirection specified by the Euler angles ϕ, ϑ, χ ψ = ψ ψ ψ − = √ ρe iξ ˆ U ( ϕ, ϑ, χ ) (1)When rotating the coordinate system by the Euler angles ϕ, ϑ, χ , the spin functions and the spin matrices for thespin F = 1 are transformed via the rotation operatorˆ U ( ϕ, ϑ, χ ) = e − iϕ ˆ F z e − iϑ ˆ F y e − iχ ˆ F z , (2)and in the representation of the cyclic basis, the operatorˆ U has the formˆ U = cos ϑ e − i ( ϕ + χ ) − sin ϑ √ e − iϕ sin ϑ e − i ( ϕ − χ )sin ϑ √ e − iχ cos ϑ − sin ϑ √ e iχ sin ϑ e i ( ϕ − χ ) sin ϑ √ e iϕ cos ϑ e i ( ϕ + χ ) . (3)In this way, the general order parameter of the ferromag-netic BEC is given by ψ ψ ψ − = √ ρe i ( ξ − χ ) cos ϑ e − iϕ sin ϑ √ sin ϑ e iϕ = √ ρe iξ z . (4)The order parameter (4) is normalized to the number ofparticles N Z d x ψ † ψ = N, (5)and the spinor z is normalized to unity z † z = 1. B. Many-particle quantum hydrodynamics description
Let us consider a system of N bosonic neutral atomswith spin-1 and a short-range potential. The micro-scopic quantum dynamics of the system of bosons is de-scribed by the many-particle Schr¨odinger equation withthe Hamiltonianˆ H = N X j =1 (cid:18) m j ˆ p αj ˆ p jα + U j,ext ( r j , t ) − γ j ˆ f αj B α (cid:19) + 12 N X j,k,j = k (cid:18) U int ( | r jk | ) − γ j γ k G αβjk ( | r jk | ) ˆ f αj ˆ f βk (cid:19) , (6)where ˆ p αj is the momentum operator of the j -th atom, m j is the mass of the particle, γ j = γ is the gyromagneticratio, takes the same value for all particles of the sys-tem, U j,ext is the potential energy in the external forcefield or a spin-independent potential such as an opticalconfinement trap, U int is the short-range interaction po-tential which goes to zero at large inter-particle distances | r jk | . The first term in the Hamiltonian represents thekinetic energy operator and the third term is the linearZeeman energy of j -th particle. The last term representsthe spin-spin interactions between polarized bosons. Thespin-spin interactions between atoms are represented bythe Green function G αβjk ( | r jk | ) = 4 πδ αβ δ jk + ∂ αj ∂ βk ( 1 | r jk | ) . (7)The Hamiltonian of the ferromagnetic Bose-Einstein con-densate (6) determines the quantum dynamics of the sys-tem from the Schr¨odinger equation i ~ ∂ψ a ( R, t ) ∂t = ˆ Hψ a ( R, t ) , (8)where the many-particle wave function of the condensate ψ a ( R, t ) = ψ a ( r , r , ..., r N , t ) . (9)is a spinor function in 3 N configuration space, and a isthe spin index ( a = − F, ..., F ).The state of the condensate is characterized by the density in the neighborhood of r in a physical space as ρ ( r , t ) = Z dR N X j =1 δ ( r − r j ) ψ ∗ a ( R, t ) ψ a ( R, t ) = h ψ † ψ i . (10)The concentration function ρ ( r , t ) is thus determined asthe quantum average of the concentration operator ˆ ρ = P j δ ( r − r j ) in the coordinate representation. The spindensity vector of the polarized bosons is determined in asimilar way F a ( r , t ) = Z dR N X j =1 δ ( r − r j ) ψ ∗ b ( R, t )( ˆ f j,a ) bc ψ c ( R, t ) , (11)as the quantum average of the spin operator, where thespin matrices ˆ f j of spin-1 particles can be written asˆ f x = 1 √ , ˆ f y = 1 √ − i i − i i , ˆ f z = − . (12) C. The macroscopic quantum hydrodynamics equations
The continuity equation for the concentration of par-ticles ρ ( r , t ) can be derived by taking the time derivativeof the definition of concentration (10) and applying themany-particle Schr¨odinger equation ∂ t ρ ( r , t ) + ∂ α J α ( r , t ) = 0 , (13)where the current density has the microscopic definition J ( r , t ) = (cid:28) m j (cid:18) ψ † ˆ p j ψ + (ˆ p j ψ ) † ψ (cid:19) ( R, t ) (cid:29) . (14)The evolution equation for the current (14) is obtained bytaking the time derivative of the definition for the currentdensity (14) and invoking the Schr¨odinger equation (8) m ∂J α ∂t + ∂ β Π αβ = γF β ∂ α B β − ρ∂ α U ext − Z d r ′ ∂ α U int ( r , r ′ ) ρ ( r , r ′ , t ) + Z d r ′ ∂ α G µν ( r , r ′ ) F µν ( r , r ′ , t ) . (15)In the equation of motion (15) F β is the spin density and B β is the total magnetic field. The tensor Π αβ on the left hand side of equation (15) is the momentum flux tensorΠ αβ ( r , t ) = 14 m (cid:28) ψ † ˆ p αj ˆ p βj ψ + (ˆ p αj ψ ) † ˆ p βj ψ + h.c. (cid:29) . (16)The terms on the right hand side of equation (15) arethe force fields due to inter-particle interactions and theaction of external fields. The inter-particle short-rangeinteraction is comparable with the radius of bosons. Themain contribution in the force field of the short-rangeinteractions comes in the first order in the small parame-ter (the interaction radius). Interactions are determinedby the two-particle correlation functions, which can berepresented in the form of ρ ( r , r ′ , t ) = Z dR X j,k,j = k δ ( r − r j ) δ ( r ′ − r k ) ψ † ( R, t ) ψ ( R, t ) . (17)The spin-spin interactions are described by the last termof the equation of motion (15), where the two-spins cor-relations are characterized by the function F µν ( r , r ′ , t ) = Z dR X j,k,j = k δ ( r − r j ) δ ( r ′ − r k ) γ j γ k ψ † ˆ f µj ˆ f νk ψ, (18)where ˆ f µj are given by (12). III. THE MADELUNG DECOMPOSITION
The Madelung decomposition of the N -particle wavefunction can be formulated in terms of the amplitude a ( R, t ), the phase ξ ( R, t ) and the local spinor, defined inthe local frame of reference, normalized as z † z = 1: ψ ( R, t ) = a ( R, t ) exp (cid:18) i ~ ξ ( R, t ) (cid:19) z ( R, r , t ) . (19)Applying the decomposition (19) for the j -th particle, amicroscopic superfluid velocity can be introduced by v αj ( R, r , t ) = ~ m j ( ∇ αj ξ − i z † ∇ αj z ) , (20)and in the terms of Euler angles v αj = ~ m j ∇ αj ( ξ − χ ) − cos ϑ ∇ αj ϕ ! . (21)After the substitution of the wave function in the explicitform (19), the microscopic momentum flux tensor (16) isrecast intoΠ αβ ( r , t ) = (cid:28) ~ m ( ∂ αj a∂ βj a − a∂ αj ∂ βj a ) + a mv αj v βj + ~ m a ∂ αj f j · ∂ βj f j (cid:29) , (22)where the first term in the microscopic definition of themomentum flux tensor is the Bohm potential, the sec-ond term is the fluid pressure and the last term is the“spin stress”, produced by the self-interaction of the spinparticles, or in other words, the spin part of the Bohmpotential.We split the velocity field of the j -th particle as v αj ( R, r , t ) = z αj ( R, r , t ) + v α ( r , t ), where z αj ( R, r , t ) isthe thermal part of the velocity or the fluctuations ve-locity about the macroscopic average v α . In a similarway, the spin of the j -th particle can be represented as f αj ( R, r , t ) = w αj ( R, r , t ) + f α ( r , t ), where w αj describesthe spin fluctuations about the macroscopic average f α .The particle system is assumed to be closed and notplaced in a thermostat. The temperature is obtainedfrom the average kinetic energy of the chaotic motion ofatoms of our system. In accordance with this definition,we introduce deviations of the velocity and the spin ofquantum particles from the local average values z αj and w αj , which correspond to the ordered motion of the par-ticles. Substituting these expressions into the definitionof the momentum flux density (22), we can obtain thecontribution of the kinetic pressure and the spin-thermal effects into the dynamics of the particles.Following the conclusion of the Ref. , we write theinter-particle interaction of bosons for the finite temper-ature, represented by the third term of the equation (15),in the form − Z dR X j,k,j = k { δ ( r − r j ) − δ ( r ′ − r k ) } ×× ∂ α U int ( r jk ) ψ † ψ, r jk = r j − r k . (23)On the basis of the results obtained in we can rep-resent the density of the interaction force for bosonswith a short-range interaction potential in the form ofthe divergence of the tensor field σ αβ ( r , t ), which is thequantum stress tensor determined by the inter-particleinteraction . The inter-particle interaction potential forthe system of bosons close to the BEC state can be finallyderived in the form σ αβ ( r , t ) = −
12 Υ δ αβ (cid:18) ρ bec ρ n + 2 ρ n + ρ bec (cid:19) , (24)where ρ bec denotes the concentration of particles in theBEC state and ρ n is the concentration of excited par-ticles. The constant Υ represents the short range in-teraction at first order in the interaction radius and isdetermined by the following integralΥ = 4 π Z dr r ∂U int ( r ) ∂r . (25)In the self-consistent field approximation two-particlecorrelation functions (17) and (18) can be representedas ρ ( r , r ′ , t ) = ρ ( r , t ) ρ ( r ′ , t ) , (26) F αβ ( r , r ′ , t ) = γ F α ( r , t ) F β ( r ′ , t ) . (27)To distinguish the flow velocity in the equations of quan-tum hydrodynamics, it is necessary to substitute the ex-plicit form of the wave function (19) in the definition of the basic current density (14) J α ( r , t ) = ρ ( r , t ) v α ( r , t ) . (28)The system of the evolution equations for the bosons witha short-range interaction in the absence of excited par-ticles in terms of the particles density ρ ( r , t ), the flowvelocity v ( r , t ) and the spin density F ( r , t ) can be recastinto a more transparent form, if we substitute the par-ticle flux density (28) into the continuity equation (13)and into the equation of motion (15) in the self-field ap-proximation ∂ t ρ + ∂ α ( ρv α ) = 0 , (29) mρ ( ∂ t + v β ∂ β ) v α + ∂ β p αβ − ~ m ∂ α △ ρ + ~ m ∂ β (cid:18) ∂ α ρ∂ β ρρ (cid:19) = Υ ρ∂ α ρ − ρ∂ α U ext + ~ m ρ∂ β (cid:18) ∂ α f · ∂ β f (cid:19) + γ F ∂ α B + ∂ β Q αβ , (30)where the second term on the left hand side of the equa-tion is the gradient of the kinetic pressure, which canvanish for the BEC p αβ = (cid:28) m j a u αj u βj (cid:29) . (31)The third and fourth terms on the left hand side of equa-tion (30) comprise the quantum force density, producedby the quantum Bohm potential. The first term on theright hand side of equation (30) is the force, produced bythe short-range interaction. The second term on the righthand side is the force field density due to the influenceof an external potential. The third term describes thegradient of the spin part of the quantum Bohm potentialor the “spin stress”. And the last term on the right handside of (30) manifests the thermal-spin interactions Q αβ = − ~ m (cid:28) a ∂ αj w j · ∂ βj w j + a ∂ αj w j · ∂ βj f j + a ∂ αj f j · ∂ βj w j (cid:29) . (32)To close the equation system (29), (30) we need to derivethe equation for the spin-thermal force field (32). But,in the case of BEC we will not take into account thecontribution of thermal fluctuations. A. The spin density evolution equation
The microscopic spin density appears in the equationof motion of the bosonic fluid F α ( r , t ) = (cid:28) ψ † ˆ f αj ψ (cid:29) (33) and one requires an additional equation for the spin den-sity dynamics, which can be derived by making use of themany-particle Schr¨odinger equation (8) with the Hamil-tonian (6): ∂ t F α ( r , t ) + ∂ β ℑ αβ ( r , t ) = ε αβγ F β ( r , t ) B γ ( r , t ) . (34)Here the tensor of the spin current density has the mi-croscopic form ℑ αβ ( r , t ) = 12 m (cid:28) ( p βj ψ ) † ˆ f αj ψ + ψ † ˆ f αj p βj ψ (cid:29) . (35)In the macroscopic fluid description, the spin currentdensity reads in terms of the fluid variables ℑ αβ ( r , t ) = m (cid:28) a f αj v βj − ~ m a ε αγη f γj ∂ βj f ηj (cid:29) , (36)where the first term represents the ordinary spin cur-rent caused by the spin f j transfered by a particle withthe velocity v j , and the second term is the self-action ofthe spins. Finally, using the thermal decomposition andMadelung decomposition of the N -particle wave function,the spin density dynamical equation is recast into ∂ t F + ∂ β ( F v β ) = γ ~ F × B + ~ m ∂ β (cid:18) F ρ × ∂ β F (cid:19) + K . (37)The first term of the spin density evolution equation de-scribes the torque acting on the spin density by the mag-netic field. In a self-consistent approximation of the mag-netic spin-spin interactions, the magnetic field appears inthe integral form B α ( r , t ) = γ Z d r ′ G αβ ( r − r ′ ) F β (r ′ , t ) (38)and satisfies the Maxwell equation ∇ × B = 4 πγ ∇ × F . (39)The second term on the right hand side of the equation(37) represents the self-action of particle’s spin, whichcreates the spin self-torque effect, and the last term isthe thermal-spin interactions K = ∂ β (cid:28) a z βj f j − a f j × ∂ βj w j (cid:29) . (40)The system of equations, which contains the spin cur-rent (36) and the momentum flux (22), driven by thetexture of spins, was obtained for a system of a largenumber of interacting particles. A realistic model of aquantum spinning particle with spin- had been derivedby T. Takabayasi . It was shown, that even whenthe magnetic field is zero or the particle does not havea magnetic moment, the spin vector will experience aquantum torque, and the velocity field will be affectedby the quantum torque. The thermal corrections, char-acterized by the thermal-spin interactions (32) and (40)appear in the equations of motion (30) and the spin den-sity evolution (37). However, specifying from the generalequations to the case of a Bose-Einstein condensate, weget a closed formalism of the quantum hydrodynamics,excluding thermal corrections from consideration. IV. EXCITATIONS IN THE POLARIZED BEC
Let us consider a system of atoms at zero temperature,which is completely in the Bose condensate state. Thenwe have to neglect the thermal effects (32) and (40), and we are now in a position to derive the dispersion lawsof the waves in the Bose condensate of cold atoms fromthe obtained closed system of the macroscopic equations(29), (30), and (37).We will analyse the small perturbations in the magne-tized 3 D Bose-Einstein condensate, put in the uniformmagnetic field B = B e z . The equilibrium concentra-tion of the cold atoms is ρ . The perturbations of equi-librium state are ρ = ρ + δρ , v = δ v , B = B + δ B and F = F + δ F . The small perturbations can be repre-sented in the form δf = f ( ω, k ) exp {− iωt + i kr } , wherethe wave vector can be decomposed into two components k = k ⊥ e ⊥ + k k e z (perpendicular and along the appliedmagnetic field, respectively). The tensor of magnetic per-meability depends on the modulus of the wave vector χ αβ = γ F ~ Ω ω − Ω γ F ~ iωω − Ω − γ F ~ iωω − Ω γ F ~ Ω ω − Ω
00 0 F γ ρ m k ω − v f k , (41)where the modified cyclotron frequency,Ω( k ) = γB ~ + ~ F mρ k , (42)depends on the modulus of the wave vector due to theinfluence of spin torque ∼ F × ∂ F , and v f = ~ m k − Υ ρ m . (43)Substituting these small perturbations into the system ofequations (29), (30), (37) and (39) and neglecting non-linear terms, we obtain the dispersion equation (cid:18) w f Ω ω − Ω (cid:19) × (cid:18) k + w f Ω k k ω − Ω + 4 πγ F mρ k ⊥ ω − v f k (cid:19) − (cid:18) w f Ω ω − Ω (cid:19) k k = 0 . (44) A. Wave along to the magnetic field, k = k k e z In the case of wave propagation parallel to an externaluniform magnetic field, pure spin waves are excited withthe dispersion law ω = Ω( k ) = γB ~ + ~ F mρ k (45) B. Wave perpendicular to the magnetic field, k = k ⊥ e ⊥ Consider now the waves propagating in the xy plane,perpendicular to the direction of the external uniformmagnetic field. In this case, several types of waves can be generated. The dispersion equation can be recast into (cid:18) ω f Ω ω − Ω (cid:19)(cid:18) F ~ ρ m ω f ω − v f k (cid:19) = 0 , (46)where ω f = πγ F ~ . There are two solutions of the dis-persion equation (46) in view of its factorized form. Thefirst solution yields the dispersion law ω = ~ m k + ρ m ( | Υ | − π (cid:18) γF ρ (cid:19) ) . (47)The relation (47) characterizes the hybrid mode, wherethe first term follows from the Bohm quantum potential,and the second term is characterized by the interactionsfor the repulsive short range interaction (Υ < ω = Ω (cid:18) Ω − πγ F ~ (cid:19) , Ω = γB ~ + ~ F mρ k . (48)From the formula (48) we can see that int long wavelength limit the frequency square becomes negative,which leads to instability of spin mode. V. DISCUSSION
In this paper, we give an original derivation of theequations of quantum hydrodynamics, which are con-sistent with the existing models, on the basis of thecollective dynamics of individual particles described bythe microscopic many-particle Schrodinger equation. Wetherefore confirm and supplement the earlier obtainedresults . Starting from the many-particle Schr¨odingerequation (8), we have constructed the quantum fluidmodel for the spin-1 Bose-Einstein condensate of neu-tral atoms, which accounts for the possible spin-spinand short-range interactions when the range of the inter-particle interactions is comparable with the radius ofbosons. The essential contribution in the force field ofthe short-range interactions arises in the first order in theinteraction radius. We demonstrate that fluid’s dynam-ics for a spin-1 BEC is described by the set of equations,which encompasses the continuity equation (29), the mo-mentum balance equation (30) and the spin density evo-lution equation (37). The force fields due to the Bohmpotential and the particle spin self-interaction contributeto the momentum balance equation (30). We have shownthat the spin density evolution equation is modified bya nontrivial spin torque effect. The latter arises as a re-sult of the self-interactions between bosons’ spins in thesystem of many-interacting atoms along with the spin-spin interactions, confirming the earlier results . Butin addition, the terms responsible for the many-particlespin-spin interactions emerge in the equation of spin den-sity evolution (37) and momentum balance equation (30).In general, there are also thermal corrections in theequations of motions, characterized by the thermal-spininteractions (32) and (40). However, the latter are ex-cluded from consideration when we specialize from gen-eral equations to the case of a Bose-Einstein condensate,and we ultimately get a closed formulation of the quan-tum hydrodynamics. The study of thermal corrections isthough an important task that requires the derivation ofthe equations for the force fields characterizing the spin-temperature interactions. As an application of the formalism, we have studied thedispersion properties of the magnetized 3 D spin-1 neu-tral Bose-Einstein condensate. The contribution of theequilibrium magnetization in the Bogoliubov’s mode dis-persion relation (47) is obtained. We have also derivedthe contribution of the spin self-interaction to the spinwave spectrum, together with the influence of an exter-nal magnetic field and the spin-spin interactions betweenpolarized particles (48). It turns out that the short-rangeinteractions do not affect the dispersion of spin waves,but the spin torque due to the self-interactions of spinsleads to the dispersion of the spin waves at cyclotron fre-quency γB / ~ , where the square of the frequency ∼ k .As we can see, the dispersion relation has a non-trivialform and in the long wave length regime the instability ofthe wave can arise due to the anisotropic spin-spin dipoleinteractions. A. Acknowledgments
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