Quantum hydrodynamics of the spinor Bose-Einstein condensate at non-zero temperatures
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b Quantum hydrodynamics of the spinor Bose-Einstein condensate at non-zerotemperatures
Pavel A. Andreev ∗ Faculty of physics, Lomonosov Moscow State University, Moscow, Russian Federation, 119991. andPeoples Friendship University of Russia (RUDN University),6 Miklukho-Maklaya Street, Moscow, 117198, Russian Federation
I. N. Mosaki
Faculty of physics, Lomonosov Moscow State University, Moscow, Russian Federation, 119991.
Mariya Iv. Trukhanova † Faculty of physics, Lomonosov Moscow State University, Moscow, Russian Federation, 119991. andRussian Academy of Sciences, Nuclear Safety Institute (IBRAE),B. Tulskaya 52, Moscow, Russian Federation, 115191. (Dated: February 26, 2021)Finite temperature hydrodynamic model is derived for the spin-1 ultracold bosons by the many-particle quantum hydrodynamic method. It is presented as the two fluid model of the BEC andnormal fluid. The linear and quadratic Zeeman effects are included. Scalar and spin-spin like short-range interactions are considered in the first order by the interaction radius. It is also represented asthe set of two nonlinear Pauli equations. The spectrum of the bulk collective excitations is consideredfor the ferromagnetic phase in the small temperature limit. The spin wave is not affected by thepresence of the small temperature in the described minimal coupling model, where the thermal partof the spin-current of the normal fluid is neglected. The two sound waves are affected by the spinevolution in the same way as the change of spectrum of the single sound wave in BEC, where speedof sound is proportional to g + g with g i are the interaction constants. I. INTRODUCTION
The superfluidity of the liquid helium below the λ pointis discovered in 1938. This phenomenon is explained asthe Bose-Einstein condensation of the part of atoms. Theliquid helium is the physical object with strong interac-tion while the Bose-Einstein condensation is predictedfor the ideal Bose gas. The Bose-Einstein condensationof collections of weakly interacting atoms is achievedin 1995. The dynamics of the weakly interacting spin-0 scalar Bose-Einstein condensates (BECs) is describedby the Gross-Pitaevskii equation which is the effectivesingle-particle nonlinear Schrodinger equation [1]. In thefollowing years the Bose-Einstein condensate of atomsbeing in several spin states [2], [3]. Corresponding modelin form of the nonlinear Pauli equation is suggested.The Gross-Pitaevskii equation can be represented inthe form of quantum hydrodynamic equations. The setof hydrodynamic equations consists of the continuity andEuler equation for the potential velocity field for the spin-0 BECs. The velocity field becomes nonpotential for thespin-1 BECs. Moreover, additional hydrodynamic func-tions appear to describe the hydrodynamic properties ofthe spin-1 BECs. They are the spin density and thenematic tensor density. The nematic tensor is propor-tional to the anticommutator of the spin operators. It ∗ Electronic address: [email protected] † Electronic address: [email protected] is an independent function for the spin-1 BECs (exceptthe ferromagnetic phase, where it is reduced to the spindensity). In contrast, the product of the spin operatorsreduces to its first degree for the spin-1/2 particles. Over-all, the set of hydrodynamic equations is obtained in theregime of small collisions [4], [5], [6].The Gross-Pitaevskii equation can be derived from themicroscopic quantum mechanics represented in the sec-ond quantization form [7], which is one of realizationsfor the transition from the microscopic to the macro-scopic models. The many-particle quantum hydrody-namics presents another approach giving the represen-tation of the many-particle wave function via the set ofhydrodynamic functions [8], [9], [10], [11], [12], [13]. Thedensity-functional theory for spin-0 bosons is consideredincluding the GP equation [14].Earlier steps in the theory of weakly interacting BECsincludes the derivation of spectrum of collective excita-tions called the Bogoliubov spectrum obtained in 1947[15]. This theory is based on the second quantizationmethod. Effects of the small temperatures are includedby Popov in 1965 [16], [17], [18].Variety of waves phenomena is observed in BECs. Itincludes the density waves [19], [20], [21] solitons [22],[23], [24], vortices [25], turbulence, shock waves [26], [27].Collective excitations are considered in Fermi gas either[28]. These phenomena demonstrate more complex be-havior in the spinor BECs in comparison with spin-0BECs [4].Finite temperature effects in ultracold gases are con-sidered as well [20], [25], [29], [30], [31]. Presence of thebosons in the excited states is modeled via the secondfluid which called the normal fluid. The interaction be-tween the atoms in the condensate and the surroundinggas of noncondensed particles can be also modeled withinthe kinetic model [32]. Finite temperature effects are con-sidered in ultracold fermions [28] either.Depletion of BECs and quantum fluctuations make thedynamical properties bosons intermixing with the finitetemperature effects more complex [13], [33], [34], [35],[36], [37], [38].mean-field potential of the magnetic dipole-dipole in-teraction is included at the study of the collective exci-tations in spin-1 BECs [39].Instantaneous quench of quadratic Zeeman shift from q > q < q = 0 corresponding to the transition from polar tothe antiferromagnetic phases is made experimentally forthe antiferromagnetic spinor BEC [40].This paper is organized as follows. In Sec. II the struc-ture of the hydrodynamic equations including basic def-initions and major steps of derivation are described. InSec. III transition to the two-fluid quantum hydrody-namic model for the finite temperature spin-1 bosons ismade. In Sec. IV the spectrum of collective excitationsis considered for the ferromagnetic phase. In Sec. V abrief summary of obtained results is presented. II. HYDRODYNAMIC EQUATIONS
The dynamic of the quantum system is governed bythe Schrodinger equation ı ¯ h∂ t Ψ S = ˆ H SS ′ Ψ S ′ , (1)where the coordinate representation is chosen. There-fore, the wave function Ψ S = Ψ S ( R, t ) is the functionof coordinates of N identical particles under consider-ation R = { r , ..., r N } representing the configurationalspace of the system. The spin-1 bosons are considered.Hence, the distribution of the probability amplitude isdetermined by the wave function Ψ S along with the am-plitude of the spin projection probability. The spinorstructure of the wave function is reflected in the subindexΨ S , where spin index S is the short notation for the N indexes of spin of each particle S = { s , ..., s N } . Moreaccurately, the spin-1 particles are the vector particles,where single particle wave function is the three compo-nent column, while the spinor is the mathematical objectinvented for the spin-1/2 fermions. The spin structure ofthe wave function is the direct product of N three com-ponent columns and the Hamiltonian ˆ H = ˆ H SS ′ actingas the 3 N -rank matrix. The action of matrix ˆ H SS ′ onthe ”spinor-vector” Ψ S ′ includes summation on subindex S ′ : P S ′ = s ′ ,...,s ′ N ˆ H SS ′ Ψ S ′ , with s i = 0, ±
1, but equation(1) does not include the summation since the summationon the repeating index is assumed. To specify the physical system under consideration re-quires the explicit form of the Hamiltonian of the system:ˆ H = N X i =1 (cid:18) ˆ p i m + V ext ( r i , t ) − p ˆ F i,z + q ˆ F i,z (cid:19) + 12 X i,j = i (cid:18) U ( r i − r j ) + U ( r i − r j ) ˆ F i · ˆ F j (cid:19) , (2)where m is the mass of atom, ˆ p i = − ı ¯ h ∇ i is the mo-mentum of i-th atom. The first group of terms is thesuperposition of the singe-particle Hamiltonians, whichincludes the linear Zeeman term proportional to p andthe quadratic Zeeman term proportional to q . The lastgroup of terms in the Hamiltonian (2) contains two terms.The scalar spinless part of the short-range boson-bosoninteraction U ,ij which equally contributes in interac-tion of atoms with same spin projection. The spinde-pendent part of the short-range boson-boson interaction U ,ij demonstrates deviation from potential U ,ij for theinteraction of atoms with different spin projections. Theexplicit form of the short-range interaction is not spec-ified. No distinguish between bosons in the BEC stateand bosons in other states is made at the microscopiclevel. Separation of all bosons on BEC and normal fluidis made in terms of collective variables.Hamiltonian (2) contains the single-particle spin ma-trixes ˆ F i = { ˆ F x , ˆ F y , ˆ F z } = √ , √ − ı ı − ı ı , − . (3)The microscopic distribution of spin-1 bosons in thephysical space can be obtained by the projection of thedistribution in the configurational space [8], [10], [12],[41], [42], [43] n = Z Ψ † S ( R, t ) N X i =1 δ ( r − r i )Ψ S ( R, t ) dR. (4)This definition contains the integration over the configu-rational space and the summation over the spin indexes.Same operations are included in the definitions of otherhydrodynamic functions presented below.Each boson has probability to be in quantum statewith different projections of spin. Considering all bosonsin the quasi-classic terms we can refer to number ofbosons with different spin projections. Therefore, theconcentration of all bosons can be separated on the par-tial concentrations n = n + n + n − , where subindexes0, ± ∂ t n + ∇ · j = 0 . (5)The appearance of the continuity equation is rather obvi-ous conclusion. However, the derivation of the continuityequation provides the explicit definition for the currentin terms of the many-particle wave function: j = 12 m Z N X i =1 δ ( r − r i )[Ψ † S ( R, t )ˆ p i Ψ S ( R, t ) + h.c. ] dR, (6)where h.c. stands for the Hermitian conjugation. Vector j is the probability current in the physical space whichreduces to the current of particles in the classical limit.Moreover, vector j is proportional to the momentum den-sity of the quantum system.Next step is the derivation of the momentum balanceequation which is the preliminary form of the Euler equa-tion ∂ t j α + ∂ β Π αβ = − m n∂ α V ext + 1 m F αint , (7)where the summation on the repeating vector indexes isassumed, andΠ αβ = 14 m Z dR X i δ ( r − r i )[Ψ † S ( R, t )ˆ p αi ˆ p βi Ψ S ( R, t )+(ˆ p αi Ψ( R, t )) † S ˆ p βi Ψ S ( R, t ) + h.c. ] (8)is the momentum flux, and F αint is the force field describ-ing the interaction between bosons.The general form of the interparticle interaction forcefield appears in the following form: F αint = − Z ( ∂ α U ( r − r ′ )) n ( r , r ′ , t ) d r ′ − Z ( ∂ α U ( r − r ′ )) S ββ ( r , r ′ , t ) d r ′ , (9)where n ( r , r ′ , t ) = Z Ψ † S X i,j = i δ ( r − r i ) δ ( r ′ − r j )Ψ S dR, (10) and S αβ ( r , r ′ , t ) = Z Ψ † S ( R, t ) X i,j = i δ ( r − r i ) ×× δ ( r ′ − r j )[ ˆ F αi ˆ F βj Ψ( R, t )] S dR, (11)with [ ˆ F i · ˆ F j Ψ( R, t )] S stands for [ ˆ F s i s ′ i · ˆ F s j s ′ j Ψ s ,...,s ′ i ,...,s ′ j ,...,s N ( R, t ).Approximate calculation of F αint for the short-range in-teraction in the first order by the interaction radius [10],[11], [12], [22] leads to the following result F αint = − g ∂ α (2 n n + 4 n n n B + n B ) − g ∂ α (2 S n + 4 S n S B + S B ) , (12)where g i = Z U i ( r ) d r (13)are the interaction constants for the two parts of theshort-range interaction potential presented in the Hamil-tonian (2), the subindex B refers to the BEC, thesubindex n refers to the normal fluid, and S ( r , t ) = Z Ψ † S ( R, t ) X i δ ( r − r i )( ˆ F i Ψ( R, t )) S dR (14)is the spin density. Moreover, expression (15) is presentedin terms of two fluid model. The functions describing theBEC and the normal fluid enter this equation in nonsym-metric form, so they are not combined in the functionsdescribing all bosons simultaneously. This result for theforce field gives additional motivation to split hydrody-namic equations on two subsystems.Equation (15) has the following zero temperature limit F αint,BEC = − g n B ∂ α n B − g S B ∂ α S B . (15)Presence of the spin density (14) leads to the necessityto derive the equation for its evolution ∂ t S α + ∂ β J αβ = − p ¯ h ε αzγ S γ + 2 q ¯ h ε αzγ N zγ + ε αβγ ¯ h Z U ( r − r ′ ) S γβ ( r , r ′ , t ) d r ′ , (16)where J αβ = 12 m Z N X i =1 δ ( r − r i )[Ψ † S ˆ F αi ˆ p βi Ψ S + h.c. ] dR (17)is the spin current giving the redistribution of particleswith no influence of interaction, and N αβ = 12 Z N X i =1 δ ( r − r i )Ψ † S ( ˆ F αi ˆ F βi + ˆ F βi ˆ F αi )Ψ S dR (18)is the density of the macroscopic nematic tensor. It isdefined as the quantum average of the nematic tensoroperator ˆ N αβi = (1 / F αi ˆ F βi + ˆ F βi ˆ F αi ) of the i -th par-ticle. The nematic tensor is the independent functionfor the spin-1 bosons, while for the spin-1/2 fermions itreduces to the concentration of particles.The general form of the spin evolution equation (16)contains the interparticle interaction in the last term.However, its calculation in the first order by the inter-action radius gives the zero value. Hence, equation (16)reduces to ∂ t S α + ∂ β J αβ = − p ¯ h ε αzγ S γ + 2 q ¯ h ε αzγ N zγ . (19)Therefore, no interaction gives contribution in the spinevolution equations. The nontrivial evolution of spin den-sity is related to the quadratic Zeeman effects presentedvia novel hydrodynamic function: the nematic tensor N αβ . If we want to obtain the influence of interactionon the spin density evolution we need to derive the ne-matic tensor evolution equation. Hence, we consider thetime derivative of function (18) and use the Schrodingerequation (1) with Hamiltonian (2) for the time deriva-tives of the microscopic many-particle wave function. Asthe result we find the following equation: ∂ t N αβ + ∂ γ J αβγN = p ¯ h ε zαγ N βγ + p ¯ h ε zβγ N αγ − q h ( ε zαγ δ βz + ε zβγ δ αz ) S γ + g ¯ h (cid:20) ε βγδ (cid:18) N αδn S γn + 2 N αδn S γB + 2 N αδB S γn + N αδB S γB (cid:19) + ε αγδ (cid:18) N βδn S γn + 2 N βδn S γB + 2 N βδB S γn + N βδB S γB (cid:19)(cid:21) , (20)where J αβγN = 14 m Z N X i =1 δ ( r − r i )[Ψ † S ( R, t ) ×× ( ˆ F αi ˆ F βi + ˆ F βi ˆ F αi )ˆ p γi Ψ S ( R, t ) + h.c. ] dR (21)is the flux of the nematic tensor, the subindex N refersto the nematic tensor, and the last group of terms pro-portional to the second interaction constant appears atthe calculation of the following term in the first order bythe interaction radius Z dR X i,k = i δ ( r − r i ) U ,ik Ψ † S ( R, t ) × × ( ε βγδ ˆ N αδi ˆ F γk + ε αγδ ˆ N βδi ˆ F γk )Ψ S ( R, t ) . (22)The general structure of hydrodynamic equations is ob-tained above. However, we need to introduce the velocityfield to give these equations more traditionally appear-ance. Moreover, this allows to consider the structure offluxes of all variables. A. Madelung transformation for the single-particlewave function
Before we present the structure of the many-particlewave function let us consider the single particle wavefunction. For the spin-1 boson it is the three componentcolumn presenting the vector wave function ψ ( r , t ) = ψ +1 ψ ψ − , (23)where the amplitude a ( r , t ) and phase S ( r , t ) of the wavefunction can be introduced together with the unit vectorshowing the spin state of the particle ψ = ae iS ˆ z (Θ) , (24)where we have unit vector ˆ z (Θ) describing the possiblespin states.Making a step ahead we consider the example of theunit vector ˆ z (Θ) applied for the description of the ferro-magnetic phase of spin-1 BECsˆ z = e − ıχ cos ( θ/ e − ıϕ √ sin( θ )sin ( θ/ e ıϕ , (25)with Θ = { χ, θ, ϕ } , and χ ( r , t ), θ ( r , t ), ϕ ( r , t ) are thescalar fields, and ˆ z † ˆ z = 1.Vector (25) can be obtain by rotation of vectorˆ z = , (26)where the rotation is made by matrix e − ıχ cos θ e − ıϕ − e − ıϕ √ sin θ e ıχ cos θ e − ıϕ e − ıχ √ sin θ cos θ e ıχ √ sin θe − ıχ sin θ e ıϕ e ıϕ √ sin θ e ıχ sin θ e ıϕ . (27)The second phase of the spin-1 BECs is the polarphase. It corresponds to the unit vector obtained by therotation of vector ˆ z = , (28)which leads to ˆ z = − √ sin( θ ) e − ıϕ cos( θ ) √ sin( θ ) e ıϕ , (29)However, these are examples of unit vector ˆ z from themacroscopic physics. Or it can be considered as the wavefunction for the single particle. We need to make stepback to the microscopic description. B. Introduction of the velocity field and theaverage spin field via the Madelung transformation
For the many-particle wave function we have 3 N di-mensional configurational space. Moreover, the spin partof the wave function is the direct product of N columnsof three component each (23). Here, we can express theamplitude a ( R, t ) and phase S ( R, t )Ψ S ( R, t ) = a ( R, t ) e ıS ( R,t ) N O i =1 ˆ z i (30)where N Ni =1 ˆ z i = ˆ z ⊗ ˆ z ⊗ ... ⊗ ˆ z N , and ˆ z i = ˆ z i (Θ i ), withΘ i = Θ i ( r i , t ) is the set of three angular fields defined inthree dimensional subspace of the configurational space.Generalized Madelung decomposition (30) gives thefollowing representation of the concentration: n = Z dR N X i =1 δ ( r − r i ) a ( R, t ) . (31)It is independent of the explicit form of unit vectors ˆ z i .Other hydrodynamic functions contain the contribu-tion of unit vectors ˆ z i . The generalized Madelung de-composition (30) provides the modified expression for thecurrent j = ¯ hm Z dR N X i =1 δ ( r − r i ) a ( R, t ) ×× ( ∂ αi S ( R, t ) − ı ˆ z † i ∂ αi ˆ z i ) , (32)where the gradient of phase S ( R, t ) existing for the spin-less particles is shifted by the gradient of the unit vectors.We have the following specification of equation (32) forthe ferromagnetic phase j = ¯ hm Z dR N X i =1 δ ( r − r i ) a ( R, t ) ×× ( ∂ αi S ( R, t ) − ∂ αi χ i − cos θ i ∂ αi ϕ i ) , (33)where the velocity of i -th particle can be introduced interms of components of unit vector ˆ z i (25): v αi = ∂ αi S − ∂ αi χ i − cos θ i ∂ αi ϕ i . For the arbitrary unit vector ˆ z i we alsointroduce the velocity of i -th particle v αi = ∂ αi S − ı ˆ z † i ∂ αi ˆ z i .Substitution of the wave function via the arbitrary unitvector ˆ z i contains the contribution of spin of i -th particle s i = ˆ z † i ˆ F i ˆ z i under the integral on the configurationalspace S = Z dR N X i =1 δ ( r − r i )(ˆ z † i ˆ F i ˆ z i ) a ( R, t ) . (34)Other unit vectors ˆ z j = i exclude themselves via the nor-malization.At the full polarization of the spin of each particle s i appear via the unit vector: S = Z dR N X i =1 δ ( r − r i ) n i a ( R, t ) , (35)where s i = n i = n ( r i , t )= { cos ϕ i sin θ i , sin ϕ i sin θ i , cos θ i } is the unit vectorwhich shows the contribution of the spin of i -th particlein the spin density vector field. We have unit moduleof each spin, but spin can have different directionsrelatively each other. Hence, it can correspond to thepartial spin polarization. Moreover, the partial spinpolarization of the system can be caused by the partialspin polarization of each particle. The ferromagneticphase exists at the full polarization of the spin of eachparticle s i which have same spin polarization. In thepolar phase s i = 0. Hence, S p = 0, where subindex p stands for the indication of the polar phase.The last in the line of major hydrodynamic func-tions necessary for the description of the spin-1 BECsis the density of the nematic tensor N αβ . It canbe represented by the single-particle nematic tensor n αβi =(1 / z † i ( ˆ F αi ˆ F βi + ˆ F βi ˆ F αi )ˆ z i under the integral on theconfigurational space N αβ = 12 Z dR N X i =1 δ ( r − r i )(ˆ z † i ( ˆ F αi ˆ F βi + ˆ F βi ˆ F αi )ˆ z i ) a ( R, t ) . (36)Let us consider the structure of the single-particle ne-matic tensor n αβi . At the full polarization of the spinof each particle s i = n i we have the following sim-plification of the single-particle nematic tensor n αβi =( δ αβ + n αi n βi ) /
2. Or it can be rewritten via the spin of i -th particle n αβi = ( δ αβ + s αi s βi ) /
2. The transition tothe nematic tensor density requires to consider the quan-tum average of this function on the many-particle wavefunction. Hence, in general, the second term does notgive the product of two spin densities S . Correspondingnematic tensor is discussed in this section below. C. Transformation of the fluxes at the Madelungtransformation
Above we consider the time evolution of the concentra-tion n , the current j , the spin density S , and the nematictensor N αβ . The interaction influence their evolution.However, the fluxes of these functions give their redis-tribution even at the zero interaction. To understandthe structure of the quantum hydrodynamic equations weneed to consider the representation of the fluxes of ba-sic hydrodynamic variables at the generalized Madelungdecomposition (30). The current j is the flux of concen-tration n . Hence, one flux is considered. We need toconsider the momentum flux Π αβ (8), the spin current J αβ (17), and the flux of nematic tensor J αβγN (21).We start this part of analysis with the representationof the spin current J αβ (17): J αβ = ¯ hm Z dR N X i =1 δ ( r − r i ) s αi ( ∂ βi S ( R, t )) a ( R, t )+ 12 m Z dR N X i =1 δ ( r − r i ) a ( R, t )( z † i ˆ F αi ˆ p βi z i + h.c. ) . (37)The first term in the spin current contains the spin of i -th particle, but it contains a part of the velocity of i -thparticle. Let us represent ∂ βi S ( R, t ) via the velocity v αi .Hence, we have term which has clear physical meaning ofthe spin current via the product of spin s αi and velocityof the same particle v βi : J αβ = Z dR N X i =1 δ ( r − r i ) s αi v βi a ( R, t )+ ı ¯ hm Z dR N X i =1 δ ( r − r i ) s αi ( z † i ∂ βi z i ) a ( R, t )+ 12 m Z dR N X i =1 δ ( r − r i ) a ( R, t )( z † i ˆ F αi ˆ p βi z i + h.c. ) . (38)Hence, the first term can be called the quasi-classic partof the spin-current J αβqc , while other terms are the quan-tum corrections.The second example of the fluxes is the momentumflux Π αβ (8). Let us present the result of application ofMadelung decomposition (30):Π αβ = Z dR N X i =1 δ ( r − r i ) a ( R, t ) v αi v βi + ¯ h m Z dR N X i =1 δ ( r − r i )( ∂ αi a · ∂ βi a − a∂ αi ∂ βi a ) + ¯ h m Z dR N X i =1 δ ( r − r i ) a ( ∂ αi z † i · ∂ βi z i + ∂ βi z † i · ∂ αi z i + 4 z † i ∂ αi z i · z † i ∂ βi z i ) . (39)The first term presents the quasi-classic part of the mo-mentum flux Π αβqc which exists in the classic limit(if¯ h → J αβγN = Z dR N X i =1 δ ( r − r i ) a ( R, t ) n αβi v γi + ı ¯ hm Z dR N X i =1 δ ( r − r i ) n αβi ( z † i ∂ γi z i ) a ( R, t )+ 12 m Z dR N X i =1 δ ( r − r i ) a ( R, t )( z † i ˆ N αβi ˆ p γi z i + h.c. ) . (40)It also has the quasi-classical and quantum parts.The ”velocities” of different quantum particles give dif-ferent contributions in the combined current j (32). Thecurrent j allows to introduce the average velocity (thevelocity field) v = j /n , which is the vector field showinglocal average velocity. The deviation of velocity of eachparticle from the average velocity u i = v i − v is intro-duced. Chaotic variations from the average velocity canbe associated with the thermal motion,while the temper-ature itself is the average square of velocity of the chaoticpart of motion. Hence, the temperature is proportionalto the average of u i .Similar steps can be made for the analysis of struc-ture of the spin density (34). The average spin canbe introduced s = S /n . Therefore, the deviation canbe introduced t i = s i − s as well. Moreover, sameseparation can be made for the nematic tensor N αβ : n αβi = N αβ /n + ν αβi , where ν αβi is the deviation of thenematic tensor of i -th particle from the average value N αβ /n . The definitions of the spin density (14) and thenematic tensor density (18) show that the average valuesof the deviations t i and ν αβi are equal to zero.Let us consider the structure of the nematic tensordensity in the ferromagnetic phase. Above we find theexpression for the nematic tensor of i -th particle via thespin of i -th particle in the ferromagnetic phase of thespin-1 bosons n αβi = ( δ αβ + s αi s βi ) /
2. We substitute thespin via the average spin s = S /n and the deviation fromthe average t i = s i − S /n . Hence, the nematic tensor of i -th particle transforms to n αβi = (1 / δ αβ + S α S β /n +( S α /n ) t βi + ( S β /n ) t αi + t αi t βi ). The last term is con-structed of the parameters defined in the local frame.So, it is associated with the ν αβi , which is the deviationof the nematic tensor of i -th particle from the averagevalue N αβ /n We use it to find the nematic tensor den-sity N αβ = R dR P i δ ( r − r i ) n αβi a . Functions S α , n canbe easily extracted from under the integral. We also in-clude R dR P i δ ( r − r i ) t αi a = 0. Hence, we find N αβ =(1 / n ( δ αβ + S α S β /n ) +(1 / R dR P i δ ( r − r i ) ν αβi a .It is considered above that the module of the spin of eachparticle s αi is equal to one in the ferromagnetic phase.Moreover, if we need to get the ferromagnetic state con-structed of number of spins all of them should be parallelto each other. Hence, there is no deviation of each spinfrom the average t αi = 0. Therefore, the last term in theexpression for the nematic tensor density N αβ is equalto zero. Finally we obtain the well-known expression N αβ = (1 / nδ αβ + S α S β /n ) (see for instance [44] intext after equation 31).The first term in equation (39) is the quasi-classic partof the momentum flux. We substitute the velocity viatwo terms v i = u i + v . Next, we use that the averageof the velocity in the comoving frame u i is equal to zero.Therefore, the quasi-classic part of the momentum fluxappears as the sum of two terms:Π αβqc = nv α v β + p αβ , (41)where p αβ is the pressure with the following definition p αβ = Z dR N X i =1 δ ( r − r i ) a ( R, t ) u αi u βi . (42)The nonequilibrium temperature scalar field is the aver-age of the square of the velocity of the chaotic motion.Hence, equation (42) shows that the temperature is pro-portional to the trace of pressure p αβ : temperature is T = p ββ / n , where p ββ = p xx + p yy + p zz , and p ββ = 3 p for the equal equal diagonal elements presented by p .Same separation appears for the quasi-classic parts ofthe spin current and the nematic tensor flux J αβqc = S α v β + J αβth , (43)where we have the thermal part of the spin current tensor J αβth = Z dR N X i =1 δ ( r − r i ) a ( R, t ) t αi u βi , (44) and J αβγN,qc = N αβ v γ + J αβγN,th , (45)where we have the thermal part of the nematic tensorflux J αβγN,th = Z dR N X i =1 δ ( r − r i ) a ( R, t ) ν αβi u γi . (46)Tensors J αβth and J αβγN,th similarly to the pressure p αβ are related to the thermal motion of the quantum parti-cles. However, the thermal spin current and the thermal nematic tensor flux includes the chaotic motion of spinorientation.Below, we present the set of two non-linear Schrodingerequations for the BEC and the normal fluid. It presentsthe minimal coupling model similar to the set of non-linear Schrodinger equations existing in literature for thespin-0 bosons [1]. So, the contribution of the pressure ofthe normal fluid is neglected as well.Equations (41), (43) and (45) presents the flux of mo-mentum, the flux of spin, and the flux of the nematictensor in the quasi-classic limit, so the quantum termsare neglected there. These expressions are obtained forthe arbitrary temperatures. However, we are interestedin the small temperature limit. First, consider the zerotemperature limit. Hence, no deviations of the velocity,spin, and the nematic tensor from the local average exist.In our notations the zero temperature leads to u αi = 0, t αi = 0, and ν αβi = 0. It also leads to p αβB = 0, J αβth,B = 0,and J αβγth,B = 0, where the subindex B refers to the BECstate. In the nonzero temperature regime in the smalltemperature limit functions p αβ , J αβth , and J αβγth havenonzero value. Moreover, they are associated with thenormal fluid. The value of parameters u αi , t αi , and ν αβi have the following behavior. If we consider their valuefrom the quasi-classic point of view we can state that forsome particles being in the BEC state these parametersare equal to zero. While for other particles being in theexcited states these parameters are non zero, and it leadsto nonzero value of functions p αβ , J αβth , and J αβγth . How-ever, more accurate quantum picture is a bit different.Each boson at small temperature has probability to bein the BEC state and in some excited states. Hence, pa-rameters u αi , t αi , and ν αβi can be nonzero for all particles,but their values are reduced by the partial probability tofind each particle in the excited states. III. TRANSITION TO THE TWO-FLUIDHYDRODYNAMICS
Basic hydrodynamic functions are additive on the par-ticles on the system. Therefore, they are additive on thesubsystems. So, we have the separation of the concen-tration n = n B + n n , where subindex B refers to theBose-Einstein condensate, and subindex n refers to thenormal fluid. Same is true for the current j = j B + j n ,the momentum flux Π αβ = Π αβB + Π αβn , the spin den-sity S = S B + S n , the spin current J αβ = J αβB + J αβn ,the nematic tensor N αβ = N αβB + N αβn , and the flux ofthe nematic tensor J αβγN = J αβγN,B + J αβγN,n . Logically, firstwe separate the basic hydrodynamic functions on twosubsystems. Second, we introduce the velocity field ineach subsystem in accordance with the method describedabove.It leads to the following set of equations ∂ t n a + ∇ · j a = 0 , (47) ∂ t j αa + ∂ β Π αβa = − m n a ∂ α V ext + 1 m F αint,a , (48) ∂ t S αa + ∂ β J αβa = − p ¯ h ε αzγ S γa + 2 q ¯ h ε αzγ N zγa , (49)and ∂ t N αβa + ∂ γ J αβγN,a = p ¯ h ε zαγ N βγa + p ¯ h ε zβγ N αγa − q h ( ε zαγ δ βz + ε zβγ δ αz ) S γa + F αβN,a , (50)where a stands for B and n , F αint,a is the force field actingon species a being caused by both species, F αβN,a is the sec-ond rank force field tensor giving the contribution of theshort-range interaction in the nematic tensor evolutionequation which acts on species a .Partial fluxes can be represented via the partial ve-locity fields in accordance with equations derived above: j = j B + j n = n B v B + n n v n , Π αβa = n a v αa v βa + p αβa , p αβB = 0, J αβa = S αa v βa , J αβγN,a = N αβa v γa , where the ther-mal part of the spin current, and the nematic tensor fluxare neglected.The force fields existing in the partial Euler equationsappear from the expression (15), where the function un-der the derivative appears as the source of the force, whilethe second function in the term present the subsystemmoving under action of this force F αint,B = − g n B ∂ α n B − g n B ∂ α n n − g S B ∂ α S B − g S B ∂ α S n , (51)and F αint,n = − g n n ∂ α n B − g n n ∂ α n n − g S n ∂ α S B − g S n ∂ α S n . (52)The second rank force field tensors have the followingform in accordance with the separation of equation (20): F αβN,B = g ¯ h (cid:20) ε βγδ N αδB (cid:18) S γn + S γB (cid:19) + ε αγδ N βδB (cid:18) S γn + S γB (cid:19)(cid:21) , (53)and F αβN,n = 2 g ¯ h (cid:20) ε βγδ N αδn (cid:18) S γn + S γB (cid:19) + ε αγδ N βδn (cid:18) S γn + S γB (cid:19)(cid:21) . (54) In equations (53) and (54) the spin densities S B and S n present the source of the force field acting on the nematictensor of BEC in (53) and the normal fluid in (54).Further calculation of the quantum terms in (38), (39),(40) allows to obtain their approximate explicit formsin term of the hydrodynamic functions. Correspondingresults for the BEC can be found in Ref. [44] (see eq.24 for the spin current, eq. 26 for the current of thenematic tensor, the divergence of the momentum flux ispresented in the Euler equation (28), its quantum part isproportional to the square of the Planck constant). Samestructures can be approximately used for the normal fluidas the equations of state for the corresponding functions.It is necessary to mention that the quantum terms arerequired to get NLSEs we need to use the explicit formof the quantum terms. A. Nonlinear Schrodinger equations
The Gross-Pitaevskii equation is the famous exampleof the non-linear Schrodinger equations used for the mod-eling of spin-0 neutral BECs.If we consider the microscopic dynamics of spin-0BECs we can derive the quantum hydrodynamic equa-tions by the method presented above (see also [10]).Further construction of the macroscopic wave functionusing the hydrodynamic functions allows to derive theGross-Pitaevskii equation from the quantum hydrody-namic equations. This step complites the derivation ofthe Gross-Pitaevskii equation from the microscopic quan-tum motion by the quantum hydrodynamic method [8],[9], [10].If we consider particles with the nonzero spin we donot derive the non-linear Schrodinger equation from thesingle fluid (if we consider the BEC only) model. How-ever, we construct the non-linear Schrodinger equationin order to give same hydrodynamic equations like thequantum hydrodynamics obtained from the microscopicmotion. Same method is used to obtain the pair of thenon-linear Schrodinger equations for the BEC and thenormal fluid of the spin-1 ultracold bosons.Therefore, the non-linear Schrodinger equations havethe following form: ı ¯ h∂ t ˆΦ B = − ¯ h ∇ m + V ext − p ˆ F z + q ˆ N zz + g ( n B + 2 n n ) + g ( S B + 2 S n ) ˆ F ! ˆΦ B , (55)and ı ¯ h∂ t ˆΦ n = − ¯ h ∇ m + V ext − p ˆ F z + q ˆ N zz +2 g ( n B + n n ) + 2 g ( S B + S n ) ˆ F ! ˆΦ n , (56)where the three component vector macroscopicwave functions have the following structureˆΦ a = √ n a e ımφ a / ¯ h ˆ z a . The concentrations n a is thesquare of module of the macroscopic wave function n a = ˆΦ † a ˆΦ a . The spin densities S a are also representedvia the macroscopic wave function S a = ˆΦ † a ˆ F ˆΦ a . Otherhydrodynamic functions like j α , Π αβ , N αβ , J αβ , and J αβγN are also expressed via the macroscopic wavefunctions ˆΦ a . Corresponding equations are shown inAppendix. IV. COLLECTIVE EXCITATIONS FOR THEFERROMAGNETIC PHASE: QUASI-CLASSICREGIME
We focused on the part of the spectrum of the bulkcollective excitations caused by the interparticle interac-tion. Hence, the spin textures related to the quantumkinematic motion is neglected.The ferromagnetic equilibrium state (phase) is consid-ered. Hence, the nematic tensor density is constructed ofthe concentration of bosons n a and the spin density S a . A. Zero temperature limit
Before we present the spectrum of the collective excita-tions existing at the finite temperatures we calculate thezero temperature spectrum appearing from the presentedmodel of spin-1 BECs.Let us consider the equilibrium state of uniform n B = const and the macroscopically motionless v B = 0BECs. In the ferromagnetic phase the spin density isalso nonzero S B = 0.In order to study the small amplitude collective exci-tations we present the approximate form of the hydro-dynamic equations containing terms up to first order onthe small amplitude: ∂ t δn B + n B ∇ δ v B = 0 , (57) mn B ∂ t δ v B = − g n B ∇ n B − g S β B ∇ δS γB , (58)and ∂ t δS αB + S α B ( ∇ δ v B )= − p ¯ h ε αzγ ( S γ B + δS γB ) + 2 q ¯ h ε αzγ ( N zγ B + δN zγB ) , (59)where N αβB = 12 δ αβ n B + 12 S αB S βB /n B . (60) It gives the equilibrium expression N αβ B = 12 δ αβ n B + 12 S α B S β B /n B , (61)and the linear perturbation δN αβB = 12 δ αβ δn B + 12 S α B δS βB /n B + 12 S β B δS αB /n B −
12 ( S α B S β B /n B ) δn B . (62)Equilibrium part of equation (59) existing on the right-hand side and has the following form ε αzγ ( − pS γ B + qδ zγ + qS z B S γ B /n B ) . (63)The second term is equal to zero since ε αzγ δ zγ = ε αzz =0. The rest should be equal to zero in order to satisfythe hydrodynamic equations with the chosen equationof state. It is satisfied for S B = S B e z . Moreover, allspin are parallel to each other in the ferromagnetic phase,hence S B = n B .We consider the propagation of the plane waves whichtravel in the arbitrary direction k = { k x , k y , k z } . Hence,the perturbations have the following structure illustratedwithin the concentration δn = N · e − ıωt + ı kr , where N isthe constant amplitude. Therefore, equations are trans-formed to the algebraic equations: ωδn B = n B ( k δ v B ) , (64) mn B ωδ v B = n B k ( g δn B + g δS zB ) , (65)and ωδS zB = S z B ( k δ v B ) , (66) and two other equations − ıωδS xB = ( p − q ) δS yB , (67)and − ıωδS yB = − ( p − q ) δS xB . (68)Set of hydrodynamic equations splits on two sets ofequation. One for δn B , δ v B , and δS zB which gives theacoustic wave spectrum: ω = ( g + g ) n k /m. (69)Another set is for δS xB and δS yB which give the constantfrequency in the quasi-classic limit: ω = | q − p | / ¯ h. (70)Since the nematic tensor density is reduced to the con-centration and the spin density we do not consider thenematic tensor evolution equation. However, the nematictensor evolution equation provides additional wave solu-tion for the partially spin polarized BECs, where the de-formation mode appears. It is related to elements N xx , N yy , and N xy . Moreover, the nematic tensor evolutioncontributes in the dynamics of the spin wave via the evo-lution of elements N xz and N yz for the partially spinpolarized BECs.0 B. Spectrum in two-fluid hydrodynamics of theBEC and normal fluid
We consider the plane wave small amplitude perturba-tions of the equilibrium state of the ferromagnetic phaseof the finite temperature spin-1 bosons. The equilib-rium state is characterized by two constant concentra-tions n B , n n , zero velocity fields v B = 0, v n = 0,and the spin densities S B = S B e z and S n = S n e z ,where S B = n B and S n = n n . Hence, we have thefollowing set of equations for the perturbations: ωδn a = n a ( k δ v a ) , (71) mn B ωδ v B = n B k ( g δn B + g δS zB )+2 n B k ( g δn n + g δS zn ) , (72) mn n ωδ v n = 2 n n k ( g δn B + g δS zB )+2 n n k ( g δn n + g δS zn ) , (73)and ωδS za = S z a ( k δ v a ) , (74) and for equations for the spin projection evolution − ıωδS xa = ( p − q ) δS ya , (75)and − ıωδS ya = − ( p − q ) δS xa . (76)The spin waves for the BEC and for the normal fluidhave same frequency in the ferromagnetic phase: ω = | q − p | / ¯ h. (77)We have the following set of equations for the pertur-bations of the concentrations of the BEC and the normalfluid mω δn B = n B k ( g + g )[ δn B + 2 δn n ] , (78)and mω δn n = 2 n n k ( g + g )[ δn B + δn n ] . (79)Presence of the spin density projection in the directionof the equilibrium spin polarization δS za = δn a changesthe speed of two sounds via the shift of the interactionconstant to g + g in the same way as it is happens forthe zero temperature. V. CONCLUSION
The spin-1 BECs have been actively studied over 20years. Corresponding non-linear Schrodinger equationshave been used in the large part of these studies. This pa-per has been focused on the derivation of the well-knownmodel from the microscopic motion of the quantum par-ticles, where the many-particle Schrodinger equation inthe coordinate representation has been used as the start-ing point of the derivation. Moreover, the generalizationof the existing minimal coupling model has been madeto include the contribution of the small temperatures.Therefore, the system of spin-1 bosons in presented as thesystem of two quantum liquids (more like two collectivedegrees of freedom of single species) the BEC presentingparticles in the quantum states with the lowest energyand the normal quantum fluid presenting particles beingin the quantum states with energies above the minimalenergy.Hydrodynamic equations are derived for the arbitraryspin state of liquid. However, the small temperature fer-romagnetic phase and the polar phase are discussed asthe limit regimes of the general model.The set of hydrodynamic equations obtained from theSchrodinger equation leads to the set of 2 × N × N is the number of particles, the multiplier ”2” comesfrom the fact that spinless part of evolution of each par-ticle is described by single complex function identical totwo real functions, the multiplier ”3” comes from threeprojections of spin for each particle. So, it is almost”infinite” number of equations. Therefore, it should betruncated.The step to make the truncation is made in accordancewith the existing with the minimal coupling model givingthe non-linear Schrodinger equation with three compo-nent vector-spinor macroscopic wave function. There-fore,the evolution of the concentration, velocity field,spin density, and the nematic tensor density are derived.These equations are derived for the arbitrary temper-ature and the arbitrary interaction strength. Next, theterm describing the interaction are calculated in the smalltemperature limit for the weakly interacting particleswith the small range of action of the interaction poten-tial.Finally, each equation is separated on two parts: onefor the particles in the BEC state, another for the nor-mal fluid. Hence, the two-fluid model of the small-temperature spin-1 bosons has been obtained.These equations have been used to study the bulk smallamplitude collective excitations in the form of the planewaves in the infinite medium. VI. ACKNOWLEDGEMENTS
Work is supported by the Russian Foundation for BasicResearch (grant no. 20-02-00476). This paper has been1supported by the RUDN University Strategic AcademicLeadership Program.
VII. APPENDIX: HYDRODYNAMICFUNCTIONS IN TERMS OF MACROSCOPICWAVE FUNCTIONS
During the derivation of the hydrodynamic equationsfrom the microscopic model we present the definitions ofthe hydrodynamic functions via the many-particle micro-scopic wave functions. However, the introduction of thenon-linear Schrodinger equations for the effective macro-scopic wave function requires the presentation of the re-lation of the hydrodynamic functions with this effectivewave function. The concentration and the spin densityare presented above, while other functions are given herein the appendix: j αa = 12 m ( ˆΦ † a ˆ p α ˆΦ a + h.c. ) , (80) Π αβa = 14 m ( ˆΦ † a ˆ p α ˆ p β ˆΦ a + (ˆ p α ˆΦ) † a ˆ p β ˆΦ a + h.c. ) , (81) N αβa = 12 ˆΦ † a ( ˆ F α ˆ F β + ˆ F β ˆ F α ) ˆΦ a , (82) J αβa = 12 m ( ˆΦ † a ˆ F α ˆ p β ˆΦ a + h.c. ) , (83)and J αβγN,a = 14 m ( ˆΦ † a ( ˆ F α ˆ F β + ˆ F β ˆ F α )ˆ p γ ˆΦ a + h.c. ) . (84) [1] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S.Stringari, ”Theory of Bose-Einstein condensation intrapped gases”, Rev. Mod. Phys. , 463 (1999).[2] D. 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