One-Dimensional Integrable Spinor BECs Mapped to Matrix Nonlinear Schr"odinger Equation and Solution of Bogoliubov Equation in These Systems

T ypes et with jpsj3.cls < ver.1.1 > Shor t Note One-Dimensiona l In tegrable Spinor BECs Mapp ed to Matrix Nonlinear Sc hr¨ odinger Equation and Solution of Bogoliub o v Equation in These Systems Daisuke A. T ak ahashi ∗ Dep a rtment of Basi c Scienc e, the University of T okyo, T okyo 153-8902 KEYW ORDS: mat rix nonlinear Schr¨ odinger equation, spino r Bose-Einstein condensate, Bogo liub ov equa- tion, squa red Jost funct ion. Nonlinear Sc hr¨ odinger equation(NLSE) is one of the oldest in tegrable nonlinear equations, which was solved by the inv erse sc a ttering method(ISM). 1) In the con- text of condensed matter physics, NLSE is also called Gross-P itaevskii equation, and it describ es the dynamics of one-dimensional(1D) Bose-Einstein condensate(BEC). Recently , in tegrable 1D BECs with internal spin degree of freedom (spinor BE C s ) ha v e been dis cov ered. 2, 3) First, the spin-1 in tegrable system has been discovered 2) by finding a mapping to the matrix NLSE(MNLSE): 4) i ∂ t Q = − ∂ 2 x Q ± 2 QQ † Q. (1) Here Q is a square matrix, and the min us(plus) sign cor- resp onds to the self-(de)focusing case. In the con text of BEC, the minus(plus) sig n represents the system o f at- tractive(repulsive) bo sons. Subsequently , integrable BEC systems ar e found for every int eger spin- n . 3) The ener gy functional of integrable spinor BECs ar e given by 3) H = Z d x  n X m = − n     ∂ ψ m ∂ x     2 ± 1 2 n − 1  ρ 2 − 1 2 | Θ | 2   , (2) where ψ m is a condens ate wav efunction with m agnetic quantum n um ber m , ρ = P n m = − n | ψ m | 2 is a n um b e r density , and Θ = P n m = − n ( − 1) m ψ m ψ − m is a singlet pair amplitude. The co efficient 2 n − 1 can b e eliminated by a choice of normalizatio n, but we keep it for la ter conv e- nience. In this s hort note, we show that the above integrable systems (2) are all ma pped to MNLSE, a nd solve the Bogoliub ov equation of these systems. While the in tegra- bilit y itself has b een already prov ed b y using a different Lax pair , 3) the mapping to MNLSE has the following adv an tages: (i) The explicit express io n for N -soliton so- lution is alr eady g iven under both v a nis hing 4) and non- v a nishing 5) bo undary co nditions. (ii) With the aid of the theory of squared eig enfunction(or squared Jost func- tion), 6, 7) solutions of Bogoliub ov equation can be ob- tained. Since there exis t v arious k inds of lo w-energy ex - citations in spinor BEC systems, 8) it would b e an in- teresting future w ork to solv e the scattering problem of low-energy e x citations in the pr esence of an exter nal po - ten tial 9) by using the so lution given in this short note. Mapping to MNLSE— W e first wr ite the mapping ma- ∗ E-mail address: tak ahashi@v ortex.c.u-toky o.ac .jp trix and its (anti)symmetrization, and next expla in how to find it. The mapping from spin- n in tegrable s pino r BEC to 2 n -dimensional MNLSE is given b y Q = 1 2 n/ 2  n X m =1 h ψ m ( − 1) m I ⊗ ( n − m ) ⊗ σ + ⊗ σ ⊗ ( m − 1) z + ψ − m I ⊗ ( n − m ) ⊗ σ − ⊗ σ ⊗ ( m − 1) z i + ψ 0 σ ⊗ n z  . (3) Here σ i ( i = x, y , z ) ar e P auli matrice s , σ ± = 1 √ 2 ( σ x ± i σ y ), I is 2 × 2 identit y matr ix , and A ⊗ i := A ⊗ · · · ⊗ A | {z } i time s . (for i = 0, we define A ⊗ 0 := 1.) The energy functional which yields E q. (1) is H = Z d x  tr  ∂ Q † ∂ x ∂ Q ∂ x  ± tr( Q † QQ † Q )  . (4) Substituting the matrix (3) to this functional, one ca n obtain the functional (2). F urther more, we can (anti)symmetrize the above ma - trix by the unitary matrix V = ( (i σ y ⊗ σ x ) ⊗ n/ 2 ( n : even) (i σ y ⊗ σ x ) ⊗ ( n − 1) / 2 ⊗ (i σ y ) ( n : o dd) . (5) The matrix ˜ Q = Q V then s atisfies ˜ Q T = ( − 1) 1 2 n ( n − 1) ˜ Q . W e briefly summarize how to construct the matrix (3). In the theo ry o f Lie a lgebra, 10) writing the elements of Cartan subalgebra a s H 1 , . . . , H r ( r : r ank of this algebra ) and ra ising or lo w ering o p e rators a s E α (whic h c hanges weigh t fro m µ to µ + α ), an ir reducible represe ntation D is determined from the following relations: H i | µ, D i = µ i | µ, D i , (6) E α | µ, D i = N α,µ | µ + α, D i . (7) Here µ = ( µ 1 , . . . , µ r ) is a weight v ector, and N α,µ is a representation-dep endent cons tant. The ir reducible ten- sor oper ator 11) corres p o nding to this repr esentation, D , is characterized by the fo llowing commutation rela tions: [ H i , T µ,D ] = µ i T µ,D , (8) [ E α , T µ,D ] = N α,µ T µ + α,D . (9) Now, let us cons ider the alg ebra so (2 n + 1). W e con- struct the tensor oper ators corresp onding to 2 n + 1- dimensional fundamen tal representation (we write it a s 2 n + 1 — it is unitary equiv alen t to the definition of the algebra itse lf.). As for the matrix r epresentation of g e n- erators , H i s and E α s, w e use the 2 n -dimensional spinor representation. Therefor e tensor oper ators a ls o b ecome 2 n -dimensional matrices. In this r epresentation, genera - tors are represe n ted by the n -fo ld tenso r pro duct of Pauli matrices. 10) One ca n determine the ma tr ix elements of T µ, 2 n +1 ( µ = 0 , ± e 1 , . . . , ± e n ) thro ugh the co mm utation relations (8 ) and (9), and obta in the matrices app earing in Eq. (3). Bo goliub ov e quation — Next, w e solve the Bogoliub ov equation of thes e integrable sys tems. Since we a re in- terested in an application to the scattering problem of collective excitations, 9) we discuss the pr oblem of non- 1 2 J. Phys. Soc. Jpn. Shor t Note Author Name v a nishing b oundar y condition, 5) and consider MNLSE of the self-defo cusing ca se with a chemical po ten tial term: i ∂ t Q = − ( ∂ 2 x + µ ) Q + 2 QQ † Q. (10) The Bogoliub ov equation, though its standard deriv a - tion is diagonalization of seco nd-quantized Hamiltonian in mea n field approximation, is easily obtained by substi- tuting Q = Q + δ Q in Eq. (10) and discarding the higher order terms of δ Q . Rewriting ( δ Q, − δQ † ) = ( U , V ), one obtains i ∂ t U = − ( ∂ 2 x + µ ) U + 2  QQ † U + U Q † Q − QV Q  , i ∂ t V = ( ∂ 2 x + µ ) V − 2  Q † QV + V QQ † − Q † U Q †  . (11) ISM of MNLSE is formulated through the following ex- tended Zakha r ov-Shabat t ype eigen v a lue pr oblem: 4, 5) ∂ ∂ x  ~ f ~ g  =  − i λ Q Q † i λ  ~ f ~ g  , (12) ∂ ∂ t  ~ f ~ g  =  i( − 2 λ 2 + 1 2 µ − QQ † ) 2 λQ + i ∂ x Q 2 λQ † − i ∂ x Q † i(2 λ 2 − 1 2 µ + Q † Q )   ~ f ~ g  . (13) Here λ is a sp ectral parameter, and ~ f and ~ g , which are called Jost function, ar e n -co mpo nent vectors when Q is an n × n matrix. The co mpatibilit y co ndition ∂ x ∂ t ( ~ f , ~ g ) T = ∂ t ∂ x ( ~ f , ~ g ) T repro duces E q. (10). Assume that Q is (anti)symmetric: Q = εQ T ( ε = ± 1). W e can then show by a straight forward calculation that if ( ~ f 1 , ~ g 1 ) T and ( ~ f 2 , ~ g 2 ) T are the solutions of E qs. (12 ) and (13) with the same λ , ( U, V ) = ( ~ f 1 ~ f T 2 , − ε ~ g 1 ~ g T 2 ) sa t- isfies Eqs. (11). Th us, the squar ed Jost function gives a solution of Bo goliub ov equation. Example — As a n example, let us consider the spin-1 case in the pres ence of one dark so liton. 12) W e w an t the stationary so lution o f Bo goliub ov equatio n with eigenen- ergy ǫ , which is obtained by the replace ment i ∂ t → ǫ in Eqs. (11). Co rresp onding ly , we mu st co ns ider Eqs. (12) and (13) with i ∂ t → ǫ/ 2, since a quadratic form of ~ f and ~ g gives a solutio n o f Bogoliub ov equation. In order to simplify the mathema tical de s criptions, w e first prepare several notations for the s olutions of one- comp onent NLSE (i.e., a scalar BEC). The o ne dark s o li- ton solution in the como ving fra me is φ s ( x ; x 0 ) = − e i ϕ e i px [ p + i q tanh( q ( x − x 0 ))] , (14) where p = − λ 0 cos ϕ, q = λ 0 sin ϕ, λ 0 > 0 and ϕ ∈ (0 , π ). The subscript “s” means scalar. F o r this solution, the chemical p otential b ecomes µ = p 2 + 2 λ 2 0 . W e also de- fine φ s ( x ; −∞ ) := λ 0 e i px . The solution of Bo goliub ov equation is giv en by ( u, v ) = ( f 2 , − g 2 ). Considering the solutions in the uniform sy stem, one can show that the wa v en um be r of a n excita tio n k and the sp e ctral par ame- ter λ are related as k = − ǫ/ (2 λ − p ). The expression fo r ( f , g ) in the pres ence of one dar k soliton is given b y 7)  f s ( x ; x 0 ) g s ( x ; x 0 )  = e i kx/ 2 × (15)  e i( px + ϕ ) / 2  i q tanh( q ( x − x 0 )) + ( k / 2 ) + ( ǫ/ 2 k )  − ie − i( px + ϕ ) / 2  i q tanh( q ( x − x 0 )) + ( k / 2 ) − ( ǫ/ 2 k )   , where the wav en um ber k satisfies the disp er s ion re- lation ( ǫ − 2 k p ) 2 = k 2 ( k 2 + 4 λ 2 0 ). W e als o define f s ( x ; −∞ ) := e i[( k + p ) x + ϕ ] / 2 [i q + ( k / 2) + ( ǫ/ 2 k )] and g s ( x ; −∞ ) := − ie i[( k − p ) x − ϕ ] / 2 [i q + ( k / 2) − ( ǫ/ 2 k )]. Having pr epared the notations, let us mov e on to the spin- 1 system. The symmetrized matrix is giv en by Q =  ψ 1 ψ 0 / √ 2 ψ 0 / √ 2 ψ − 1  , a nd cor resp ondingly , Bo g oli- ubov wa v efunctions become U =  u 1 u 0 / √ 2 u 0 / √ 2 u − 1  and V =  v 1 v 0 / √ 2 v 0 / √ 2 v − 1  . The one dar k so lito n s o lution 12) can be diagonalized as Q = D  φ s ( x ; x 1 ) 0 0 φ s ( x ; x 2 )  D T . (16) Here D is a rotation matrix of spin- 1/2. (Note that the rightmost term is not D † but D T . 12) ) If either x 1 or x 2 equals to −∞ , Eq. (16) represents a ferro magnetic soli- ton. 12) If b o th are finite, it r epresents a po lar soliton. 12) Jost functions a re given by  ~ f ~ g  =  f s ( x ; x i ) D~ e i g s ( x ; x i ) D ∗ ~ e i  , ( i = 1 , 2) , (17) where ~ e 1 = (1 , 0) T and ~ e 2 = (0 , 1) T . F rom these J o st functions, we can construct the solutio ns of the Bogo li- ubov equation as  U V  =  f s ( x ; x 1 ) 2 D ( I + σ z ) D T − g s ( x ; x 1 ) 2 D ∗ ( I + σ z ) D †  ,  f s ( x ; x 2 ) 2 D ( I − σ z ) D T − g s ( x ; x 2 ) 2 D ∗ ( I − σ z ) D †  ,  f s ( x ; x 1 ) f s ( x ; x 2 ) D σ x D T − g s ( x ; x 1 ) g s ( x ; x 2 ) D ∗ σ x D †  . (18) In the integrable case, the disp ersio n rela tions o f excita- tions of the spin fluctuation a nd the density fluctuation are degener ate. (See Ref. 8 for the genera l case ; Note that the asymptotic for m o f Eq. (16 ) is alwa ys p olar state, re- gardless o f whether the soliton is ferro magnetic or p olar.) Therefore any linear com bination of the above Eq. (18 ) bec omes the solutio n. How ev er, if o ne wan ts to separate the s pin fluctua tio n part a nd the density fluctua tion part, one nee ds to mak e a sp ecial linear combination of them. 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