Fordy-Kulish models and spinor Bose-Einstein condensates

Journal of Nonlinear Mathematical P hysics V olume *, Numb er * (20**), 1– 8 Ar ticle F ordy-Kulish mo del and spino r Bose-Ei nstein condens ate V. A. Ata nasov 1 , 2 , V. S. Ger djikov 1 , G. G. Gr aho vski 1 , 3 , N. A. Kostov 1 1 Institute for Nucle ar R ese ar ch and Nucle ar Ener gy, Bulgarian A c ademy of Scienc es, 72 Tsarigr ad sko chauss ´ ee, 1784 Sofia, Bulgaria 3 L ab or a toir e de Physique Th ´ eorique et Mo d´ elisation, Universit´ e d e Cer gy-Pontoise, 2 avenue A. Chauvin, F-95302 Cer gy-Pontoise Ce dex, F r anc e 3 Scho ol of E le ctr onic E ngine ering, Du b lin City University, Gla snevin, Dublin 9, Ir eland E-mail: victor@inrne.b as.b g, gerjikov@inrne.b as.b g, gr ah@inrne.b as.b g, nakostov@inrne.b as.b g R e c eive d Mo nth *, 200*; A c c epte d Month *, 200 * Abstract A three-comp onent nonlinear Schrodinge r-t yp e mo del whic h describ es spinor Bose- Einstein condens a te ( BEC ) is considered. This mo del is in tegrable by the in verse scattering method and using Zakharov-Shabat dres sing metho d we obtain three types of soliton solutio ns. The m ulti-comp onent no nlinear Schr¨ odinger type mo dels related to symmetric spaces C . I ≃ Sp(4) / U(2) is studied. 1 In tro du ction The dynamics of spinor BEC is d escrib ed by a thr ee-c omp onen t Gross-Pitaevskii (GP) system of equations. In the one-dimensional ap p ro ximation the GP sy s tem go es into the follo wing nonlinear Sc hr¨ odinger (MNLS) equation in (1D) x -space [1]: i∂ t Φ 1 + ∂ 2 x Φ 1 + 2( | Φ 1 | 2 + 2 | Φ 0 | 2 )Φ 1 + 2Φ ∗ − 1 Φ 2 0 = 0 , i∂ t Φ 0 + ∂ 2 x Φ 0 + 2( | Φ − 1 | 2 + | Φ 0 | 2 + | Φ 1 | 2 )Φ 0 + 2Φ ∗ 0 Φ 1 Φ − 1 = 0 , (1) i∂ t Φ − 1 + ∂ 2 x Φ − 1 + 2( | Φ − 1 | 2 + 2 | Φ 0 | 2 )Φ − 1 + 2Φ ∗ 1 Φ 2 0 = 0 . W e consider BEC of alcali atoms in the F = 1 hyper fi ne state, elongated in x direction and confined in the transverse d irections y , z by purely optical means. Th us th e assem bly of atoms in the F = 1 hyperfi ne state can b e describ ed by a normalized spinor w av e ve ctor Φ ( x, t ) = (Φ 1 ( x, t ) , Φ 0 ( x, t ) , Φ − 1 ( x, t )) T (2) whose comp onents are labelled b y the v alues of m F = 1 , 0 , − 1. The a b o ve mo del is in tegrable by means of in v erse scattering transform metho d [1]. It also allo ws an exact description of the dynamics and in teraction of b righ t solitons with spin degrees of freedom. Matter-w a ve solitons are exp ected to b e us eful in atom laser, atom in terferometry and coheren t atom transp ort. It could con tribu te to the realization of quantum in formation pro cessing or computation, as a part of new field of atom optics. Lax p airs and geometric in terpretation of the mo del (1) are given in [4]. Darb oux tran s formation for th is sp ecial in tegrable mo del is d ev elop ed in [5]. The aim of pr esen t p ap er is to sho w that the system (1) is related to sym m etric space C . I ≃ Sp (4) / U(2) (in the Cartan classification [8]) with canonical Z 2 -reduction and has natur al Lie algebraic in terpretation. The mo del allo ws also a sp ecial class of soliton solutions. W e will show that they can b e obtained b y a suitable mo difi catio n of the generalization of the so-call ed “dressing metho d”, prop osed in [9]. 2 V. A. Ata naso v, V. S. Gerd jik o v, G. G. Gr ah ovski and N. A. Kosto v 2 Solving the mo del: algebraic and analytic asp ects. The mo del (1) b elongs to the class of m u lticomponent NLS equations th at can b e solv ed b y the inv erse scattering metho d [7 , 6]. It is a particular case of the MNLS related to the C . I typ e symmetric space Sp (4) / U( 2) [4]. These MNLS systems allo w Lax repr esen tation with the generalized Zakharov–Shabat system as the Lax op erator: Lψ ( x, t, λ ) ≡ i dψ dx + ( Q ( x, t ) − λJ ) ψ ( x, t, λ ) = 0 . (3) where J and Q ( x, t ) are 4 × 4 matrices: J = diag (1 , 1 , − 1 , − 1) and Q ( x, t ) is a blo c k-off- diagonal matrix: Q ( x, t ) =  0 q ( x, t ) p ( x, t ) 0  , q ( x, t ) =  Φ 0 ( x, t ) − Φ 1 ( x, t ) Φ − 1 ( x, t ) − Φ 0 ( x, t )  , p ( x, t ) =  Φ ∗ 0 ( x, t ) Φ ∗ − 1 ( x, t ) − Φ ∗ 1 ( x, t ) − Φ ∗ 0 ( x, t )  . (4) Solving the direct and the inv erse scattering pr oblem for L uses the Jost solutions φ = ( φ + , φ − ) and ψ = ( ψ − , ψ + ) of (3) wh ich are defin ed by , see [14] and the references therein: lim x →−∞ φ ( x, t, λ ) e iλJ x = 1 1 , lim x →∞ ψ ( x, t, λ ) e iλJ x = 1 1 (5) These definitions are compatible with the c lass of sm ooth potenti als Q ( x, t ) v anish ing sufficien tly rapidly at x → ± ∞ . It can b e shown th at φ + and ψ + (resp. φ − and ψ − ) comp osed b y 4 rows and 2 columns are analytic in the upp er (resp . low er) half plane of λ . The scattering matrix asso ciated to (3) is defin ed as T ( t, λ ) = ( ψ ( x, t, λ )) − 1 φ ( x, t, λ ) =  a + ( t, λ ) − b − ( t, λ ) b + ( t, λ ) a − ( t, λ )  , ( T ( t, λ )) − 1 =  c − ( t, λ ) d − ( t, λ ) − d + ( t, λ ) c + ( t, λ )  , (6) where a ± ( t, λ ) and b ± ( t, λ ) are 2 × 2 blo c k matrices. The blo cks a ± , b ± , c ± and d ± satisfy a n um b er of relations [11, 12]; for example a + ( λ ) c − ( λ ) + b − ( λ ) d + ( λ ) = 1 1 , a + ( λ ) d − ( λ ) − b − ( λ ) c + ( λ ) = 0 , (7) etc. The fu ndamen tal analytic solutions (F AS) χ ± ( x, t , λ ) of L ( λ ) are analytic f unctions of λ for Im λ ≷ 0 and are related to the J ost solutions by: χ ± ( x, t , λ ) = φ ( x, t, λ ) S ± J ( t, λ ) = ψ ( x, t, λ ) T ∓ J ( t, λ ) . (8) Here S ± J , T ± J upp er- and low er- blo ck- triangular matrices: S + J ( t, λ ) =  1 1 d − ( t, λ ) 0 c + ( t, λ )  , S − J ( t, λ ) =  c − ( t, λ ) 0 − d + ( t, λ ) 1 1  , T + J ( t, λ ) =  1 1 − b − ( t, λ ) 0 a − ( t, λ )  , T − J ( t, λ ) =  a + ( t, λ ) 0 b + ( t, λ ) 1 1  , F ordy-Ku lish mo dels and sp inor Bose-Einstein condensates 3 satisfying T ± J ( t, λ ) ˆ S ± J ( t, λ ) = T ( t, λ ) and can b e view ed as the factors of a generalized Gauss decomp ositions of T ( t, λ ) [13]. If Q ( x, t ) ev olve s acco rding to (1) then the scattering matrix and its elemen ts satisfy the follo win g linear evo lution equations: i db ± dt + 2 λ 2 b ± ( t, λ ) = 0 , i da ± dt = 0 , (9) so the blo c k-matrices a ± ( λ ) can b e considered as generating functionals of the int egrals of motion. The fact that all 4 matrix elements of a + ( λ ) f or λ ∈ C + (resp. of a − ( λ ) f or λ ∈ C − ) generate in tegrals of motion reflect th e su p erin tegrabilit y of th e mo del and are due to the d egeneracy of the disp ersion law of (1). The system (1) can b e written in a Hamiltonian form b y introd u cing the Poi sson brac kets: { q j ( x ) , p k ( y ) } = 2 iδ k j δ ( x − y ) , { q 12 ( x ) , p 12 ( y ) } = iδ ( x − y ) , (10) and the Hamiltonian H = H kin + H int : H kin = Z ∞ −∞ dx  ∂ Φ 0 ∂ x ∂ Φ ∗ 0 ∂ x + 1 2  ∂ Φ 1 ∂ x ∂ Φ ∗ 1 ∂ x + ∂ Φ − 1 ∂ x ∂ Φ ∗ − 1 ∂ x  , (11) H int = − Z ∞ −∞ dx  ( | Φ 0 | 2 + | Φ 1 | 2 ) 2 + ( | Φ 0 | 2 + | Φ − 1 | 2 ) 2  − Z ∞ −∞ dx  | Φ 0 Φ ∗ − 1 + Φ 1 Φ ∗ 0 | 2  . As ment ioned ab ov e, one can use an y of the matrix elemen ts of a ± ( λ ) as generating functional of in tegrals of motion of our mo del. Generically su c h inte grals wo uld ha v e non-lo cal d en sities and will not b e in inv olution. The classical R -matrix appr oac h [6, 4] is an effectiv e metho d to determine the generat- ing fu nctionals of lo cal int egrals of motion w hic h are in inv olution. F rom it there follo ws that suc h integ rals are generated by expan d ing ln m ± k ( λ ) o v er the in v erse p ow ers of λ , see [13]. Here m ± k ( λ ) are the principal minors of T ( λ ); in our case m + 1 ( λ ) = a + 11 ( λ ) , m + 2 ( λ ) = d et a + ( λ ) , m − 1 ( λ ) = a − 22 ( λ ) , m − 2 ( λ ) = d et a − ( λ ) . (12) If w e consid er ln m + k ( λ ) = ∞ X s =1 λ − k I ( k ) s , then one can p ro ve th at th e densities of I ( k ) s are lo cal in Q ( x, t ). The fact that [13]: { m ± k ( λ ) , m ± j ( µ ) } = 0 , f or k , j = 1 , 2 , and for all λ, µ ∈ C ± allo w on e to conclude that { I ( k ) s , I ( j ) p } = 0 for all k , j = 1 , 2 and s, p ≥ 1. In particular, the Hamiltonian of our mo del is prop ortional to I (2) 3 , i.e. H = 8 iI (2) 3 . 4 V. A. Ata naso v, V. S. Gerd jik o v, G. G. Gr ah ovski and N. A. Kosto v 3 Soliton solutions for the spinor BEC: The so (5) con nection The soliton solutions of the sp (4) MNLS (1) we re d eriv ed b y using the d ressing metho d [3]. They can b e consid ered as p articular cases of the soliton solutions of the generic MNLS eqs., derive d through the matrix v ersion of the Gel’fand-Levitan-Marc henko equation, see [2, 1, 3]. Here w e extend furth er these resu lts and combining the ideas of [3, 15] we s p ecify three t yp es of solitons for the mo del (1). W e start our analysis with the w ell-kno wn isomorphism b etw een th e algebras sp (4 , C ) and so (5 , C ) [8]. Since the Lax representati on is of pur e algebraical n ature it is n atural to exp ect th at our mo del (1) can b e treated also by an equiv alen t Lax op erator L ′ whose p oten tial Q ′ ( x, t ) and J ′ tak e v alues in s o (5). A consequence of the ab o ve-men tioned isomorphism is that the typical rep r esen tation of sp (4) u sed ab o ve is equiv alent to the spinor r epresen tation of s o (5). So w e first remind some of our resu lts in [10, 11, 12], w h ere we h a v e constructed the fu ndamen tal analytic solutions, the dressing factors, the soliton solutions etc. f or a class of Lax op erators (includin g L ′ ), related to the simple Lie algebra g , in the t ypical represent ation of g . So we fi rst ha ve to sp ecify (if n ecessary) g ≃ so (5) and then reformula te the corresp onding results for the spinor representat ion of so (5). The main goal of th e d r essing metho d is, starting from a solution χ ± 0 ( x, t , λ ) of L 0 ( λ ) with p oten tial Q (0) ( x, t ) to construct a n ew sin gular solution χ ± 1 ( x, t , λ ) with singularities lo cated at prescrib ed p ositions λ ± 1 ; the reduction p = q † used in eq. (4) ensur es that λ − 1 = ( λ + 1 ) ∗ . T he n ew solutions χ ± 1 ( x, t , λ ) will corresp ond to a p otentia l Q (1) ( x, t ) of L ( λ ) (3) with t wo discrete eigenv alues λ ± 1 . It is related to the regular one by a d ressing factor u ( x, λ ) χ ± 1 ( x, t , λ ) = u ( x, λ ) χ ± 0 ( x, t , λ ) u − 1 − ( λ ) . u − ( λ ) = lim x →−∞ u ( x, λ ) (13) Note th at u − ( λ ) is a diagonal matrix. The d ressing factor u ( x, λ ) must satisfy the equation i du dx + Q (1) ( x ) u − uQ (0) ( x ) − λ [ J , u ( x, λ )] = 0 , (14) and the normalization condition lim λ →∞ u ( x, λ ) = 1 1. Besides χ ± i ( x, λ ), i = 0 , 1 and u ( x, λ ) m u st b elong to the corresp ond ing Lie group S p (4 , C ); in addition u ( x, λ ) b y construction has p oles and /or zero es at λ ± 1 . The construction of u ( x, λ ) is b ased on an app ropriate anzats sp ecifying explicitly the form of its λ -dep endence. u ( x, λ ) = 1 1 + ( c 1 ( λ ) − 1) P 1 ( x, t ) +  1 c 1 ( λ ) − 1  P 1 ( x, t ) , c 1 ( λ ) = λ − λ + 1 λ − λ − 1 , (15) where the pro jectors P 1 ( x, t ) and P 1 ( x, t ) are are of rank 1 and are related by P 1 ( x ) = S P T 1 ( x ) S − 1 . They must satisfy P 1 ( x, t ) P 1 ( x, t ) = P 1 ( x, t ) P 1 ( x, t ) = 0. By S w e ha v e denoted th e sp ecial matrix which en ters in the d efinition of the orthogonal algebra, i.e. X ∈ so (5) if X + S X T S − 1 = 0. In the typical representat ion of s o (5) w e hav e S = P 5 k =1 ( − 1) k +1 E k , 6 − k where ( E ij ) k m = δ ik δ j m . The construction of P 1 ( x, t ) and P 1 ( x, t ) F ordy-Ku lish mo dels and sp inor Bose-Einstein condensates 5 using the ‘p olarization’ vecto rs is outlined in [11] and we skip it. Th e new p oten tial is obtained from Q (1) ( x, t ) − Q (0) ( x, t ) = ( λ + 1 − λ − 1 )[ J, P 1 ( x, t ) − P 1 ( x, t )] . Here we show that the λ -dep enden ce of u ( x, λ ) ma y d ep end [10 ] on the choice of the represent ation of so (5 , C ) ≃ sp (4 , C )). F or so (5) it was s ho wn[10, 11, 12, 15] that there are three t yp es of solitons: • the firs t t yp e of soliton solutions are generated by dressing f acto rs of the form (15). F or generic c h oice of the p olarizati on vect ors P 1 ( x, t ) − P 1 ( x, t ) ∈ so (5). • the second typ e of soliton solutions are generated analogously with dressing factor (15), bu t d u e to a sp ecific choice of the p olarizatio n v ectors P 1 ( x, t ) − P 1 ( x, t ) ∈ so (3) ⊂ s o (5). • the third type of soliton solutions are generated again b y (15) b ut no w the corre- sp onding pro jectors P 1 ( x, t ) and P 1 ( x, t ) h a v e rank 2. Eac h of these types of soliton solutions h a ve their coun terpart relev an t to our mo d el on sp (4). T o the first t yp e of soliton solutions there corresp ond dressin g factor and p oten tial Q (1) ( x, t ) of the form [3]: ˜ u ( x, λ ) = p c 1 ( λ ) π 1 ( x ) + 1 p c 1 ( λ ) π 1 ( x ) , Q (1) ( x, t ) − Q (0) ( x, t ) = 1 2 [ J, π 1 ( x, t ) − π 1 ( x, t )] , (16) where π 1 ( x ) and π 1 ( x ) are r ank 2 pro jectors, such that π 1 ( x ) π 1 ( x ) = π 1 ( x ) , π 1 ( x ) = 0 , π 1 ( x ) + π 1 ( x ) = 1 1 . (17) This last prop erty ensur es the non-degeneracy of u ( x, λ ). No te that no w the dressing factor is not a rational function of λ bu t for the dressed F AS χ ( x, λ ) eq. (13) we get: χ ± 1 ( x, t , λ ) =  π 1 ( x, t ) + 1 c 1 ( λ ) π 1 ( x, t )  χ ± 0 ( x, t , λ )  π − 1 + c 1 ( λ ) π − 1  , π − 1 = lim x →−∞ π 1 ( x, t ) , (18) i.e., the fractional p o w er s of c 1 ( λ ) disapp ear. The second t yp e of solitons with rank 2 p ro jector P 1 ( x ) after recalculat ing to the spinor represent ation formally ke eps the same form (15) w ith P 1 ( x ) r eplaced by A 1 ( x ) w h ic h has rank 1 but generically is not a pro jector, see [3]. The third t yp e of solitons is s imilar to the second one b ut with additional constraint s on th e factor A 1 ( x ) so that A 1 ( x ) − A 1 ( x ) ∈ sp (2) ⊂ s p (4). Consider th e purely solitonic case when Q (0) = 0. F r om n o w on w e introdu ce the follo wing notations λ ± 1 = µ 1 ± iν 1 and A = − 2 i (( λ + 1 ) 2 − ( λ − 1 ) 2 ) t − i ( λ + 1 − λ − 1 ) x, B = − 2(( λ + 1 ) 2 + ( λ − 1 ) 2 ) t − ( λ + 1 + λ − 1 ) x. (19) 6 V. A. Ata naso v, V. S. Gerd jik o v, G. G. Gr ah ovski and N. A. Kosto v Here A ( x, t ) and B ( x, t ) are x and t d ep enden t real v alued fun ctions. Making use of the explicit f orm of the pro jectors P ± 1 ( x ) v alid for the t ypical represen tations of B 2 w e obtain[11]: Φ (1) ( x, t ) = 4( λ + 1 − λ − 1 ) h m | n i  n 0 , 1 m 0 , 2 e A + n 0 , ¯ 2 m 0 , ¯ 1 e − A  e iB (20) Φ (0) ( x, t ) = i 2 √ 2( λ + 1 − λ − 1 ) h m | n i  n 0 , 1 m 0 , 3 e A − n 0 , ¯ 3 m 0 , ¯ 1 e − A  e iB (21) Φ ( − 1) ( x, t ) = − 4( λ + 1 − λ − 1 ) h m | n i  n 0 , 1 m 0 , ¯ 2 e A + n 0 , 2 m 0 , ¯ 1 e − A  e iB (22) where the d enominator in th e ab ov e formula is giv en by: h m | n i = m 0 , 1 n 0 , 1 ( e 2 A ) + m 0 , ¯ 1 n 0 , ¯ 1 ( e − 2 A ) + m 0 , 2 n 0 , 2 + m 0 , ¯ 2 n 0 , ¯ 2 + m 0 , 3 n 0 , 3 . (23) and m 0 ,k , n 0 ,k are the comp on ents of the p olarizatio n vecto rs. Cho osing appr opriately the elements of the p olarization vect ors | n 0 i and | m 0 i , one can sho w th at the conjecture that the Zakharo v-Shabat d ressing pro cedure and the Gel’fand- Levitan Marc hen ko formalism lead to comparable soliton solutions is true. It is not a problem to m u ltiply th e p olarization ve ctors | n 0 i and | m 0 i by an appr opriate scalar and th u s to adjust the t wo solutions. S uc h a multiplicatio n easily go es thr ough the whole sc h eme outlined ab o ve. Th e inv olution Q † (1) = Q (1) that the p oten tial of the Lax op erator (3) asso ciated with the system (1) is sub ject to resu lts in th e follo wing relations b et ween the elemen ts of the ”p olarizatio n” v ectors | n 0 i and h m 0 | , namely n 0 ,k = m ∗ 0 ,k . Utilizing the ab o ve and a p rop er c hange of field comp onents, w e can relate the solution q ( x, t ) = 4 ν 1 C † e A + σ 2 C t σ 2 det { C † } e − A e 2 A + W + | det { C }| 2 e − 2 A e iB , (2 4) where W = (2 | c 12 | 2 + | c 1 | 2 + | c 2 | 2 ) and the ”p olarization” matrix can b e cast int o the f orm C =  c 12 c 1 c 2 − c 12  (25) In the sp ecial case when W = 1 and d et { C } = 0 w e obtain q ( x, t ) = 2 ν 1 e iB cosh A C † (26) Th us we confirm the resu lt obtained in[1], aquaired with the help of GLM formalism and the solution (20), deriv ed within the generalized Zakharo v-Sh abat dressing pro cedure, pro vided w e mak e s ure that th e extra condition on the vecto r | m i : − 2 m 0 , 1 m 0 , ¯ 1 + 2 m 0 , 2 m 0 , ¯ 2 = ( m 0 , 3 ) 2 , (27) and analogous one for | n i holds true. Setting m 0 , 1 = 1 , m 0 , ¯ 1 = − ( c ∗ 12 ) 2 − c ∗ 1 c ∗ 2 , m 0 , 2 = ic ∗ 1 , m 0 , ¯ 2 = ic ∗ 2 , m 0 , 3 = m 0 , ¯ 3 = − √ 2 c ∗ 12 w e establish the equ iv alence b et we en the tw o solutions. F ordy-Ku lish mo dels and sp inor Bose-Einstein condensates 7 4 Conclusions W e ha ve deriv ed th e soliton solutions of the th r ee-co mp onen t system of NLS typ e on the symmetric sp ace Sp(4) / U(2) whic h is related to spin or Bose-Einstein cond ensate mo del (with F = 1). F u r thermore, we ha v e describ ed br iefly the Hamiltonian prop erties of the mo del and the in tegrals of motion. Usin g the classical r -matrix app roac h, we sho wed that the int egrals of motion, that b elong to the principal series are in inv olition. The r eduction of the m u lti-co mp onent nonlinear Schr¨ odinger (NLS) equations on sym- metric space C . I ≃ Sp(2p) / U(p) for p = 2 is r elate d to spinor mo del of Bose-Einstein condensate. 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