Exactly solvable models for multiatomic molecular Bose-Einstein condensates

Exactly solv able mo dels for m ultiatomic molecular Bose-Ei nstein condens ates G. San tos 1 1 Instituto de F ´ ısica da UFR GS Av. Bento Gon¸ calves, 9500 - Agro nomia - Porto Alegre - RS - Braz il E-mail: gfilh o@if.uf rgs.br, gfilh o@cbpf.b r Abstract. I in tr oduce tw o family o f exactly solv able models fo r multiatomic hetero- nu clear and homo-nuclear molecula r B o se-Einstein condensa tes thro ugh the a lgebraic Bethe ansatz metho d. The conser v ed quantities of the resp ective mo dels are also show ed. 1. In t ro duction One of the most intere sting recen t exp erimen tal achie v emen ts in phys ics is the one that led to realizations of Bose-Einstein condensates (BEC), by taking dilute alk ali gases to extremely lo w temp eratures [1, 2]. Since then, a great effort has b een dev oted t o the comprehension of new phenomena in volving this state of matt er as w ell as its prop erties, either exp erimentally or theoretically . On the experimen ta l side I could men tion a molecular BEC comp o und t hat has b een obtained combining differen t tec hniques [3], leading this kind of research also in the direction of a c hemistry of BEC, where, for instance, b y F esh bach resonances [4 – 6] or photo-asso ciation [7, 8] the a t omic constituen ts ma y form molecules . Man y comp ounds of diatomic homo-n uclear molecular BECs [9 – 11] hav e b een pro duced since the first realization [12]. Also, diatomic hetero-n uclear molecular BECs hav e b een detected using these techn iques [13 – 19]. Actually , due to the r apid tec hnological dev elopmen ts in the field o f ultra- cold systems, it is b eliev ed that some of these exp erimen ts ma y b e just the daw n of t he study of multiatomic molecules [20, 21]. More recen tly the experimen tal evidence f o r Efimo v states in an ultra-cold cesium gas [22 – 24] and a mixture of ultra-cold p ota ssium and rubidium gases [25] provid es a ph ysical ground f o r the in ve stigation of t r ia tomic and tetratomic homo-n uclear and triatomic hetero-n uclear molecular BECs. These results b o o sted the searc h f o r solv able mo dels that could describe some of the BEC prop erties [26 – 42]. The rationale b eneath these studies is that through exactly solv able mo dels it is p ossible to fully take in to accoun t quantum fluctuations, going b ey ond the usual mean field a pproximations. Then, I exp ect that this approach ma y pro vide some impact in this area, as w ell as a con tributio n to the field of integrable Exactly solvable mo dels fo r multiatomic mole cular Bose-Einstein c ondensates 2 systems itself [4 3 , 44]. In this pap er I will use the algebraic Bethe ansatz metho d. The algebraic formulation of the Bethe ansatz, and the a ssociated quantum in v erse scattering metho d (QISM), w as primarily dev elop ed b y the group of mathematical ph ysicists in St. P etersburg [45 – 4 9]. The QISM could b e used to study the one-dimensional spin c hains, quan tum field theory in o ne- dimensional b osons in teracting systems [50] and t wo-dimensional latt ice mo dels [51], systems of strongly correlated elec trons [52, 53], conformal field t heory [54], as w ell as precipitated the notion of quan tum algebras (deformations o f univ ersal en veloping a lgebras of Lie algebras) [55 – 58]. F or a p edagog ical and historical review see [59]. Owing to recen t insigh ts in the understanding of t he construction of Lax op erators it is p ossible to obtain solv able mo dels suitable for the effectiv e description of t he in terconv ersion in teractions o ccurring in the BEC. Inspired b y some of these ideas I presen t, in the presen t pap er, the construction of t wo complete family of Bethe-ansatz solv able mo dels f or b oth homo-nucle ar and hetero-n uclear molecular BECs obtained through a comb ination of three Lax op erators constructed using sp ecial realizations of the su (2) Lie algebra and of the Heisen b erg-W eyl Lie algebra, as we ll as a multibosonic represen tation of the sl (2) Lie algebra, discussed recen tly in [60]. Notice that the mo dels obtained thro ugh this construction do not ha v e spatial degrees of freedom. The pap er is org anized as follo ws: In Section 2, I will review shortly the algebraic Bethe ansatz metho d and presen t the La x op erators and the transfer matrix fo r b ot h mo dels. In Section 3, I will presen t a family of multiatomic homo- n uclear mo dels and their solutions. In Section 4, I will presen t a family of multiatomic hetero-n uclear mo dels and their solutions. In Section 5 , I will mak e m y remarks. 2. Algebraic Bethe ansatz method In this sec tion w e will shortly review the a lgebraic Bethe ansatz metho d and presen t the transfer matrix used to get the solution of the mo dels [28 , 61]. W e b egin with the g l (2)-inv ariant R -matrix, depending on t he sp ectral parameter u , R ( u ) =      1 0 0 0 0 b ( u ) c ( u ) 0 0 c ( u ) b ( u ) 0 0 0 0 1      , (1) with b ( u ) = u/ ( u + η ) and c ( u ) = η / ( u + η ) . Ab ov e, η is an arbitrary parameter, to b e c hosen later. It is easy to c heck that R ( u ) satisfies the Y ang-Baxter equation R 12 ( u − v ) R 13 ( u ) R 23 ( v ) = R 23 ( v ) R 13 ( u ) R 12 ( u − v ) . (2) Here R j k ( u ) denotes the matrix acting non-trivially o n the j -th and the k -th spaces and as the iden tity on the remaining space. Next we define the mono drom y matrix T ( u ), T ( u ) = A ( u ) B ( u ) C ( u ) D ( u ) ! , (3) Exactly solvable mo dels fo r multiatomic mole cular Bose-Einstein c ondensates 3 that satisfy the Y ang- Baxter algebra, R 12 ( u − v ) T 1 ( u ) T 2 ( v ) = T 2 ( v ) T 1 ( u ) R 12 ( u − v ) . (4) In what follows w e will c ho ose differen t realizations for the mono dromy matrix π ( T ( u )) = L ( u ) to obtain solutions of t w o family of mo dels for m ultiatomic hetero- n uclear a nd homo-n uclear molecular BECs. In this construction, the Lax op erators L ( u ) hav e t o satisfy the relation R 12 ( u − v ) L 1 ( u ) L 2 ( v ) = L 2 ( v ) L 1 ( u ) R 12 ( u − v ) . (5) where we use the notation, L 1 = L ( u ) ⊗ I and L 2 = I ⊗ L ( u ) . (6) Then, defining the tra nsfer matrix, as usual, through t ( u ) = tr π ( T ( u )) = π ( A ( u ) + D ( u )) , (7) it follows from (4) that the transfer matrix commutes for differen t v alues of the sp ectral parameter; i. e., [ t ( u ) , t ( v )] = 0 , ∀ u , v . (8) Consequen tly , the mo dels deriv ed from this transfer mat r ix will b e in tegrable. Another consequenc e is that the co efficien ts C k in the transfer mat r ix t ( u ), t ( u ) = X k C k u k , (9) are conserv ed quan tities or simply c -n um b ers, with [ C j , C k ] = 0 , ∀ j, k . (10) If the transfer matrix t ( u ) is a p olynomial function in u , with k ≥ 0, it is easy to see that , C 0 = t (0) and C k = 1 k ! d k t ( u ) du k     u =0 . (11) W e will use three solutions o f the equation (5): (i) The L S ( u ) Lax op erator: L S ( u ) = 1 u u − η S z − η S + − η S − u + η S z ! , (12) in terms of the su ( 2 ) Lie algebra with generators S z and S ± sub ject to the comm utation relations [ S z , S ± ] = ± S ± , [ S + , S − ] = 2 S z . (13) Exactly solvable mo dels fo r multiatomic mole cular Bose-Einstein c ondensates 4 (ii) The L j ( u ) Lax op erator: L j ( u ) = u + η N j j j † η − 1 ! , (14) in terms o f the Heisen b erg-W eyl Lie a lg ebra with generators N j , j , j † and I , sub ject to the comm utation relations [ N j , j ] = − j, [ N j , j † ] = + j † , [ j, j † ] = I and [ I , ⋆ ] = 0 , (15) where ⋆ means N j , j or j † . (iii) The L A ( u ) Lax op erator: L A ( u ) = u + η 2 A 0 η A − − η A + u − η 2 A 0 ! , (16) in terms of the sl (2) Lie alg ebra with generators A 0 and A ± , sub j ect to the comm utation relations [ A − , A + ] = A 0 , [ A 0 , A ± ] = ± 2 A ± . (17) Using the co-multiplication prop erties of t he Lax op erator and the g l (2) in v ariance of the R -matrix, w e can obtain differen t realizations for the mono drom y matrix: (i) Multiatomic homo-n uclear: F or the m ultiatomic homo-n uclear molecular BEC mo del w e c ho ose π ( T ( u )) = L ( u ) = η − 1 GL j ( u − δ − η − 1 ) L A ( u + ω ) , (18) with, G = diag ( − , +), fr o m whic h w e find t he follow ing tr a nsfer matrix t ( u ) = − η − 1 ( u + ω + η 2 A 0 )( u − δ − η − 1 + η N b ) + η − 2 ( u + ω − η 2 A 0 ) + bA + + b † A − , (19) with t (0) = η − 1 ω ( δ + 2 η − 1 ) − ω N b + 1 2 ( δ − η N b ) A 0 + bA + + b † A − , (20) and, discarding c -n um b er terms, the conserv ed quantities are, C 0 = t (0) , (21) C 1 = 1 2 A 0 + N b . (22) Exactly solvable mo dels fo r multiatomic mole cular Bose-Einstein c ondensates 5 (ii) Multiatomic hetero-n uclear: F o r the m ultiatomic hetero-n uclear molecular BEC mo dels w e c ho ose π ( T ( u )) = L ( u ) = η − 1 u − GL S ( u − ) L A ( u + ) , (23) with u ± = u ± ω , G = diag (+ , − ), from whic h w e find the follo wing transfer matrix t ( u ) = u − A 0 − 2 u + S z + η ( S + A + + S − A − ) , (24) and the conserv ed quan tities, C 0 = t (0) , (25) C 1 = A 0 − 2 S z . (26) In the next sections w e will describ e the mo dels and its in tegra bilit y b y the algebraic Bethe ansatz metho d, using differen t realizations of the algebras (1 3), (15) and (17). The Hamiltonians are written in the F o ck space using the standard notatio n. W e are considering t he coupling para meters real, suc h that the Hamiltonians are Hermitian. In the diag onal part of t he Hamiltonians, the U j parameters describ e the atom-atom, atom-molecule and molecule-molecule S -w av e scatterings and the µ j parameters are the externals p oten tials. The op erators N j are the num b er op erato r s of a toms or molecules. In the off diagonal part of t he Hamiltonians the parameter Ω is the amplitude for in terconv ersion of a toms and molecules. 3. Multiatomic homo-n uclear molecular mo dels In this section w e presen t the integrabilit y of a new family of Hamiltonians describing m ultiato mic homo-n uclear molecular BEC. The Hamiltonians that describ es the in terconv ersion of homogeneous molecules lab elled b y b with l ato ms of t yp e a are giv en b y H = U a N 2 a + U b N 2 b + U ab N a N b + µ a N a + µ b N b + Ω(( a † ) l b α − ( N a ) + α − ( N a ) b † ( a ) l ) , (27) where α − ( N a ) is a f unction of N a that con tr o ls the amplitude o f in t erconv ersion Ω. This indicates that the density of a toms N a has some influence in the g eneration o f a b ound-state comp osed b y l iden tical atoms. The l = 3 case w as studied in [35]. The total num b er of particles N = N a + l N b is a conserv ed quan tity . There is a m ultib osonic realization of the sl (2) Lie algebra [6 0] A 0 = α 0 ( N ) , A − = α − ( N ) a l , A + = ( a † ) l α − ( N ) , (28) with Exactly solvable mo dels fo r multiatomic mole cular Bose-Einstein c ondensates 6 α 0 ( N ) = 2 l ( N − R ) + α 0 ( R ) , (29) α − ( N ) = s N ! ( N + l )! ( 1 l ( N − R ) + α 0 ( R ))( 1 l ( N − R ) + 1) , (30) where N = a † a and l ∈ N . The op erator R is R =        0 for l = 1, l − 1 2 + l − 1 X m =1 e − (2 πm/l ) N e (2 πm/l ) − 1 for l > 1, (31) and acts on t he states {| n i} as R | n i = n mod l | n i . The function α 0 ( R ) is a p ositiv e function of the sp ectrum of R defined b y initia l conditions. F or n = r < l , w e ha ve 1 l ( N − R ) | r i = 0 | r i , (32) with A 0 = α 0 ( R ) suc h that α 0 ( R ) | r i = α 0 ( r ) | r i and R | r i = r | r i . No w w e will use this realizatio n to show ho w to construct the Hamiltonian (27) from the transfer matrix (19) and presen t their exact Bethe ansatz solution. It is straigh tf orw ard to ch ec k that the Hamiltonian (27) is related with the transfer matrix t (0) (20), or with the conserv ed quan tity C 0 (21), thro ug h H = Ω t (0) , (33) where we ha ve the f ollo wing iden tification η = l 2 U a − lU ab + U b Ω , (34) θ = 2 l U ab − 4 U b , ξ = 2 l 2 µ a − 2 lµ b , ( 3 5) 2 l Ω( ω + δ ) = (2Ω η + θ ) N − Ω ρη + ξ , (36) 4 l 2 Ω ω δ η + 8 l 2 Ω ω = η 2 [Ω ρ 2 η + 4 U b N 2 − θN ρ + 4 l (Ω ω + µ b ) N + (2 l Ω ω − ξ ) ρ ] , (37) with ρ ≡ ρ ( R ) = l α 0 ( R ) − 2 R . It is easy to see that ρ is a conserv ed quan tit y using t he total n um b er of atoms, N , to write the conserv ed quan tit y C 1 (22) as, C 1 = 1 l N + 1 2 l ρ. (38) Exactly solvable mo dels fo r multiatomic mole cular Bose-Einstein c ondensates 7 W e can apply the algebraic Bethe a nsatz method, using as the pseudo-v acuum the pro duct state ( | 0 i = | 0 i b ⊗ | r i A , with | 0 i b denoting the F ock v acuum state and | r i A denoting the lo w est weigh t stat e of the sl (2) Lie algebra, where r = 0 , 1 , ... , l − 1 , are the eigenv alues of R for N = nl + r , with n ∈ N , t o find t he Bethe a nsatz equations (BAE) (1 − η v i + η δ )( v i + ω + η 2 α 0 ( r )) v i + ω − η 2 α 0 ( r ) = M Y i 6 = j v i − v j − η v i − v j + η , i, j = 1 , ..., M , (39) and the eigen v alues of the Hamiltonian (27 ), E = Ω η − 1 ( δ + η − 1 )  ω + η 2 α 0 ( r )  M Y i =1 v i − η v i + Ω η − 2  ω − η 2 α 0 ( r )  M Y i =1 v i + η v i . (40) The parameters δ and ω are a r bit r a ry a nd can b e chosen conv enien tly . In the limit without scatterings, U j → 0, the BAE (39) can b e write as, M X i =1 1 v i + ω = 1 α 0 ( r ) M X i =1 v i − M α 0 ( r ) δ , (41) and fo r ω = 0 the eigenv alues (40) b ecome, E =  1 2 α 0 ( r ) + M  Ω δ − Ω M X i =1 v i . (42) No w, the relation betw een t he intercon v ersion par a meter a nd the externals po ten tials is simply , Ω = lµ a − µ b δ . (43) 4. Multiatomic hetero-n uclear molecular mo dels In this section w e presen t the integrabilit y of a new family of Hamiltonians describing m ultiato mic hetero-n uclear molecular BEC. The Hamilto nians that describ es the in terconv ersion of heterogeneous molecules lab elled b y c with l ato ms of type a and one ato m of ty p e b are give n b y H = U a N 2 a + U b N 2 b + U c N 2 c + U ab N a N b + U ac N a N c + U bc N b N c + µ a N a + µ b N b + µ c N c + Ω(( a † ) l b † c α − ( N a ) + α − ( N a ) c † b ( a ) l ) , (44) Exactly solvable mo dels fo r multiatomic mole cular Bose-Einstein c ondensates 8 where α − ( N a ) is a function of N a that con tro ls the amplitude of inte rcon ve rsion Ω. In the same w ay of the Hamiltonians (27), this indicates that the densit y of atoms N a has some influence in t he generation of a b ound-state comp osed by l iden t ical at oms. The imbalance b et w een the num b er o f atoms a and the n umber of atoms b , J ab = N a − lN b , (45) is a conserv ed quan tity and the total n umber of atoms, N = N a + N b + ( l + 1) N c , can b e writes with the other t w o conserv ed quantities I ac = N a + l N c , (46) I bc = N b + N c . (47) Using J ab and N , the S -wa v e diagonal part of t he Hamiltonian (44) can b e writte as, α J 2 ab + β N 2 + γ N J ab , (48) where we ha ve used the following iden tification for the coupling constan ts U a = α + β + γ , U b = αl 2 + β − γ l , U c = β ( l + 1) 2 , (49) U ab = − 2 lα + 2 β − γ ( l − 1) , U ac = 2 β ( l + 1) + γ ( l + 1 ) , (50) U bc = 2 β ( l + 1 ) − γ l ( l + 1) . (51) No w, using the following realization for the su (2) Lie algebra, S + = b † c, S − = c † b, S z = N b − N c 2 , (52) and the m ultib osonic realization of the sl (2) Lie algebra (28), A 0 = α 0 ( N ) , A − = α − ( N ) a l , A + = ( a † ) l α − ( N ) , (53) with α 0 ( N ) = 2 l ( N − R ) + α 0 ( R ) , (54) α − ( N ) = s N ! ( N + l )! ( 1 l ( N − R ) + α 0 ( R ))( 1 l ( N − R ) + 1) , (55) where N = a † a and l ∈ N , it is straig h tforward to c hec k that the Hamiltonian (44) is related with the transfer matrix t ( u ) (24), or with the conserv ed quan tit y C 0 (25) if u = 0 , through H = σ + α J 2 ab + β N 2 + γ N J ab + t ( u ) , (56) Exactly solvable mo dels fo r multiatomic mole cular Bose-Einstein c ondensates 9 where the follo wing identific ation has b een made for the parameters µ a = 2 u − l , µ c = − µ b = u + , Ω = η , σ = − u − l ρ, (57) with ρ ≡ ρ ( R ) = l α 0 ( R ) − 2 R . W e also can use the conserv ed quantities J ab and I ac to write the conserv ed quantit y C 1 (26) as, C 1 = 1 l ( J ab + I ac ) + 1 l ρ, (58) sho wing that ρ is also a conserv ed quan tity . W e can apply the algebraic Bethe ansatz metho d, using as t he pseudo-v acuum the pro duct state ( | 0 i = | r i A ⊗ | φ i , with | r i A denoting the lo wes t w eight state of the sl (2 ) Lie algebra where r = 0 , 1 , ... , l − 1 , a r e the eigen v alues of R for N = nl + r , with n ∈ N and | φ i denoting the highest w eight state of the su (2 ) Lie algebra with w eight m z ), to find the Bethe ansatz equations (BAE) − ( v i − ω − η m z )( v i + ω + η 2 α 0 ( r )) ( v i − ω + η m z )( v i + ω − η 2 α 0 ( r )) = M Y i 6 = j v i − v j − η v i − v j + η , i, j = 1 , ..., M , (59) and the eigen v alues of the Hamiltonian (44 ) E = σ + α J 2 ab + β N 2 + γ N J ab + ( u − ω − η m z )( u + ω + η 2 α 0 ( r )) M Y i =1 u − v i + η u − v i − ( u − ω + η m z )( u + ω − η 2 α 0 ( r )) M Y i =1 u − v i − η u − v i . (60) The eigenv alues ( 60) are indep enden t of the spectral parameter u a nd o f the parameter ω , that are arbitrary . 5. Summary I ha v e in tro duced t wo new family of multiatomic molecular BEC mo dels for homo- n uclear and hetero-nucle ar molecules and deriv ed the Bethe ansatz equations and the eigen v alues. The conserv ed quan t it ies a re also deriv ed. The multiatomic homo- n uclear and hetero-nuc lear molecular BEC mo dels we re obta ined through a combin ation of Lax op erators constructed using sp ecial realizations of the su ( 2 ) Lie a lg ebra and Heisen b erg- W eyl Lie algebra, as w ell as a m ultib osonic represen tat io n of the sl (2) Lie a lg ebra. The dep endence of the parameters with the size o f the molecules is explicit. Exactly solvable mo dels fo r multiatomic mole cular Bose-Einstein c ondensates 10 A cknow le d gments The author ac knowle dge suppor t from CNPq (Conselho Nacional de D esen v olvimen to Cien t ´ ıfico e T ecnol´ ogico). The a uthor also would lik e to thank A. F o erster and I. Ro diti for interes t ing discussions. References [1] E. A. Cornell and C. E. Wieman, Rev. Mo d. Phys. 74 (2002) 875. [2] J. R. Anglin and W. K etterle, Nature 416 (20 02) 21 1. [3] P . Zoller , Natu r e 417 (20 02) 493. [4] S. Inouye, M. R. Andrews, J. Steng er, H.-J. Miesner , D. M. Stamper -Kurn and W. Ketterle , Natur e 392 (1998) 1 5 1. [5] T. Kohler, K. Goral, and P . S. Julienne, R ev. Mo d. Phys. 78 (200 6) 131 1. [6] Cheng Chin, Rudolf Grimm, Paul Julienne and E ite Ties inga, R ev. Mo d. Phys. 82 , (2010 ) 12 25. [7] K. M. Jones, E. Tiesinga, P . D. Lett, and P . S. Julienne, R ev. Mo d. Phys. 78 (2006) 483 . [8] C. R. Menegatti, B. S. Marango ni a nd L. G. Marca ssa, L aser Physics 18 (2008 ) 1305 . [9] J. Herbig et al, Scienc e 301 (20 03) 15 10. [10] S. Durr et al, Phys. R ev. L et t. 92 (2004) 0204 06. [11] C. Chin, T. Kr aemer, M. Mark , J . Herbig , P . W aldburger, H.-C. N¨ agerl and R. Grimm, Phys. Rev. L ett. 94 (20 05) 1232 01. [12] E . A. Donley , N. R. Claussen, S. T. Thompson, and C. E . Wieman, Natur e 417 , (2002 ) 529. [13] B. Damski, L. Santos, E. Tiemann, M. Lewenstein, S. Koto chigo v a, P . Julienne, and P . Zoller , Phys. R ev. L ett . 90 (2003) 1104 01. [14] M.W. Ma ncini, G. D. T elles, A. R. L. Caires, V. S. Bagnato and L. G. Marcas sa, Phys. R ev. L et t . 92 (2 0 04) 1332 03. [15] C. Haimberger , J. Kleinert, M. B hattac harya and N. P . Bigelow, Phys. Rev. A 70 (2004 ) 0214 02(R). [16] J . M. Sage, S. Sainis, T. Bergeman, a nd D. DeMille, Phys. R ev. L ett. 94 (200 5) 2 03001. [17] S. B. Papp and C. E. Wieman, Phys. Rev. L ett. 97 (200 6) 180 404. [18] C. W eb er, G. Bar on tini, J. Catani, G. Thalha mmer, M. Inguscio a nd F. Minardi, Phys. R ev. A 78 (2008) 0 61601R. [19] K . Pilc h, A. D. Lang e, A. Pra n tner, G. Kerner, F. F erlaino, H.-C. N¨ ag erl and R. Grimm, Phys. R ev. A 79 (2009) 0 42718. [20] E . Braa ten, H.-W. Hammer, Annals of Physic s 322 (20 0 7) 120 . [21] B. D. Esry and C. Greene, Natur e 440 (2006) 2 89. [22] T. Kra e mer, M. Mar k, P . W aldbur ger, J. G. Danzl, C. Chin, B. Engeser, A. D. Lange, K. Pilc h, A. J aakkola, H.-C. Nagerl, R. Grimm, N atur e 440 (200 6) 315 . [23] S. Kno op, F. F erlaino, M. Berninger, M. Ma rk, H.-C. N¨ agerl a nd R. Gr imm, Journal of Physics: Conf. Ser. 194 (200 9) 01206 4. [24] F. F erlaino, S. Kno op, M. Ber ninger, W. Har m, J. P . D’Incao, H.-C. N¨ agerl and R. Gr imm, P h y s. Rev. Lett. 102 (200 9) 140 401. [25] G. Barontini, C. W eber, F. Raba tti, J. Ca tani, G. Thalhammer, M. Ing us cio and F. Minardi, Phys. R ev. L ett. 103 (200 9) 0 43201. [26] H.-Q . Zhou, J. Links, M. Gould and R. McK enzie, J. Math. Phys. 44 (2003 ) 46 90. [27] H.-Q . Zhou, J. Links, R. H. McK enzie, Int. Jour. Mo d. Phys. B 17 (2003 ) 5819 . [28] J . Links, H.-Q. Zhou, R. H. McK enzie and M. D. Gould, J. Phys. A 36 (2003 ) R63 . [29] A. F o erster, J . Links, H.-Q. Zhou, in Classic al and quantum nonline ar inte gr able systems: t he ory and appl ic ations , edited by A. Kundu (IOP Publishing, Bristol and Phila de lphia , 2003) pp. 208-2 33. [30] J . Dukelsky , G. Dussel, C. E sebbag and S. Pittel, Phys. Rev. L ett. 93 (200 4) 050 403. [31] G. O r tiz, R. So mma, J . Dukelsky and S. Rombouls, Nucle ar Physics B 707 (2005 ) 42 1. Exactly solvable mo dels fo r multiatomic mole cular Bose-Einstein c ondensates 11 [32] A. Kundu, The or etic al and Mathematic al Physics 151 (2 007) 831 . [33] A. F oer s ter and E. Ragoucy , Nu cle ar Physics B 77 7 (2007 ) 37 3. [34] J on Links, Angela F o er s ter, Arle i P restes T onel and Gilb erto Sa n tos , Ann. Henri Poinc ar ´ e 7 (2006) 1 591. [35] G. Santos, A. F o erster, I. Roditi, Z. V. T. Santos and A. P . T onel, J. Phys. A: Math. The or. 4 1 (2008) 2 95003. [36] G. Santos, A. T o ne l, A. F oer ster and J. Links, Phys. Re v. A 73 (2006) 0 23609. [37] G. Santos, A. F o erster , J. Links, E. Mattei and S. R. Dahmen, Phys. R ev. A 73 (201 0) 0 23609. [38] A. P . T onel, C. C . N. Kuhn, G. Santos, A. F o erster, I. Roditi and Z. V. T. Sant os, Phys. R ev. A 79 (2 0 09) 0136 24. [39] L. Zhou, W. Zhang , H. Y. Ling, L. Jiang, and H. Pu, Phys. R ev. A 75 (2007) 0436 03. [40] L. Zhou, J. Qian, H. Pu, W. Z hang, and H. Y. Ling, Phys. R ev. A 78 (2008) 053 6 12. [41] A. V ardi, V. A. Y urovsky and J. R. Anglin, Phys. R ev. A 64 (200 1 ) 0 6 3611. [42] M. Duncan, A. F o erster , J . Links, E. Mattei, N. Oelkers and A. T onel, Nucle ar Physics B 767 [FS] (2007) 2 2 7. [43] M. H´ er itier, Natur e 41 4 (2001 ) 31 . [44] M. T. Batchelor, Physics T o day 60 (2 007) 36. [45] L. D. F addeev , E. K. Sklyanin and L. A. T akhta jan, The or. Math. Phys. 40 (19 79) 194. [46] P . P . Kulish and E. K. Sklyanin, L e ct. Notes Phys. 151 (198 2) 6 1. [47] L. A. T a k h ta jan, L e ct. Notes Phys. 370 (1 990) 3. [48] V. E. Korepin, N. M. Bogoliub ov and A. G. Izer gin, Quantu m inverse sc attering metho d and c orr elation functions (Cambridge University Pr ess, Cambridge, 199 3). [49] L. D. F addeev , Int . J . Mo d. Phys. A 10 (1 995) 184 5 . [50] A. G. Izer gin a nd V. E. Kor epin, L ett. Math. Phys. 6 (19 82) 28 3. [51] A. G. Izer gin a nd V. E. Kor epin, Nu c. Phys. B 205 (198 2) 401. [52] F. H. L. Essler and V. E. K orepin (eds.) Exactly solvabl e mo dels of str ongly c orr elate d ele ctr ons (W orld Scientific, Singap ore, 1994 ). [53] F abian H. L. E ssler, Holger F ra hm, F rank G¨ o hmann, Andrea s Kl ¨ umper and Vladimir E . Kor epin, The one-dimensional Hubb ar d Mo del (Cambridge University P ress, Cambridge, 200 5). [54] V. Ba zhanov, S. Luk y a no v and A. B. Z amolo dc hikov, Commun. Math. Phys. 177 (1996 ) 38 1. [55] M. Jimbo, L et t. Math. Phys. 10 (198 5) 63. [56] M. Jimbo, L e ct. Notes Phys. 24 6 (1986 ) 3 3 5. [57] V. G. Drinfeld, Quantum groups Pr o c. Int. Congr ess of Mathematici ans ed A M Glea son (Providence, RI: American Mathematical So ciet y ) (1986 ) 798. [58] N. Y u Reshetikhin, L. A. T akhta ja n and L. D. F addeev, L eningr ad Math. J. 1 (1990) 193 . [59] L. D. F addeev, 40 Y ea rs in Mathematical Physics World Scientific S eries in 20th Century Mathematics , vol. 2 , (W or ld Scientific Publishing Co . Pte. Ltd., Sing apor e, 199 5). [60] T. Goli ´ nski, M. Horowski, A. Odzijewicz, A. Sli ˙ zewsk a, Jour. Math. Phys. 48 (2007) 0 23508. [61] I. Ro diti, Br azilian Journal of Physics 30 (2000 ) 35 7.