Polynomial algebras and exact solutions of general quantum non-linear optical models I: Two-mode boson systems
aa r X i v : . [ m a t h - ph ] M a r Polynomial algebras and exact solutions of generalquantum non-linear optical models I: Two-modeboson systems
Yuan-Harng Lee a , Wen-Li Yang b and Yao-Zhong Zhang a a. School of Mathematics and Physics, The University of Queensland, Brisbane, Qld4072, Australia b. Institute of Modern Physics, Northwest University, Xi’an 710069, China
Abstract
We introduce higher order polynomial deformations of A Lie algebra. We con-struct their unitary representations and the corresponding single-variable differentialoperator realizations. We then use the results to obtain exact (Bethe ansatz) solu-tions to a class of 2-mode boson systems, including the Boson-Einstein Condensatemodels as special cases. Up to an overall factor, the eigenfunctions of the 2-modeboson systems are given by polynomials whose roots are solutions of the associatedBethe ansatz equations. The corresponding eigenvalues are expressed in terms ofthese roots. We also establish the spectral equivalence between the BEC modelsand certain quasi-exactly solvable Sch¨ordinger potentials.
PACS numbers : 02.20.-a; 02.20.Sv; 03.65.Fd; 42.65.Ky.
Keywords : Polynomial algebras, quasi-exactly solvable models, Bethe ansatz.
Polynomial algebras are non-linear deformations of Lie algebras and have recently foundwidespread applications in theoretical physics whereby they appear in diverse topics suchas quantum mechanics, Yang-Mills type gauge theories, quantum non-linear optics, inte-grable systems and (quasi-)exactly solvable models, to name a few (see e.g. [1]-[10]).One of the reasons for their increasing prevalence stems from the realization thattraditional linear Lie algebras describe only a very restrictive subset of linear symmetriesand that many physical systems do in fact possess non-linear symmetries such as those inwhich the commutations of the symmetry algebra generators return polynomial terms.Due to their importance, a number of studies have been undertaken to investigate themathematical properties of these algebras [11]. In particular, differential realizations of1ertain quadratic and cubic algebras have been explored in [12, 13] and also in [14, 15] inconnection with the theory of quasi-exact integrability [16, 17, 18].In this article, we introduce a novel class of higher order polynomial deformationsof the classical A Lie algebra and construct their unitary representations in terms ofboson operators and single-variable differential operators. We will then use the differentialrealizations of these algebras and the Functional Bethe Ansatz method (see e.g. [19, 20])to obtain one of the main results of this paper, that is the exact eigenfunctions and energyeigenvalues of the following class of Hamiltonians H = X i w i N i + X i,j w ij N i N j + g (cid:16) a † s a r + a s a † r (cid:17) , r, s ∈ N , (1.1)where and throughout a i ( a † i ) are boson or photon annihilation (creation) operators withfrequencies w i , N i = a † i a i are number operators, w ij and g are real coupling constants.Without loss of generality, in the following we will identify w with w . Hamiltonians(1.1) appear in the description of various physical systems of interest such as non-linearoptics [8, 9] and Bose-Einstein Condensates (BECs) [21]-[24]. For instance, the non-diagonal terms in (1.1) describe processes of multi-photon scattering and higher-orderharmonic generation in quantum nonlinear optics. Let us point out that (1.1) is a two-mode version of the more general multi-mode Hamiltonian considered in [25] in which thequasi-exact solvability of the multi-mode system was established by a different procedureand without giving exact solutions (see also [26] for the case of third-order harmonicgeneration) ∗ . Hamiltonians for the special cases of s = r = 1 and s = 2 , r = 1 have alsobeen studied using Algebraic Bethe Ansatz (ABA) method [27].This paper is organized as follows, in section 2 we propose a class of generalizedpolynomial su (1 ,
1) algebras and derive their boson realizations. In section 3, we use thesedeformed su (1 ,
1) algebras as base elements to generate higher order polynomial algebrasvia a Jordan-Schwinger like construction method. We then identify these algebras asthe dynamical algebra of the Hamiltonian (1.1) in section 4 and solve for the eigenvalueproblem in general via the Functional Bethe Ansatz method. In section 5, we presentexplicit results for the Hamiltonian (1.1) when r, s ≤ r = s = 3. In section 6, weestablish the spectral correspondence of these specific models with quasi-exactly solvable(QES) Schr¨odinger potentials. Finally we summarize our results in section 7 and discussfurther avenues of investigation. ∗ We became aware of these three references after submitting our work. We thank one referee forpointing them out. Polynomial deformations of su(1,1) algebra
Let k be a positive integer, k = 1 , , · · · . We start off by proposing a class of polynomialalgebras of degree k − Q , Q ± ] = ± Q ± , [ Q + , Q − ] = φ ( k ) ( Q ) − φ ( k ) ( Q − , (2.1)where φ ( k ) ( Q ) = − k Y i =1 (cid:18) Q + ik − k (cid:19) + k Y i =1 i − kk − k ! (2.2)is a k th -order polynomial in Q . The algebra admits Casimir operator of the followingform C = Q − Q + + φ ( k ) ( Q ) = Q + Q − + φ ( k ) ( Q − . (2.3)For k = 1 and k = 2, (2.1) reduces to the oscillator and su (1 ,
1) algebras, respectively.Thus, the algebra (2.1) can be viewed as polynomial extensions of the linear su (1 ,
1) andoscillator algebras.Similar to the su (1 ,
1) algebra case, unitary representations of (2.1) are infinite dimen-sional. In this section, we shall concentrate on the following one-mode boson realizationof the algebra, Q + = 1( √ k ) k ( a † ) k , Q − = 1( √ k ) k ( a ) k , Q = 1 k (cid:18) a † a + 1 k (cid:19) . (2.4)In this realization, the Casimir (2.3) takes the particular value, C = k Y i =1 i − kk − k ! . (2.5)We now construct the unitary representations corresponding to the realization (2.4)in the Fock space H b . There are k lowest weight states, | i , ( a † ) | i , ..., ( a † ) k − | i . (2.6)Writing these lowest weight states as | q, i using the Bargmann index q , we have Q | q, i = q | q, i , Q − | q, i = 0 . (2.7)It follows from (2.3) and (2.5) that Q ki =1 (cid:16) q + i − kk − k (cid:17) = 0, from which we get q = 1 k , k + 1 k , k + 1 k , · · · , ( k − k + 1 k . (2.8)This means that the boson realization (2.4) corresponds to the infinite dimensional unitaryrepresentation with particular q values (2.8). In other words, the H b decomposes into thedirect sum H b = H k b ⊕ · · · ⊕ H ( k − k +1 k b of k irreducible components H k b , ..., H ( k − k +1 k b .3oting kq − k = 0 , , .., k − q values given in (2.8), we can write thelowest weight states (2.6) as | q, i = ( a † ) kq − k | i . The general Fock states | q, n i ∼ Q n + | q, i in the irreducible representation space H qb are then given by | q, n i = a † k ( n + q − k ) rh k (cid:16) n + q − k (cid:17)i ! | i . (2.9)It is easy to show that Q , Q ± and C act on these states as follows Q | q, n i = ( q + n ) | q, n i ,Q + | q, n i = k Y i =1 n + q + ik − k ! | q, n + 1 i ,Q − | q, n i = k Y i =1 n + q − ( i − k + 1 k ! | q, n − i ,C | q, n i = k Y i =1 i − kk − k ! | q, n i ,n = 0 , , · · · , q = 1 k , k + 1 k , · · · , ( k − k + 1 k . (2.10) The unitary representations of the polynomial algebras discussed in the preceding sectionare all infinite dimensional. In this section, we shall employ a Jordan-Schwinger likeconstruction [12, 13], to derive polynomial algebras that have finite dimensional unitaryrepresentations. Towards this end, we consider two mutually commuting polynomialalgebras introduced in the preceding section, n Q (1)+ , Q (1) − , Q (1)0 o of degree ( k −
1) and n Q (2)+ , Q (2) − , Q (2)0 o of degree ( k − k , k = 1 , , · · · .Introduce new generators, Q + = Q (1)+ Q (2) − , Q − = Q (2)+ Q (1) − , Q = 12 (cid:16) Q (1)0 − Q (2)0 (cid:17) . (3.1)We can easily show that Q , ± form a polynomial algebra of degree ( k + k −
1) whichclose under the following commutation relations:[ Q , Q ± ] = ±Q ± , [ Q + , Q − ] = ϕ ( k + k ) ( Q , L ) − ϕ ( k + k ) ( Q − , L ) , (3.2)where L = 12 (cid:16) Q (1)0 + Q (2)0 (cid:17) (3.3)4s the central element of the algebra, [ L , Q ± , ] = 0 (3.4)and ϕ ( k + k ) ( Q , L ) = − k Y i =1 L + Q + ik − k ! k Y j =1 L − ( Q + 1) + jk − k ! (3.5)is a ( k + k ) th -order polynomial in Q and the central elements L . The Casimir operatorof (3.2) is given by C = Q − Q + + ϕ ( k + k ) ( Q , L ) = Q + Q − + ϕ ( k + k ) ( Q − , L ) . (3.6)For k + k = 2, i.e. k = k = 1, the polynomial algebra (3.2) reduces to the linear su (2)algebra. So the algebras defined by (3.2) are polynomial deformations of su (2).In terms of two sets of mutually commuting boson operators acting on the tensorproduct of the Fock spaces, we have the realization ( i = 1 , Q ( i )+ = 1( √ k i ) k i ( a † ) k i , Q ( i ) − = 1( √ k i ) k i ( a ) k i , Q ( i )0 = 1 k i (cid:18) N i + 1 k i (cid:19) . (3.7)This realization gives rise to finite dimensional representations of the polynomial algebra(3.2). To show this, let | q , n i , | q , n i be the one-mode Fock states of the algebras n Q (1)0 , ± o , n Q (2)0 , ± o respectively, where n , n = 0 , , · · · , and q = k , k +1 k , · · · , ( k − k +1 k and q = k , k +1 k , · · · , ( k − k +1 k . The representations of {Q , ± } corresponding to therealization (3.7) are then given by the two-mode Fock states | q , n i| q , n i . Since L is a central element of the algebra, it must be a constant, denoted as l below, on anyirreducible representations. This imposes a constraint on the values of n and n , L| q , n i| q , n i = 12 ( q + n + q + n ) | q , n i| q , n i = l | q , n i| q , n i . (3.8)That is n + n = 2 l − ( q + q ). Thus obviously 2 l − q − q take only positive integervalues, i.e. 2 l − q − q = 0 , , · · · . (3.9)It follows that the Fock states corresponding to the realization (3.7) are | q , q , n, l i = | q , n i| q , l − q − q − n i = ( a † ) k (cid:16) n + q − k (cid:17) ( a † ) k (cid:16) l − q − n − k (cid:17)r(cid:16) k ( n + q − k ) (cid:17) ! r(cid:16) k (2 l − q − k − n ) (cid:17) ! | i ,n = 0 , , · · · , l − q − q , (3.10)5oting that 2 l − q − q is always less than or equal to 2 l − q − k . This gives us the2 l − q − q + 1 dimensional irreducible representation of (3.2), Q | q , q , n, l i = ( q − l + n ) | q , q , n, l i , Q + | q , q , n, l i = k Y i =1 l − q − n − k ( i −
1) + 1 k ! × k Y j =1 n + q + jk − k ! | q , q , n + 1 , l i , Q − | q , q , n, l i = k Y i =1 l − q − n + ik − k ! × k Y j =1 n + q − ( j − k + 1 k ! | q , q , n − , l i . (3.11)By using the Fock-Bargmann correspondence, a † i −→ z i , a i −→ ddz i , | n i i −→ z n i i √ n i ! , (3.12)we can make the following association | q , q , n, l i −→ z k ( n + q − k )1 z k (2 l − q − n − k )2 r(cid:16) k ( n + q − k ) (cid:17) ! r(cid:16) k (2 l − q − k − n ) (cid:17) ! . (3.13)Now since l, q , q , k , k are constants, we can map the states | q , q , n, l i above to themonomials in z = z k /z k ,Ψ q ,q ,n,l ( z ) = z n r(cid:16) k ( n + q − k ) (cid:17) ! r(cid:16) k (2 l − q − k − n ) (cid:17) ! ,n = 0 , , · · · , l − q − q . (3.14)The corresponding single-variable differential operator realization of (3.2) takes the fol-lowing form Q = z ddz + q − l, Q + = z ( √ k ) k ( √ k ) k k Y j =1 l − q − ( j − k + 1 k − z ddz ! , Q − = z − ( √ k ) k ( √ k ) k k Y j =1 z ddz + q − ( j − k + 1 k ! . (3.15)These differential operators form the same 2 l − q − q + 1 dimensional representations inthe space of polynomials as those realized by (3.7) in the corresponding Fock space. Weremark that because Q k j =1 (cid:16) q − ( j − k +1 k (cid:17) ≡ q values there is no z − term in Q − above and thus the differential operator expressions (3.15) are non-singular.6 Exact solution of the 2-mode boson systems
We now use the differential operator realization (3.15) to exactly solve the 2-mode bosonHamiltonian (1.1).By means of the Jordan-Schwinger type construction (3.1) and the realization (3.7),identifying k with s and k with r , we may express the Hamiltonian (1.1) in terms of thegenerators of the polynomial algebra (3.2), H = X i w i N i + X i,j w ij N i N j + g √ s s r r ( Q + + Q − ) (4.1)with the number operators having the following expressions in Q and L N = s ( Q + L ) − s , N = r ( L − Q ) − r . (4.2)Keep in mind that {Q ∓ , } in (4.1) as realized by (3.7) (and (3.1)) form the (2 l − q − q )+1dimensional representation of the polynomial algebra (3.2). This representation is alsorealized by the differential operators (3.15) acting on the (2 l − q − q ) + 1 dimensionalspace of polynomials with basis n , z, z , ..., z l − q − q o . We can thus equivalently represent(4.1) (i.e. (1.1)) as the single-variable differential operator of order max { s, r, } , H = X i w i N i + X i,j w ij N i N j + gz r Y j =1 r l − q − ( j − r + 1 r − z ddz ! + gz − s Y j =1 s z ddz + q − ( j − s + 1 s ! (4.3)with N = s ( z ddz + q ) − s , N = r (2 l − q − z ddz ) − r . (4.4)We will now solve for the Hamiltonian equation Hψ ( z ) = E ψ ( z ) (4.5)by using the Functional Bethe Ansatz method, where ψ ( z ) is the eigenfunction and E isthe corresponding eigenvalue. It is easy to verify Hz m = z m +1 g r Y j =1 r l − q − ( j − r + 1 r − m ! + lower order terms , m ∈ Z + . (4.6)This means that the differential operator (4.3) is not exactly solvable. However, it is quasiexactly solvable, since it has an invariant polynomial subspace of degree (2 l − q − q ) + 1: H V ⊆ V , V = span { , z, ..., z l − q − q } , dim V = 2 l − q − q + 1 . (4.7)7his is easily seen from the fact that when m = 2 l − q − q the first term on the r.h.s.of (4.6) becomes z l − q − q +1 g Q rj =1 r (cid:16) q − ( j − r +1 r (cid:17) which vanishes identically for all theallowed q values. We remark that the quasi-exact solvability of the system is connectedwith its quantum integrability, i.e. with the fact that there exists quantum operatorcoinciding with a linear combination of the operators N and N which commutes withthe Hamiltonian (1.1).As (4.3) is a quasi exactly solvable differential operator preserving V , up to an overallfactor, its eigenfunctions have the form, ψ ( z ) = M Y i =1 ( z − α i ) , (4.8)where M ≡ l − q − q (= 0 , , · · · ), and { α i | i = 1 , , · · · , M } are roots of the polynomialwhich will be specified later by the associated Bethe ansatz equations (4.14) below. Wecan rewrite the Hamiltonian (4.3) as H = max { r,s, } X i =1 P i ( z ) ddz ! i + P ( z ) (4.9)where P ( z ) = zg r Y i =1 r l − q − ( i − r + 1 r ! + w (cid:18) sq − s (cid:19) + w (cid:18) r (2 l − q ) − r (cid:19) +2 w (cid:18) sq − s (cid:19) (cid:18) r (2 l − q ) − r (cid:19) + w (cid:18) sq − s (cid:19) + w (cid:18) r (2 l − q ) − r (cid:19) (4.10)and P i ( z ) are the coefficients in front of d i /dz i in the expansion of (4.3) (see the Appendix), P i ( z ) = g s s z i − s X k = i s X l <... 1) = 0 , , · · · . That is l − , , , · · · . The differential operator representation of the Hamiltonian (5.1) is H = P ( z ) d dz + P ( z ) ddz + P ( z ) , (5.2)where P ( z ) = A z ,P ( z ) = − gz + B z + g,P ( z ) = 2( l − gz + D (5.3)with A = w + w − w = 0 ,B = w − w + w + (5 − l ) w + (4 l − w ,D = 2( l − w + 4( l − w . (5.4)10he Bethe ansatz equations are given by l − X i = p α i − α p = g + B α p − gα p A α p , p = 1 , , · · · , l − 1) (5.5)and the energy eigenvalues are E = 4 w ( l − + 2 w ( l − − g l − X i =1 α i . (5.6) B. s = 2 , r = 1The Hamiltonian is H = X i w i N i + X i,j w ij N i N j + g (cid:16) a † a + a a † (cid:17) . (5.7)This is the homo-atomic-molecular BEC model and has been solved by the ABA method[27]. Specializing the general results in the preceding section to this case, we have q = or , and q = 1. The differential operator representation of the Hamiltonian (5.7) is thus H = P ( z ) d dz + P ( z ) ddz + P ( z ) (5.8)where P ( z ) = A z + 4 gz,P ( z ) = − gz + B z + 8 gq ,P ( z ) = g (2 l − q − z + D (5.9)with A = 4 w + w − w ,B = 2 w − w + 2 w (1 + 4 q ) + w (3 + 2 q − l )+ w ( − − q + 8 l ) ,D = 2 w (cid:18) q − (cid:19) + w (2 l − q − w (cid:18) q − (cid:19) + w (2 l − − q ) +4 w (cid:18) q − (cid:19) (2 l − − q ) . (5.10)The Bethe ansatz equations are l − − q X i = p α i − α p = 8 gq + B α p − gα p α p ( A α p + 4 g ) , p = 1 , , · · · , l − − q (5.11)11nd the energy eigenvalues are given by E = 2 w (cid:18) l − (cid:19) + 4 w (cid:18) l − (cid:19) − g l − − q X i =1 α i . (5.12) C. s = 2 , r = 2The Hamiltonian is H = X i w i N i + X i,j w ij N i N j + g (cid:16) a † a + a a † (cid:17) . (5.13)This gives another model of the atom-molecule BECs. To our knowledge, this model hasnot been exactly solved previously. Applying the general results in the preceding section,we have in this case q = , and q = , . The differential operator representation ofthe Hamiltonian (5.13) is H = P ( z ) d dz + P ( z ) ddz + P ( z ) , (5.14)where P ( z ) = 4 gz + 4 A z + 4 gz,P ( z ) = B z + D z + 8 gq ,P ( z ) = F z + G , (5.15)with A = w + w − w ,B = 8 g (1 + q − l ) ,D = 2 w − w + 2 w (1 + 4 q ) + 2 w (3 − l + 4 q )+8 w ( − − q + 2 l ) ,F = 4 g (cid:18) l − q − (cid:19) (cid:18) l − q − (cid:19) ,G = 2 w (cid:18) q − (cid:19) + 2 w (cid:18) l − q − (cid:19) + 4 w (cid:18) q − (cid:19) +8 w (cid:18) q − (cid:19) (cid:18) l − q − (cid:19) + 4 w (cid:18) l − q − (cid:19) . (5.16)Note that (2 l − q − q )(2 l − q + q − ≡ (2 l − q − / l − q − / 4) for q = 1 / , / l − q − q X i = p α i − α p = 8 gq + D α p − B α p α p ( gα p + A α p + g ) , p = 1 , , · · · , l − q − q (5.17)12nd the energy eigenvalues are E = 4 w (cid:18) l − q − (cid:19) + 4 w (cid:18) q − (cid:19) +8 w (cid:18) l − q − (cid:19) (cid:18) q − (cid:19) + 2 w (cid:18) l − q − (cid:19) +2 w (cid:18) q − (cid:19) − g (cid:18) q + 14 (cid:19) (cid:18) q + 34 (cid:19) l − q − q X i =1 α i . (5.18) D. s = 3 , r = 3The considered examples with s, r ≤ H = X i w i N i + X i,j w ij N i N j + g (cid:16) a † a + a a † . (cid:17) . (5.19)This is a non-linear optical model with third-order harmonic generation. Specializing thegeneral results in the preceding section to this case, we have q , q = , , or . Thedifferential operator representation of the Hamiltonian (5.19) is H = P ( z ) d dz + P ( z ) d dz + P ( z ) ddz + P ( z ) , (5.20)where P ( z ) = 27 g ( − z + z ) P ( z ) = A z + B z + D z,P ( z ) = F z + G z + K P ( z ) = R z + S (5.21)with A = 9 g (18 l − q − ,B = 9( w + w − w ) ,D = 9 g (9 q + 5) ,F = 9 g (cid:18) − l − 769 + 34 l + 36 lq − q − q (cid:19) ,G = 3 w − w + w (7 + 18 q ) + 2 w ( − − q + 18 l ) + w (7 + 18 q ) ,K = 9 g (cid:18) q + 9 q + 49 (cid:19) ,R = 27 g (cid:18) l − q − (cid:19) (cid:18) l − q − (cid:19) (cid:18) l − q − (cid:19) ,S = 9 w (cid:18) q − (cid:19) + 9 w (cid:18) l − q − (cid:19) + 18 w (cid:18) q − (cid:19) (cid:18) l − q − (cid:19) +3 w (cid:18) q − (cid:19) + 3 w (cid:18) l − q − (cid:19) . (5.22)13he Bethe ansatz equations read l − q − q X i 16 ( gz + A z + gz )+( B − F ) z + D − A − G = X i =1 c i (cid:16) g − ℘ ( g x ) − A g (cid:17) i ℘ ′ ( g x ) + ( B − F ) g − ℘ ( g x )+ A ( B − F )3 g + D − A − G , (6.19)where c = 2 g (4 q − D − A ) + 2 g (4 q − D − A ) ,c = 2 g (4 q − B + 2 g (4 q − B − g ) + ( D − A )( D − A ) ,c = B ( D − A ) + ( B − g )( D − A ) ,c = B ( B − g ) . (6.20)Here we have used 4( gz + A z + gz ) = 4 ℘ ( g x ) − g ℘ ( g x ) − g = ℘ ′ ( g x ) and(1 − q )(3 − q ) = 0 for the two allowed q values q = or . Let us now quickly summarize the work. We began by constructing the boson represen-tation of a class of su (1 , 1) polynomially deformed algebras (2.1), deriving their infinitedimensional Fock space realization and lowest weight state parametrization. We thenused the Jordan-Schwinger like construction to get the polynomial algebra (3.2) whichpossesses finite dimensional irreducible representations. We used the differential realiza-tion of (3.2) to rewrite the Hamiltonian (1.1) as QES differential operators acting onthe finite dimensional monomial space. The exact eigenfunctions and eigenvalues of theHamiltonian were then found by employing the Functional Bethe Ansatz technique. Asexamples, we provided some explicit expressions for the BEC models which correspondto the r, s ≤ H = k + k ′ X i w i N i + k + k ′ X i,j w ij N i N j + g (cid:16) a † m · · · a † m k k a m k +1 k +1 · · · a m k + k ′ k + k ′ + a m · · · a m k k a † m k +1 k +1 · · · a † m k + k ′ k + k ′ (cid:17) . (7.1)Results on on this and other models of physical interest will be presented elsewhere. Acknowledgments: This work was supported by the Australian Research Council. Theauthors would like to thank Ryu Sasaki for very valuable comments and suggestions whichlead to significant improvement of the presentation of the paper. In this appendix, we work out the expansion coefficients in front of d i dz i in the xpansion of Q mi =1 (cid:16) z ddz + A i (cid:17) .First, we see m Y i =1 z ddz + A i ! = m Y i =1 A i + m X j =1 m Y i = j A i z ddz + m X j