Polynomial algebras and exact solutions of general quantum non-linear optical models II: Multi-mode boson systems

This paper is the second of a series of three devoting to polynomial algebraic structures and exact solutions of general quantum non-linear optical models. In the previous work [1], we exactly solved a class of two-mode boson systems by identifying and applying the underlying dynamical polynomial algebra symmetries. In this article, we generalize the results of [1] and deal with a general class of multi-mode boson systems. There has been constant interest in the study of deformed Lie algebras and their applications to physics. This has been motivated somewhat by the realization that the traditional linear Lie algebras and their affine counterparts may be too restrictive as there is no reason for symmetries to be linear in the first place. Polynomial algebras are a particular type of non-linear deformations of Lie algebras [2]. Their applications span across a diverse field of theoretical physics whereby they appear in topics such as quantum mechanics, Yang-Mills type gauge theories, quantum nonlinear optics, integrable systems and (quasi-)exactly solvable models, to name a few (see e.g. [3]- [13]). Different realizations of polynomial algebras have been extensively studied in [10,11,12] (see also [14,15,16,17] for differential operator realizations of certain quadratic and cubic algebras in connection with quasi-exact integrability [18,19,20]). One of the main results of this paper is the derivation of the exact eigenfunctions and energy eigenvalues of the multi-boson Hamiltonian, where and throughout a i (a † i ) and N i = a † i a i are bosonic annihilation (creation) and number operators, respectively, and w i , w ij and g are real coupling constants. Hamiltonians of the form (1.1) appear in the description of various physical systems of interest such as non-linear optics and laser physics. Using the Bargmann representation, it was shown in [21] that the multi-boson system is quasi-exactly solvable (see also [22]). Some special cases of the system have been studied in e.g. [23,24] by means of the Algebraic Bethe Ansatz method. Here we determine the underlying dynamical polynomial algebra symmetries and the exact solutions of the general Hamiltonian (1.1). The polynomial algebras and representations constructed in this paper are of interest in their own right, in view of their potential applications in other fields. This paper is organized as follows. In sections 2 we introduce higher order polynomial algebras with multi-mode boson realizations. We construct their unitary representations in the Fock space and the corresponding single-variable differential operator realizations. We then identify the polynomial algebras as the dynamical symmetries of Hamiltonian (1.1) in section 3, and solve for the eigenvalue problem via the Functional Bethe Ansatz method (see e.g. [25,26,27]). In section 4, we present explicit results for models corresponding to the special cases r + s ≤ 4, providing a unified treatment of the so-called Bose-Einstein Condensate (BEC) models. We summarize our results in section 5 and discuss further avenues of investigation. We first briefly review the polynomial algebras defined in [1]. Let k be a positive integer, where is a polynomial in Q 0 of degree k. The algebra admits Casimir operator of the form As was shown in [1], algebra (2.1) has an infinite dimensional irreducible unitary representation given by the following one-mode boson realization, In this realization, the Casimir (2.3) takes fixed value which corresponds to the common eigenvalue [1] for the direct sum of k irreducible representations with quantum number in each case. Note that for k = 2, C becomes 3/16 and q reduces to the well-known quantum number K = 1/4, 3/4 used in su(1, 1). Therefore quantum number q is a very natural choice for labeling the Fock states, denoted as |q, n , of the irreducible representations of (2.1). Explicitly, The action of Q 0 , Q ± on these states are as follows Now let us take r mutually commuting copies of algebra (2.1), {Q + , Q Consider the algebra generated by (2.9) It can be shown that Q 0,± form the polynomial algebra of degree ( r i=1 k i -1), (2.10) In the above, {L} stands for the set being the r -1 central elements of algebra (2.10), [L j , Q 0,± ] = 0, and is a polynomial in Q 0 and L i of degree k i . From (2.11), we can show that Unitary irreducible representations of polynomial algebra (2.10) are infinite dimensional. The corresponding Fock states take the form r i=1 |q i , n i , where . By means of the central elements (2.11), we can show that the Fock states become where n = 0, 1, • • •, and We now construct polynomial algebra which has finite dimensional unitary irreducible representations and thus can be identified as the dynamical symmetry algebra of the multiboson Hamiltonian (1.1). We do so by considering two mutually commuting algebras {Q (1) 0,± } of degree r i=1 k i -1 and {Q (2) 0,± } of degree r+s i=r+1 k i -1, with (2.15) Introducing new generators, We can easily show that P 0,± form a polynomial algebra of degree ( M 1 +M 2 i=1 k i -1) which close under the following commutation relations: (2) ), (2.17) where {L} (1) (2.11) are the central elements of (2.17), i.e. [L j , P 0,± ] = 0, is the (r + s -1)-the central element of the algebra, [K, P 0,± ] = 0, and finally ϕ Unitary irreducible representations of polynomial algebra (2.17) are finite dimensional. To show this, let |{q} (1) , n (1) , {l} (1) and |{q} (2) , n (2) , {l} (2) be the Fock states of the algebras Q (1) 0,± and Q (2) 0,± respectively, where n (1) , n (2) = 0, 1, • • •, and {q} (1) ≡ {q 1 , • • • , q r }, {q} (2) ≡ {q r+1 , • • • , q r+s }. The representations of {P 0,± } are then given by the Fock states |{q} (1) , n (1) , {l} (1) |{q} (2) , n (2) , {l} (2) . Since K is a central element of the algebra, it must be a constant, denoted as κ below, on any irreducible representations. This imposes the following constraint where Let N = 2κq rq r+st. Then obviously N = 0, 1, 2, • • • , and the Fock states become |{q} (1) , {q} (2) , n, {l} (1) , {l} (2) , κ = |{q} (1) , n, {l} (1) |{q} (2) , Nn, where n = 0, 1, • • • , N . This gives us the N + 1 dimensional irreducible representation of (2.17), P 0 |{q} (1) , {q} (2) , n, {l} (1) , {l} (2) , κ  |{q} (1) , {q} (2) , n, {l} (1) , {l} (2) , κ , P + |{q} (1) , {q} (2) , n, {l} (1) , {l} (2) , κ |{q} (1) , {q} (2) , n + 1, {l} (1) , {l} (2) , κ , P -|{q} (1) , {q} (2) , n, {l} (1) , {l} (2) , κ |{q} (1) , {q} (2) , n -1, {l} (1) , {l} (2) , κ . (2.23) By using the standard Fock-Bargmann correspondence, we can map the Fock states |{q} (1) , {q} (2) , n, {l} (1) , {l} (2) , κ to the monomials in z = r i=1 z i -1 The corresponding single-variable differential operator realization of (2.17) is j , (2.25) These differential operators form the same (N + 1) dimensional representations in the space of monomials as those realized by (2.15) in the corresponding Fock space. We remark that because r j=1 k j i=1 q r + s j - ≡ 0 for all the allowed q r values (noting s (1) r = 0) there is no z -1 term in P -above and thus the differential operator expressions (2.25) are non-singular. We now use the differential operator realization (2.25) to exactly solve the multi-mode boson Hamiltonian (1.1). By (2.16) and the multi-boson realization (2.15), we may express the Hamiltonian (1.1) in terms of the generators of the polynomial algebra (2.17), with the number operators having the following expressions in P 0 and L In deriving the above expressions for N i , we have used the relationships between Q (1,2) 0 and 0 similar to those given in (2.13). Keep in mind that {P ±,0 } in (3.1) as realized by (2.15) (and (2.16)) form the N + 1 dimensional representation of the polynomial algebra (2.17). This representation is also realized by the differential operators (2.25) acting on the N + 1 dimensional space of polynomials with basis 1, z, z 2 , ..., z N . We can thus equivalently represent (3.1) (i.e. (1.1)) as the single-variable differential operator of order with We will now solve for the Hamiltonian equation by using the Functional Bethe Ansatz method, where ψ(z) is the eigenfunction and E is the corresponding eigenvalue. It is easy to verify This means that the differential operator (3.3) is not exactly solvable. However, it is quasi exactly solvable, since it has an invariant polynomial subspace of degree N + 1: This is easily seen from the fact that when n = N , the first term on the r.h.s. of (3.6) becomes z N +1 g r+s j=r+1 k j i=1 k j q r+s + s (2) j - which vanishes identically for all the allowed q r+s values (noting s (2) r+s = 0). As (3.3) is a quasi exactly solvable differential operator preserving V, up to an overall factor, its eigenfunctions have the form, where The l.h.s. of (3.10) is a constant, while the r.h.s is a meromorphic function in z with at most simple poles. For them to be equal, we need to eliminate all singularities on the r.h.s of (3.10). We may achieve this by demanding that the residues of the simple poles, z = α i , i = 1, 2, ..., N should all vanish. This leads to the Bethe ansatz equations for the roots {α i } : The wavefunction ψ(z) (3.8) becomes the eigenfunction of H (3.3) in the space V provided that the roots {α i } of the polynomial ψ(z) (3.8) are the solutions of (3.11). Let us remark that the Bethe ansatz equation (3.11) is the necessary and sufficient condition for the r.h.s. of (3.10) to be independent of z. This is because when (3.11) is satisfied the r.h.s. of (3.10) is analytic everywhere in the complex plane (including points at infinity) and thus must be a constant by the Liouville theorem. To get the corresponding eigenvalue E, we consider the leading order expansion of ψ(z), It is easy to show that P ±,0 ψ(z) have the expansions, Substituting these expressions into the Hamiltonian equation (3.5) and equating the z N terms, we arrive at i ) - i ) - i ) - i ) - j ) - i ) - i ) - where {α i } satisfy the Bethe ansatz equations (3.11). This gives the eigenvalue of the 2-mode boson Hamiltonian (1.1) with the corresponding eigenfunction ψ(z) (3.8). In this section we give explicit results on the Bethe ansatz equations and energy eigenvalues of the Hamiltonian (1.1) for the special cases of r + s ≤ 4. These models arise in the description of Josephson tunneling effects and atom-molecule conversion processes in the context of BECs. Our approach provides a unified treatment of their exact solutions. The Hamiltonian is This is the so-called hetero-atom-molecule BEC model and has been solved in [23,24] via the Algebraic Bethe Ansatz method. Here for completeness we present the exact solution derived from our approach without providing any details. The Bethe ansatz equations are given by with N ≡ 2κ -2 -l 1 2 and the energy eigenvalues are Here The Hamiltonian is This yields a model of three-mode atomic-molecular BECs which has not been exactly solved previously. Specializing the general results in the preceding section to this case, we have q 2 = 1, The Bethe ansatz equations are given by and the energy eigenvalues are given by where The Hamiltonian is This gives a model of four-mode atom-molecule BECs. This model has not been exactly solved previously. Applying the results in the preceding section gives q 2 = 1 = q 4 , t = 1 2 (s 1 + s 3 ) = 1 2 (l 1 + l 3 ) and N = 2κ -2 -1 2 (l 1 + l 3 ). The differential operator representation of the Hamiltonian (4.11) thus reads where We have introduced the higher order polynomial algebra (2.17) and identified it as the dynamical symmetry algebra of the multi-boson system (1.1). We have constructed its unitary irreducible representation (2.23) and the corresponding single-variable differential realization (2.25). We have then used the differential realization to rewrite the Hamiltonian (1.1) as a QES differential operator (3.3) acting on the finite dimensional monomial space, thus providing an algebraization of the higher order Hamiltonian differential equation (3.5). Exact eigenfunctions and eigenvalues of the Hamiltonian (1.1) have been found by employing the Functional Bethe Ansatz technique. As examples, we have provided explicit expressions for the BEC models which correspond to the r + s ≤ 4 cases of (1.1). It would be interesting to adapt the techniques of this paper to obtain exact solutions of the generalized Lipkin-Meshkov-Glick and Tavis-Cummings type model defined by where J ±,0 are the generators of the su(2) algebra. Also we plan to explore the q-boson counterparts of the multi-boson systems and their underlying q-deformed polynomial algebraic structures. Lastly, it is of interest to further explore the connection between the higher order ODEs and integrable models (i.e. the ODE/IM correspondence) along the line of [28]. Results in those directions will be presented elsewhere. 3 + A 21 z 2 + gz, P 1 (z) = B 21 z 2 + D 21 z + g(l 1 + 1), P 0 (z) = F 21 z (4.7)with A 3 + A 21 z 2 + gz, P 1 (z) = B 21 z 2 + D 21 z + g(l 1 + 1), P 0 (z) = F 21 z (4.7)