GitBucket
Newer
zeynalig
<?xml version="1.0" ?>
<tei>
<teiHeader>
<fileDesc xml:id="_1"/>
</teiHeader>
<text xml:lang="en">
<titlePage>Exact solutions for a family of spin-boson systems<lb/> This article has been downloaded from IOPscience. Please scroll down to see the full text article.<lb/> 2011 Nonlinearity 24 1975<lb/> (http://iopscience.iop.org/0951-7715/24/7/004)<lb/> Download details:<lb/> IP Address: 145.64.134.241<lb/> The article was downloaded on 27/05/2011 at 09:38<lb/> Please note that terms and conditions apply.<lb/> View the table of contents for this issue, or go to the journal homepage for more<lb/> Home Search Collections Journals About Contact us My IOPscience<lb/> </titlePage>
<front>IOP PUBLISHING<lb/> NONLINEARITY<lb/> Nonlinearity 24 (2011) 1975–1986<lb/> doi:10.1088/0951-7715/24/7/004<lb/> Exact solutions for a family of spin-boson systems<lb/> Yuan-Harng Lee, Jon Links and Yao-Zhong Zhang<lb/> School of Mathematics and Physics, The University of Queensland, Brisbane, Qld 4072, Australia<lb/> Received 15 December 2010<lb/> Published 24 May 2011<lb/> Online at stacks.iop.org/Non/24/1975<lb/> Recommended by S Nonnenmacher<lb/> Abstract<lb/> We obtain the exact solutions for a family of spin-boson systems. This<lb/> is achieved through application of the representation theory for polynomial<lb/> deformations of the su(2) Lie algebra. We demonstrate that the family of<lb/> Hamiltonians includes, as special cases, known physical models which are<lb/> the two-site Bose–Hubbard model, the Lipkin–Meshkov–Glick model, the<lb/> molecular asymmetric rigid rotor, the Tavis–Cummings model and a two-mode<lb/> generalization of the Tavis–Cummings model.<lb/> Mathematics Subject Classification: 83C15, 81U15, 81V80, 17B80, 81R12<lb/> PACS numbers: 02.30.Ik, 03.65.Fd<lb/></front>
<body>1. Introduction<lb/> The study of polynomial deformations of Lie algebras is an area of research which has found<lb/> many applications in systems involving non-linear interactions [1–3]. In recent publications [4]<lb/> we have formulated such methods for the analysis of a class of multi-boson systems. The<lb/> approach of [4] is to express the Hamiltonian of the systems in terms of the generators of<lb/> polynomial deformations of the su(2) Lie algebra, through the explicit construction of Fock-<lb/>space representations. By utilizing a correspondence between the Fock-space representations<lb/> and differential operator realizations, it was shown that exact solutions are obtained in terms<lb/> of a system of coupled equations. These equations can be viewed as providing a Bethe ansatz<lb/> type of solution for the calculation of the energy spectrum and associated eigenstates. The<lb/> generality of this approach allows for application on a wider level. The work described below<lb/> is concerned with extending these methods to the study of a family of Hamiltonians which<lb/> couple multi-boson degrees of freedom to a spin degree of freedom. In this manner we unify<lb/> the problem of exactly solving spin-boson Hamiltonians to a particular class which contains<lb/> within it a number of models which are already known in the literature, as we will discuss.<lb/> The main result of this paper is the derivation of the exact eigenfunctions and energy<lb/> eigenvalues of the infinite family of spin-boson systems defined by the Hamiltonian<lb/> H =<lb/> M<lb/> <lb/> i=1<lb/> w i N i + g J s<lb/> 0 + g<lb/> J r<lb/> + a k 1<lb/> 1 · · · a k M<lb/> r + J r<lb/> − a k 1<lb/> 1 · · · a k M<lb/> r<lb/> ,<lb/> (1.1)<lb/></body>
<front>0951-7715/11/071975+12$33.00 © 2011 IOP Publishing Ltd & London Mathematical Society Printed in the UK & the USA<lb/></front>
<page>1975<lb/> </page>
<page>1976<lb/> </page>
<note place="headnote">Y-H Lee et al<lb/> </note>
<body>where throughout r, s ∈ Z + , M, k 1 , . . . , k M ∈ N, a i , a<lb/> i and N i = a<lb/> i a i are bosonic<lb/> annihilation, creation and number operators, respectively, J ±,0 are the generators for the su(2)<lb/> spin algebra, and w i , w ij , g are real coupling constants. The Hamiltonian of the form (1.1)<lb/> appears in the description of various physical systems of interest in atomic, molecular, nuclear<lb/> and optical physics. We will explicitly demonstrate that (1.1) includes, as special cases,<lb/> several known models which are the two-site Bose–Hubbard model [5], the Lipkin–Meshkov–<lb/> Glick model [6], the molecular asymmetric rigid rotor [7], the many-atom Tavis–Cummings<lb/> model [8, 9] and a two-mode generalized Tavis–Cummings model [10].<lb/> This paper is organized as follows. In section 2 we introduce new higher order<lb/> polynomial algebras which are dynamical symmetry algebras of Hamiltonian (1.1), which<lb/> enable an algebraization of the spin-boson systems. We construct finite-dimensional unitary<lb/> representations of the dynamical symmetry algebras in section 3 and the corresponding single-<lb/>variable differential operator realizations in section 4. This leads to the higher order differential<lb/> operator realizations of the Hamiltonian (1.1). In section 5 we establish the quasi-exact<lb/> solvability of this differential operator [11–13] and solve for the eigenvalue problem via the<lb/> functional Bethe ansatz method (see, e.g. [14–16]). In section 6 we present explicit results<lb/> for several special cases, thus providing a unified derivation of exact solutions to the widely<lb/> studied models mentioned above. We summarize our results in section 7 and discuss further<lb/> avenues for investigation.<lb/> 2. Algebraization<lb/> In this section we introduce new higher order polynomial deformations of sl(2) and give<lb/> an algebraization of the Hamiltonian (1.1). Our approach extends previous studies [1–4]<lb/> where more restricted classes of systems have been exactly solved using polynomial algebra<lb/> structures.<lb/> We introduce generators<lb/> P + = P +<lb/> M<lb/> <lb/> i=1<lb/> Q (i)<lb/> − ,<lb/> P − = P −<lb/> M<lb/> <lb/> i=1<lb/> Q (i)<lb/> + ,<lb/> P 0 =<lb/> P 0 −<lb/> M<lb/> i=1 Q (i)<lb/> 0<lb/> M + 1<lb/> , (2.1)<lb/> where<lb/> P 0 =<lb/> J 0<lb/> r ,<lb/> P + = J r<lb/> + ,<lb/> P − = J r<lb/> − ,<lb/> Q (i)<lb/> + =<lb/> a k i<lb/> i<lb/> √ k i<lb/> k i<lb/> , Q (i)<lb/> − =<lb/> a k i<lb/> i<lb/> √ k i<lb/> k i<lb/> , Q (i)<lb/> 0 =<lb/> 1 k i<lb/> <lb/> a i a i +<lb/> 1 k i<lb/> <lb/> are M + 1 mutually commuting operators. It can be shown that P ±,0 satisfy the following<lb/> commutation relations:<lb/> P 0 , P ±<lb/> = ± P ± ,<lb/> P + , P −<lb/> = ψ (2r) (K, P 0 − 1, C)<lb/> M<lb/> <lb/> i=1<lb/> φ (k i ) (K, P 0 − 1, {L})<lb/> − ψ (2r) (K, P 0 , C)<lb/> M<lb/> <lb/> i=1<lb/> φ (k i ) (K, P 0 , {L}) ,<lb/> (2.2)<lb/> where C is the Casimir operator of su(2),<lb/> K =<lb/> MP 0 +<lb/> M<lb/> ν=1 Q (ν)<lb/> 0<lb/> M + 1<lb/> , L i = Q (i)<lb/> 0 − Q (i+1)<lb/> 0<lb/> , i= 1 · · · M − 1,<lb/> (2.3)<lb/>
<note place="headnote">Exact solutions for a family of spin-boson systems<lb/> </note>
<page>1977<lb/> </page>
are M central elements of (2.2) and<lb/> ψ (2r) (K, P 0 , C) = −<lb/> r<lb/> <lb/> i=1<lb/> [C − (rK + rP 0 + r − i + 1)(rK + rP 0 + r − i)] ,<lb/> φ (k i ) (K, P 0 , {L}) = −<lb/> k i<lb/> <lb/> i=1<lb/> <lb/> K<lb/> M − (P 0 + 1) −<lb/> 1 M<lb/> M−1 <lb/> µ=1<lb/> µL µ +<lb/> M−1 <lb/> µ=i<lb/> L µ +<lb/> ik i − 1<lb/> k 2<lb/> i<lb/> <lb/> <lb/> are polynomial functions of degree 2r and k i , respectively. Thus (2.2) defines a polynomial<lb/> algebra of degree 2r +<lb/> M<lb/> i=1 k i − 1.<lb/> In terms of the generators of the polynomial algebra (2.2), the Hamiltonian (1.1) can be<lb/> written as<lb/> H =<lb/> M<lb/> <lb/> i=1<lb/> w i N i + g r s (P 0 + K) s + g<lb/> M<lb/> <lb/> i=1<lb/> <lb/> k i<lb/> k i<lb/> <lb/> (P + + P − )<lb/> (2.4)<lb/> with the number operators having the following expression in P 0 , K and L i :<lb/> N i = k i<lb/> <lb/> −P 0 +<lb/> K<lb/> M −<lb/> 1 M<lb/> M−1 <lb/> µ=1<lb/> µL µ +<lb/> M−1 <lb/> µ=i<lb/> L µ<lb/> <lb/> −<lb/> 1 k i<lb/> .<lb/> It follows that the polynomial algebra (2.2) is the dynamical symmetry algebra of the<lb/> Hamiltonian (1.1).<lb/> 3. Unitary irreducible representations<lb/> Irreducible representations of the polynomial algebra (2.2) can be constructed in the tensor<lb/> product space of the representation space of P ±,0 and the Fock spaces of {Q (i)<lb/> ±,0 }. As shown<lb/> in [4], the Fock states for irreducible representations of {Q (i)<lb/> ±,0 } are labelled by quantum numbers<lb/> q i = 1<lb/> k 2<lb/> i<lb/> , k i +1<lb/> k 2<lb/> i<lb/> , . . . , (k i −1)k i +1<lb/> k 2<lb/> i<lb/> , through<lb/> |q i , m i =<lb/> a k i (m i +q i −k −2<lb/> i )<lb/> [k i (m i + q i − k −2<lb/> i )]!<lb/> |0.<lb/> m i = 0, 1, . . . .<lb/> (3.1)<lb/> The action of Q (i)<lb/> ±,0 on these states is<lb/> Q (i)<lb/> 0 |q i , m i = (q i + m i )|q i , m i ,<lb/> Q (i)<lb/> + |q i , m i =<lb/> k i<lb/> <lb/> j =1<lb/> <lb/> m i + q i +<lb/> jk i − 1<lb/> k 2<lb/> i<lb/> 1/2<lb/> |q i , m i + 1,<lb/> Q (i)<lb/> − |q i , m i =<lb/> k i<lb/> <lb/> j =1<lb/> <lb/> m i + q i −<lb/> (j − 1)k i + 1<lb/> k 2<lb/> i<lb/> 1/2<lb/> |q i , m i − 1.<lb/> The irreducible representations of P ±,0 can be deduced from the su(2)-module V j ,<lb/> j = 0, 1<lb/> 2<lb/> , 1, . . . as follows. First, it can be shown that P 0,± satisfy the relations<lb/> P 0 , P ±<lb/> = ±P ± ,<lb/> P + , P −<lb/> = ψ (2r) (P 0 , C) − ψ (2r) (P 0 − 1, C),<lb/> (3.2)<lb/>
<page>1978<lb/> </page>
<note place="headnote">Y-H Lee et al<lb/> </note>
where<lb/> ψ (2r) (P 0 , C) = −<lb/> r<lb/> <lb/> i=1<lb/> (C − (rP 0 + r − i + 1)(rP 0 + r − i))<lb/> (3.3)<lb/> is a polynomial in P 0 and C of degree 2r. Thus (3.2) is a polynomial algebra of degree 2r − 1.<lb/> The Casimir operator of (3.2) takes fixed value<lb/> r<lb/> i=1 (C − i(i − 1)).<lb/> It is easily verified that there are min{r, 2j + 1} lowest weight states,<lb/> |j, 0; p ∼ J<lb/> p<lb/> + |j, 0,<lb/> p= 0, 1, . . . , min{r − 1, 2j },<lb/> where |j, 0 is the lowest weight state of su(2). This implies that finite-dimensional<lb/> irreducible representations of (3.2), denoted as V j,p , are labelled by quantum numbers j and<lb/> p, j = 0, 1<lb/> 2<lb/> , 1, . . ., and p = 0, 1, . . . , min{r − 1, 2j }. Thus we have the branching rule from<lb/> su(2) representation V j into V j,p of (3.2):<lb/> V j = ⊕<lb/> min{(r−1),2j }<lb/> p=0<lb/> V j,p .<lb/> General basis vectors in the irreducible representation space V j,p are given by |j, n; p ∼<lb/> (P + ) n |j, 0; p. Explicitly,<lb/> |j.n; p =<lb/> (2j − p − rn)!<lb/> (p + rn)!(2j)!<lb/> J<lb/> p+rn +<lb/> |j, 0.<lb/> (3.4)<lb/> The action of P 0,± on these vectors is given by<lb/> P 0 |j, n; p =<lb/> p − j<lb/> r + n<lb/> |j, n; p,<lb/> P + |j, n; p =<lb/> r<lb/> <lb/> i=1<lb/> (p + i + rn)(2j − p − i + 1 − rn) |j, n + 1; p,<lb/> P − |j, n; p =<lb/> r<lb/> <lb/> i=1<lb/> (p − i + 1 + rn)(2j − p + i − rn) |j, n − 1; p.<lb/> (3.5)<lb/> It can also be shown that<lb/> P − |j, 0; p = 0,<lb/> P + |j,<lb/> 2j − p − λ<lb/> r ; p = 0,<lb/> where λ is a non-negative integer taking specific values λ = 0, 1, . . . , min{r − 1, 2j }<lb/> according to j and p. Moreover, 2j −p−λ<lb/> r<lb/> is always a non-negative integer. Therefore<lb/> n = 0, 1, . . . , 2j −p−λ<lb/> r<lb/> , and (3.5) is a finite-dimensional representation of (3.2) with dimension<lb/> 2j −p−λ<lb/> r<lb/> + 1.<lb/> We now construct irreducible representation of (2.2) in the tensor space V j,p ⊗ H (1)<lb/> q 1<lb/> · · · ⊗<lb/> H (M)<lb/> q M<lb/> , where V j,p is the representation space of P ±,0 and H (i)<lb/> q i<lb/> is the Fock space of Q (i)<lb/> ±,0 . From<lb/> (2.3) we have<lb/> Q (i)<lb/> 0 = Q (M)<lb/> 0<lb/> + M−1 <lb/> µ=i<lb/> L µ ,<lb/> (M+ 1)K = MP 0 + MQ (M)<lb/> 0<lb/> + M−1 <lb/> µ=1<lb/> µL µ .<lb/> This implies that for any irreducible representation of (2.2) defined by basis states |j, n; p ⊗<lb/> M<lb/> i=1 |q i , m i ,<lb/> m i = m M + q M − q i +<lb/> M−1 <lb/> µ=i<lb/> l µ ,<lb/> i= 1, . . . , M − 1,<lb/> n + m M =<lb/> M + 1<lb/> M κ − q M −<lb/> 1 M<lb/> M−1 <lb/> µ=1<lb/> µl µ −<lb/> p − j<lb/> r ,<lb/>
<note place="headnote">Exact solutions for a family of spin-boson systems<lb/> </note>
<page>1979<lb/> </page>
where κ and l µ denote the eigenvalues of central elements K and L µ , respectively. It follows<lb/> that q i q M +<lb/> M−1<lb/> µ=i l µ and<lb/> n + m i = A i ,<lb/> A i =<lb/> M + 1<lb/> M κ −<lb/> 1 M<lb/> M−1 <lb/> µ=1<lb/> µl µ +<lb/> M−1 <lb/> µ=i<lb/> l µ −<lb/> p − j<lb/> r − q i .<lb/> Clearly A i always take non-negative integer values, i.e. A i = 0, 1, . . .. Thus, the irreducible<lb/> representation of (2.2) has basis states<lb/> |j, n, p, {q}, {l}, κ ≡ |j, n; p ⊗<lb/> M<lb/> <lb/> i=1<lb/> |q i , m i <lb/> = J<lb/> p+rn +<lb/> |j, 0<lb/> √ (p + rn)!(2j − p − rn)!<lb/> M<lb/> <lb/> i=1<lb/> a k i (A i +q i −k −2<lb/> i −n)<lb/> i<lb/> |0 <lb/> [k i (A i + q i − k −2<lb/> i − n)]!<lb/> , (3.6)<lb/> where<lb/> n =<lb/> <lb/> <lb/> 0, 1, . . . , min<lb/> <lb/> A M ,<lb/> 2j − p − λ<lb/> r<lb/> <lb/> for M > 0,<lb/> 0, 1, . . .<lb/> 2j − p − λ<lb/> r for M = 0.<lb/> The action of (2.1) on these states is given by<lb/> P 0 |j, n, p, {q}, {l}, κ =<lb/> p − j<lb/> r + n − κ<lb/> |j, n, p, {q}, {l}, κ,<lb/> P + |j, n, p, {q}, {l}, κ =<lb/> r<lb/> <lb/> i=1<lb/> (p + i + rn)(2j − p − i + 1 − rn)<lb/> ×<lb/> M<lb/> <lb/> i=1<lb/> k i<lb/> <lb/> µ=1<lb/> <lb/> A i + q i −<lb/> (µ − 1)k i + 1<lb/> k 2<lb/> i<lb/> − n<lb/> 1/2<lb/> × |j, n + 1, p, {q}, {l}, κ,<lb/> P − |j, n, p, {q}, {l}, κ =<lb/> r<lb/> <lb/> i=1<lb/> (p − i + 1 + rn)(2j − p + i − rn)<lb/> ×<lb/> M<lb/> <lb/> i=1<lb/> k i<lb/> <lb/> µ=1<lb/> <lb/> A i + q i +<lb/> µk i − 1<lb/> k 2<lb/> i<lb/> − n<lb/> 1/2<lb/> |j, n − 1, p, {q}, {l}, κ.<lb/> This gives an N + 1 dimensional representation of the polynomial algebra (2.2), where<lb/> N =<lb/> <lb/> <lb/> min<lb/> <lb/> A M ,<lb/> 2j − p − λ<lb/> r<lb/> <lb/> for M > 0,<lb/> 2j − p − λ<lb/> r for M = 0.<lb/> 4. Differential operator realization<lb/> The finite-dimensional irreducible representations in the proceeding section can be realized<lb/> by differential operators acting on N + 1-dimensional space of monomials with basis<lb/>
<page>1980<lb/> </page>
<note place="headnote">Y-H Lee et al<lb/> </note>
{1, z, z 2 , . . . , z N }, by mapping the basis vectors (3.6) into monomials in z:<lb/> |j, n, p, {q}, {l}, κ −→<lb/> z n<lb/> (p + rn)!(2j − p − rn)!<lb/> M<lb/> i=1 [k i (A i + q i − 1<lb/> k 2<lb/> i<lb/> − n)]!<lb/> .<lb/> The corresponding single-variable differential operator realization of (2.1) in the monomial<lb/> space takes the following form<lb/> P 0 = z<lb/> d dz<lb/> − κ +<lb/> p − j<lb/> r ,<lb/> P + = z<lb/> r<lb/> <lb/> i=1<lb/> <lb/> 2j − p − i + 1 − rz<lb/> d dz<lb/> ×<lb/> M<lb/> <lb/> i=1<lb/> k i<lb/> <lb/> ν=1<lb/> k i<lb/> <lb/> A i + q i −<lb/> (ν − 1)k i + 1<lb/> k 2<lb/> i<lb/> − z<lb/> d dz<lb/> <lb/> ,<lb/> P − =<lb/> z −1<lb/> M<lb/> µ=1<lb/> k µ<lb/> k µ<lb/> r<lb/> <lb/> i=1<lb/> <lb/> rz d<lb/> dz + p − i + 1<lb/> <lb/> . (4.1)<lb/> Note that P − contains no singularities as<lb/> r<lb/> i=1 (p − i + 1) = 0 for all allowed p values.<lb/> We can thus equivalently represent Hamiltonian (2.4) as the single-variable differential<lb/> operator of order M ≡ max{r +<lb/> M<lb/> i=1 k i , s},<lb/> H =<lb/> M<lb/> <lb/> i=1<lb/> w i N i + g <lb/> <lb/> rz d<lb/> dz − j + p<lb/> s<lb/> + gz −1<lb/> r<lb/> <lb/> i=1<lb/> <lb/> rz d<lb/> dz + p − i + 1<lb/> <lb/> + gz<lb/> r<lb/> <lb/> i=1<lb/> <lb/> 2j − p − i + 1 − rz<lb/> d dz<lb/> M<lb/> <lb/> i=1<lb/> k i<lb/> <lb/> ν=1<lb/> k i<lb/> <lb/> A i + q i −<lb/> (ν − 1)k i + 1<lb/> k 2<lb/> i<lb/> − z<lb/> d dz<lb/> <lb/> (4.2)<lb/> with<lb/> N i = k i<lb/> <lb/> −z<lb/> d dz<lb/> + M + 1<lb/> M κ −<lb/> p − j<lb/> r +<lb/> M−1 <lb/> µ=i<lb/> l µ −<lb/> 1 M<lb/> M−1 <lb/> µ=1<lb/> µl µ<lb/> <lb/> −<lb/> 1 k i<lb/> .<lb/> 5. Exact solutions<lb/> We will now solve for the Hamiltonian equation<lb/> H ψ(z) = E ψ(z)<lb/> (5.1)<lb/> for the differential operator realizations by using the functional Bethe ansatz method [14–16],<lb/> where ψ(z) is the eigenfunction and E is the corresponding eigenvalue. It is straightforward<lb/> to verify<lb/> H z n = z n+1 g<lb/> r<lb/> <lb/> i=1<lb/> (2j − p − i + 1 − rn)<lb/> M<lb/> <lb/> i=1<lb/> k i<lb/> <lb/> ν=1<lb/> k i<lb/> <lb/> A i + q i − n −<lb/> (ν − 1)k i + 1<lb/> k 2<lb/> i<lb/> <lb/> + lower order terms,<lb/> n∈ Z + .<lb/> (5.2)<lb/> This means that the differential operator (4.2) is not exactly solvable. However, it is quasi-<lb/>exactly solvable, since when n = N the first term (∼ z n+1 ) on the rhs of (5.2) is vanishing.<lb/> That is H preserves an invariant polynomial subspace of degree N ,<lb/> H V ⊆ V,<lb/> V = span{1, z, ..., z N }<lb/> (5.3)<lb/>
<note place="headnote">Exact solutions for a family of spin-boson systems<lb/> </note>
<page>1981<lb/> </page>
Thus up to an overall factor, the eigenfunctions of (4.2) have the form<lb/> ψ(z) =<lb/> N<lb/> <lb/> i=1<lb/> (z − α i ) ,<lb/> (5.4)<lb/> where {α i | i = 1, 2, . . . , N } are roots of the polynomial which will be specified by the<lb/> associated Bethe ansatz equations (5.7) below. We can rewrite the Hamiltonian (4.2) as<lb/> H =<lb/> M<lb/> <lb/> i=1<lb/> P i (z)<lb/> d<lb/> dz<lb/> i<lb/> + P 0 (z),<lb/> (5.5)<lb/> where P 0 (z) and P i (z) are polynomials in z determined from the expansion of the products<lb/> in (4.2).<lb/> Dividing the Hamiltonian equation H ψ = Eψ by ψ gives us<lb/> E =<lb/> H ψ<lb/> ψ =<lb/> M<lb/> <lb/> i=1<lb/> P i (z)i!<lb/> N<lb/> <lb/> n 1 <n 2 <···<n i<lb/> 1 (z − α n 1 ) · · · (z − α n i )<lb/> + P 0 (z).<lb/> (5.6)<lb/> The lhs of (5.6) is a constant, while the rhs is a meromorphic function in z with at most simple<lb/> poles. For them to be equal, we need to eliminate all singularities on the rhs of (5.6). We<lb/> may achieve this by demanding that the residues of the simple poles, z = α i , i = 1, 2, . . . , N<lb/> should all vanish. This leads to the Bethe ansatz equations for the roots {α i } :<lb/> M<lb/> <lb/> i=2<lb/> N<lb/> <lb/> n 1 <n 2 <...<n i−1 =µ<lb/> P i (α µ )i!<lb/> (α µ − α n 1 ) · · · (α µ − α n i−1 )<lb/> + P 1 (α µ ) = 0,<lb/> µ= 1, 2, . . . , N .<lb/> (5.7)<lb/> The wavefunction ψ(z) (5.4) becomes the eigenfunction of H (4.2) in the space V provided<lb/> that the roots {α i } of the polynomial ψ(z) (5.4) are the solutions of (5.7).<lb/> Let us remark that the Bethe ansatz equation (5.7) is the necessary and sufficient condition<lb/> for the rhs of (5.6) to be independent of z. This is because when (5.7) is satisfied the rhs of<lb/> (5.6) is analytic everywhere in the complex plane (including points at infinity) and thus must<lb/> be a constant by Liouville's theorem.<lb/> To obtain the corresponding eigenvalue E, we consider the leading order expansion<lb/> of ψ(z),<lb/> ψ(z) = z N − z N −1<lb/> N<lb/> <lb/> i=1<lb/> α i + · · · .<lb/> It can be directly shown that the P ±,0 ψ(z) have the expansions<lb/> P + ψ = − z N g<lb/> r<lb/> <lb/> i=1<lb/> (2j − p − i + 1 − r(N − 1))<lb/> ×<lb/> M<lb/> <lb/> i=1<lb/> k i<lb/> <lb/> ν=1<lb/> k i<lb/> <lb/> A i + q i − N + 1 −<lb/> (ν − 1)k i + 1<lb/> k 2<lb/> i<lb/> N<lb/> <lb/> i=1<lb/> α i + · · · ,<lb/> P − ψ ∼ z N −1 + · · · ,<lb/> P 0 ψ = z N<lb/> N +<lb/> p − j<lb/> r − κ<lb/> <lb/> + · · · .<lb/>
<page>1982<lb/> </page>
<note place="headnote">Y-H Lee et al<lb/> </note>
Substituting these expressions into the Hamiltonian equation (5.1) and equating the z N terms,<lb/> we arrive at<lb/> E =<lb/> M<lb/> <lb/> i=1<lb/> w i<lb/> <lb/> k i<lb/> <lb/> M + 1<lb/> M κ −<lb/> p − j<lb/> r − N −<lb/> 1 M<lb/> M−1 <lb/> µ=1<lb/> l µ +<lb/> M−1 <lb/> µ=i<lb/> l µ<lb/> <lb/> −<lb/> 1 k i<lb/> <lb/> <lb/> + g (rN − j + p) s − g<lb/> r<lb/> <lb/> i=1<lb/> (2j − p − i + 1 − r(N − 1))<lb/> ×<lb/> M<lb/> <lb/> i=1<lb/> k i<lb/> <lb/> ν=1<lb/> k i<lb/> <lb/> A i + q i − N + 1 −<lb/> (ν − 1)k i + 1<lb/> k 2<lb/> i<lb/> N<lb/> <lb/> i=1<lb/> α i ,<lb/> (5.8)<lb/> where {α i } satisfy the Bethe ansatz equations (5.7). This gives the eigenvalue of the<lb/> Hamiltonian (1.1) with the corresponding eigenfunction ψ(z) (5.4).<lb/> 6. Explicit examples<lb/> In this section we give explicit results on the Bethe ansatz equations and energy eigenvalues of<lb/> the Hamiltonian (1.1) for special cases which correspond to some established models frequently<lb/> studied in the field of atomic and molecular physics, condensed matter, nuclear physics and<lb/> quantum optics.<lb/> 6.1. Two-site Bose–Hubbard model<lb/> This model corresponds to the special case with M = 0, r = 1, s = 2 and its Hamiltonian<lb/> takes the simple form<lb/> H = g J 2<lb/> 0 + g (J + + J − ) .<lb/> (6.1)<lb/> This model has been widely employed in the context of Josephson-coupled Bose–Einstein<lb/> condensates via the realization of J ±,0 in terms of two bosons, J + = b<lb/> 1 b 2 , J − = b 1 b<lb/> 2 , J 0 =<lb/> 1 2<lb/> (b<lb/> 1 b 1 − b<lb/> 2 b 2 ) (see, e.g. [5] and references therein). Exact solutions of the model in terms<lb/> of algebraic Bethe ansatz methods were first studied in [17]. From the general results in the<lb/> preceding section, in this case we have κ = 0, p = 0, and N = 2j . Thus (6.1) takes the form<lb/> H = P 2 (z)<lb/> d 2<lb/> dz 2 + P 1 (z)<lb/> d dz<lb/> + P 0 (z),<lb/> where<lb/> P 2 (z) = g z 2 ,<lb/> P 1 (z) = g z(1 − 2j) + g(1 + z 2 ),<lb/> P 0 (z) = g j 2 − 2jzg.<lb/> The Bethe ansatz equations are given by<lb/> 2j<lb/> <lb/> i =µ<lb/> 2 α i − α µ<lb/> = −<lb/> α µ g (1 − 2j) + g(1 + α 2<lb/> µ )<lb/> g α 2<lb/> µ<lb/> , µ= 1, 2, . . . , 2j<lb/> and the energy eigenvalues are<lb/> E = g j 2 − g<lb/> 2j<lb/> <lb/> i=1<lb/> α i .<lb/> This exact solution is equivalent to a case described in [18].<lb/>
<note place="headnote">Exact solutions for a family of spin-boson systems<lb/> </note>
<page>1983<lb/> </page>
6.2. Lipkin–Meshkov–Glick model<lb/> This model is the special case corresponding to M = 0, r = 2, s = 1. The Hamiltonian is<lb/> given by [6]<lb/> H = g J 0 + g<lb/> J 2<lb/> + + J 2<lb/> −<lb/> (6.2)<lb/> and continues to be studied extensively (see e.g. [19] and references therein). Exact solution<lb/> via the algebraic Bethe ansatz method is discussed in [20, 21]. Specializing the general results<lb/> of the preceding section to this case, we have p = 0, 1, κ = 0 and N = 2j −p−λ<lb/> 2<lb/> with λ = 0, 1<lb/> so that N is a non-negative integer. The differential operator representation of the Hamiltonian<lb/> (6.2) is thus<lb/> H = P 2 (z)<lb/> d 2<lb/> dz 2 + P 1 (z)<lb/> d dz<lb/> + P 0 (z)<lb/> where<lb/> P 2 (z) = 4gz 3 + 4gz,<lb/> P 1 (z) = g(6 + 4p − 8j)z 2 + 2g z + g(2 + 4p),<lb/> P 0 (z) = gz(2j − p)(2j − p − 1) + g (p − j).<lb/> The Bethe ansatz equations are given by<lb/> 2j −p−λ<lb/> 2<lb/> <lb/> i =µ<lb/> 2 α i − α µ<lb/> = −<lb/> g(3 + 2p − 4)α 2<lb/> µ + g α µ + g(1 + 2p)<lb/> 2g(α 3<lb/> µ + α µ )<lb/> , µ= 1, 2, . . . ,<lb/> 2j − p − λ<lb/> 2<lb/> and the energy eigenvalues are<lb/> E = g (j − λ) − g(λ + 1)(λ + 2)<lb/> 2j −p−λ<lb/> 2<lb/> <lb/> i=1<lb/> α i .<lb/> 6.3. Molecular asymmetric rigid rotor<lb/> Up to an additive constant, this model corresponds to the special case with M = 0, r = s = 2.<lb/> This is shown as follows. The Hamiltonian of the rigid rotor has the following form in terms<lb/> of the su(2) generators J x , J y and J z [7]:<lb/> H = aJ 2<lb/> x + bJ 2<lb/> y + cJ 2<lb/> z<lb/> where a, b, c are constants. The model has previously been discussed in [22] as a Hamiltonian<lb/> which is solvable by algebraic Bethe ansatz methods. The Hamiltonian can be rewritten as<lb/> H =<lb/> 2c − a − b<lb/> 2 J 2<lb/> 0 +<lb/> a − b<lb/> 4 (J 2<lb/> + + J 2<lb/> − ) +<lb/> a + b<lb/> 2 C,<lb/> (6.3)<lb/> where C is the Casimir element of su(2). This shows that the molecular asymmetric rigid rotor<lb/> is indeed a special case of (1.1). Note that the Hamiltonian (6.3) of the rigid rotor almost has<lb/> the same form as that of the Lipkin–Meshkov–Glick model. To our knowledge, this connection<lb/> has not been noted previously.<lb/> Specializing the general results in the preceding section to this case, we have p = 0, 1,<lb/> κ = 0 and N = 2j −p−λ<lb/> 2<lb/> , where λ = 0, 1 as in the case of the Lipkin–Meshkov–Glick model.<lb/> The differential operator representation of the Hamiltonian (6.3) is thus<lb/> H = P 2 (z)<lb/> d 2<lb/> dz 2 + P 1 (z)<lb/> d dz<lb/> + P 0 (z)<lb/>
<page>1984<lb/> </page>
<note place="headnote">Y-H Lee et al<lb/> </note>
where P 2 (z) = (a − b)z 3 + 2(2c − a − b)z 2 + (a − b)z,<lb/> P 1 (z) =<lb/> a − b<lb/> 2 (3 + 2p − 4j)z 2 + 2(2c − a − b)(1 + p − j)z +<lb/> a − b<lb/> 2 (1 + 2p),<lb/> P 0 (z) =<lb/> a − b<lb/> 4 (2j − p)(2j − p − 1)z +<lb/> 2c − a − b<lb/> 2 (p − j) 2 +<lb/> a + b<lb/> 2 j (j + 1).<lb/> The Bethe ansatz equations are given by<lb/> 2j −p−λ<lb/> 2<lb/> <lb/> i =µ<lb/> 2 α i − α µ<lb/> = −<lb/> (a − b)(3 + 2p − 4)α 2<lb/> µ + 4(2c − a − b)(1 + p − j)α µ + (a − b)(1 + 2p)<lb/> 2(a − b)(α 3<lb/> µ + α µ ) + 4(2c − a − b)α 2<lb/> µ<lb/> ,<lb/> µ = 1, 2, . . . ,<lb/> 2j − p − λ<lb/> 2 and the energy eigenvalues are<lb/> E =<lb/> 2c − a − b<lb/> 2 (j − λ) 2 +<lb/> a + b<lb/> 2 j (j + 1) −<lb/> a − b<lb/> 4 (λ + 1)(λ + 2)<lb/> 2j −p−λ<lb/> 2<lb/> <lb/> i=1<lb/> α i .<lb/> 6.4. Tavis–Cummings model<lb/> This model corresponds to the special case when M = r = s = k 1 = 1. The Hamiltonian is<lb/> given by<lb/> H = w 1 N 1 + g J 0 + g<lb/> J + a 1 + J − a<lb/> 1<lb/> .<lb/> This is one of the widely studied models in quantum optics and had been exactly solved via<lb/> the algebraic Bethe ansatz approach [23, 24]. Applying the results in the preceding section<lb/> gives q 1 = 1, p = 0 and N = min{2κ + j − 1, 2j }. The differential operator representation<lb/> of the Hamiltonian is<lb/> H = P 2 (z)<lb/> d 2<lb/> dz 2 + P 1 (z)<lb/> d dz<lb/> + P 0 (z),<lb/> where P 2 (z) = gz 3 ,<lb/> P 1 (z) = −g(3j + 2κ − 2)z 2 + (g − w 1 )z + g,<lb/> P 0 (z) = w 1 (2κ + j − 1) − g j + 2gj z(2κ + j − 1).<lb/> The Bethe ansatz equations read<lb/> N<lb/> <lb/> i =µ<lb/> 2 α i − α µ<lb/> = g(3j + 2κ − 2)α 2<lb/> µ − (g − w)α µ − g<lb/> gα 3<lb/> µ<lb/> , µ= 1, 2, . . . , N<lb/> and the energy eigenvalues are<lb/> E = w 1 (2κ + j − N − 1) + g (N − j ) − g (2j − N + 1) (2κ + j − N )<lb/> N<lb/> <lb/> i=1<lb/> α i ,<lb/> where N = min{2κ + j − 1, 2j }.<lb/> 6.5. Two-mode generalized Tavis–Cummings model<lb/> Finally, we consider the case when M = 2, r = s = k 1 = k 2 = 1. This gives the Hamiltonian<lb/> H = w 1 N 1 + w 2 N 2 + g J 0 + g<lb/> J − a<lb/> 1 a<lb/> 2 + J + a 1 a 2<lb/> ,<lb/>
<note place="headnote">Exact solutions for a family of spin-boson systems<lb/> </note>
<page>1985<lb/> </page>
which belongs to the class of su(1, 1) generalized Tavis–Cummings model discussed in<lb/> [10]. Applying the results in the preceding section gives q 1 = q 2 = 1, p = 0 and<lb/> N = min{ 3κ−l 1<lb/> 2<lb/> − 1 + j, 2j }. The differential operator representation of the Hamiltonian<lb/> thus reads<lb/> H = P 3 (z)<lb/> d 3<lb/> dz 3 + P 2 (z)<lb/> d 2<lb/> dz 2 + P 1 (z)<lb/> d dz<lb/> + P 0 (z),<lb/> where<lb/> P 3 (z) = −gz 4 ,<lb/> P 2 (z) = g(3κ + 4j − 5)z 3 ,<lb/> P 1 (z) = Az 2 + (g − w 1 − w 2 )z + g,<lb/> P 0 (z) = zB + F<lb/> with<lb/> A = g<lb/> −9jκ + 10j + 6κ +<lb/> l 2<lb/> 1<lb/> 4 − 5j 2 − 4 −<lb/> 9 4<lb/> κ 2<lb/> <lb/> ,<lb/> B = g<lb/> 9jκ<lb/> 2 + 6j 2 κ − 6jκ + 2j −<lb/> jl 2<lb/> 1<lb/> 2 + 2j 3 − 4j 2<lb/> <lb/> ,<lb/> F = (w 1 + w 2 )<lb/> 3κ<lb/> 2 − 1 + j<lb/> <lb/> + l 1<lb/> 2 (w 1 − w 2 ) − g j.<lb/> The Bethe ansatz equations assume the form<lb/> N<lb/> <lb/> µ<ν =β<lb/> 6gα 4<lb/> β<lb/> (α β − α µ )(α β − α ν )<lb/> −<lb/> N<lb/> <lb/> i =β<lb/> 2g(3κ + 4j − 5)α 3<lb/> β<lb/> α β − α i<lb/> = Aα 2<lb/> β + (g − w 1 − w 2 )α β + g,<lb/> β = 1, 2, · · · , N<lb/> and the energy eigenvalues are<lb/> E = (w 1 + w 2 )<lb/> 3κ<lb/> 2 − 1 + j − N<lb/> <lb/> + l 1<lb/> 2 (w 1 − w 2 ) + g (N − j )<lb/> − g(2j − N + 1)<lb/> <lb/> 3κ 2<lb/> + j − N<lb/> 2<lb/> − l 2<lb/> 1<lb/> 4<lb/> N<lb/> <lb/> i=1<lb/> α i ,<lb/> where N = min{ 3κ−l 1<lb/> 2<lb/> − 1 + j, 2j }.<lb/> 7. Discussions<lb/> We have derived the exact solutions of a family of Hamiltonians with the following general<lb/> form<lb/> H = F (Q 0 ) + g(Q + + Q − )<lb/> (7.1)<lb/> whereby Q ±,0 are particular polynomial deformations of the sl(2) Lie algebra and F (Q 0 ) is<lb/> a polynomial function of Q 0 with real coefficients. We have seen that via the differential<lb/> operator realization of these algebras, the block diagonal sectors of the Hamiltonians can<lb/> be realized as higher order quasi-exactly solvable differential operators. The eigenvalues<lb/> of the Hamiltonians in these sectors have been obtained via the functional Bethe ansatz<lb/> approach.<lb/> Specific cases of the general Hamiltonian have previously been solved via the algebraic<lb/> Bethe ansatz approach [17, 20–24] as mentioned earlier. Comparing both methods, it appears<lb/>
<page>1986<lb/> </page>
<note place="headnote">Y-H Lee et al<lb/> </note>
that the functional Bethe ansatz approach has some advantage over the algebraic Bethe ansatz<lb/> by requiring less algebraic machinery. This advantage manifests itself in the fact that we have<lb/> been able to give a unified solution for (1.1) through (5.7) and (5.8). Such a unified solution<lb/> presently appears beyond the limits of algebraic Bethe ansatz approaches which treat the<lb/> models on a case-by-case basis. It would therefore be interesting to see whether other classes<lb/> of exactly solvable models can be easily handled by the functional Bethe ansatz approach.<lb/> One avenue for further work would be to generalize the functional Bethe ansatz approach<lb/> to solve for other classes of Hamiltonians, such as q-deformed versions of the models<lb/> discussed above. It would also be worthwhile to explore the role of polynomial algebra<lb/> structures in connections between higher order ODEs and integrable models, i.e. the ODE/IM<lb/> correspondence [25].<lb/>
</body>
<back>
<div type="acknowledgement">Acknowledgment<lb/> This work was supported by the Australian Research Council.<lb/></div>
<listBibl>References<lb/> [1] Karassiov V P and Klimov A 1994 Phys. Lett. A 191 117<lb/> [2] Karassiov V P 1992 J. Sov. Laser Res. 13 188<lb/> Karassiov V P 1994 J. Phys. A: Math. Gen. 27 153<lb/> Karassiov V P 2000 J. Russian Laser Res. 21 370<lb/> [3] Karassiov V P, Gusev A A and Vinitsky S I 2002 Phys. Lett. A 295 247<lb/> [4] Lee Y-H, Yang W-L and Zhang Y-Z 2010 J. Phys. A: Math. Theor. 43 185204<lb/> Lee Y-H, Yang W-L and Zhang Y-Z 2010 J. Phys. A: Math. Theor. 43 375211<lb/> [5] Leggett A J 2001 Rev. Mod. Phys. 73 307<lb/> [6] Lipkin H J, Meshkov N and Glick A J 1965 Nucl. Phys. 62 188<lb/> [7] King G W, Hainer R M and Cross P C 1943 J. Chem. Phys. 11 27<lb/> [8] Tavis M and Cummings F W 1968 Phys. Rev. 170 379<lb/> [9] Hepp K and Lieb E H 1973 Ann. Phys. 76 360<lb/> [10] Rybin A, Kostelewicz G, Timonen J and Bogoliubov N M 1998 J. Phys. A: Math. Gen. 31 4705<lb/> [11] Turbiner A 1988 Commun. Math. Phys. 118 467<lb/> [12] Ushveridze A G 1994 Quasi-exactly Solvable Models in Quantum Mechanics (Bristol: Institute of Physics<lb/> Publishing)<lb/> [13] Gonzárez-López A, Kamran N and Olver P 1993 Commun. Math. Phys. 153 117<lb/> [14] Wiegmann P B and Zabrodin A V 1994 Phys. Rev. Lett. 72 1890<lb/> Wiegmann P B and Zabrodin A V 1995 Nucl. Phys. B 451 699<lb/> [15] Sasaki R, Yang W-L and Zhang Y-Z 2009 SIGMA 5 104<lb/> [16] Sasaki R and Takasaki K 2001 J. Phys. A: Math. Gen. 34 9533<lb/> [17] Enol'skii V Z, Salerno M, Kostov N A and Scott A C 1991 Phys. Scr. 43 229<lb/> Enol'skii V Z, Salerno M, Scott A C and Eilbeck J C 1992 Physica D 59 1<lb/> Enol'skii V Z, Kuznetsov V B and Salerno M 1993 Physica D 68 138<lb/> [18] Links J and Hibberd K 2006 SIGMA 2 094<lb/> [19] Ribeiro P, Vidal J and Mosseri R 2007 Phys. Rev. Lett. 99 050402<lb/> Ribeiro P, Vidal J and Mosseri R 2008 Phys. Rev. E 78 021106<lb/> [20] Pan F and Draayer J P 1999 Phys. Lett. B 451 1<lb/> [21] Morita H, Ohnishi H, da Providenica J and Nishiyama S 2006 Nucl. Phys. B 737 337<lb/> [22] Jarvis P D and Yates L A 2008 Mol. Phys. 106 955<lb/> [23] Bogoliubov N M, Bullough R K and Timonen J 1996 J. Phys. A: Math. Gen. 29 6305<lb/> [24] Amico L and Hikami K 2005 Eur Phys. J. B 43 387<lb/> Amico L, Frahm H, Osterloh A and Ribeiro G A P 2007 Nucl. Phys. B 787 283<lb/> [25] Dorey P and Tateo R 1999 J. Phys. A: Math. Gen. 32 L419</listBibl>
</back>
</text>
</tei>