Integrable Models of Interaction of Matter with Radiation

The simplified models of interaction of charged matter with resonance modes of radiation generalizing the well-known Jaynes–Cummings and Dicke models are considered. It is found that these new models are integrable for arbitrary numbers of dipole sources and resonance modes of the radiation field. The problem of explicit diagonalisation of corresponding Hamiltonians is discussed.


Introduction
In the last four decades, much attention has been paid to the problem of describing processes of interaction of charged matter with electromagnetic radiation at resonance [1,2,3,4]. This problem, being overcomplicated from the mathematical viewpoint in the general case, admits some very attractive simplifications which allow one to construct some rather simple models having even exact solutions. These models can be used for quantum statistical description of real interaction processes.
The starting point for constructing such models is the quantum Hamiltonian [3] where p and m are the momentum and mass of the electron in some atom, u(x) is the Coulomb potential of interaction of the electron with nucleus, and A is the vector potential of the secondly quantized electromagnetic field which is given by the formula where e k are polarization vectors and b k , b + k are the usual operators of annihilation and creation of the mode with wave vector k and the energy ω k . There are several steps of simplification of this general Hamiltonian. First, one neglects the term which is nonlinear in the vector potential, where σ z is the Pauli matrix. Third, one considers only small k, ka ≪ 1, where a is the typical size of an atom, e ±i k x ∼ 1. Fourth, one uses the dipole and so-called rotating wave approximations [4], which result in where σ ± are the usual Pauli matrices. And finally, one considers only the case of resonance: k → k fixed . By all these steps one can write down very simple Jaynes-Cummings Hamiltonian [2] which is linear in the operators of creation and annihilation of one resonant mode, The solution of eigenequation for the operator (1) is almost trivial, but it allows one to describetime evolution of the corresponding wave function in great detail, with transparent applications to physics of one-mode laser. This extremely simple model admits almost evident generalisations in two directions.
2 The simplest generalisations of the JC model: the n-level atom and Dicke model The first of these generalisations is the model of one n-level atom with (n − 1) modes of resonant radiation, the sets {ω α }, {g α } are energies of the modes and their coupling constants, and ∆ is so-called detuning parameter [5,6]. The extended physical motivation for this kind of the generalization of the Jaynes-Cummings model can be found in [6]. There are evident commutation relations JC ,N α = 0, which make the problem of diagonalisation of (2) at n ≥ 3 quite easy. The second way of generalisation of the Jaynes-Cummings model consists in considering an arbitrary number N of two-level sources of radiation, The solution of this so-called Dicke model which has been proposed many years ago [1] is far from being trivial. It turned out that the problem with the Hamiltonian (3) has some hidden dynamical symmetry which allows one to construct some nontrivial operators commuting with (3). What happens if one will try to use both the ways of generalisation of the Jaynes-Cummings model, i.e. consider the Hamiltonian of the problem of an arbitrary number N of multilevel sources interacting with an arbitrary number n − 1 of resonant modes of radiation, where N ≥ 2, n ≥ 3 (the detuning parameters {∆} and coupling constants {g (k) α } are assumed to be arbitrary)? The answer is not clear till now. In the following two sections we will argue that there is some simplification of the spectral problem defined by (4) in the case of equality of all the modules of the coupling constants {g}: these constants should not depend on k and their dependence on α is given by the formula g (k) α ∝ ε α , ε α = ±1. Namely, we will construct in this case explicitly some set of N − 1 nontrivial operators commuting with (4); their existence allows one to call the model (4) quantum integrable under the above choice of the coupling constants.

The models of Gaudin type
In this section, let us consider some more general operators introduced first by Gaudin [7], where T j αβ are operators of absolutely arbitrary representations of SL(n), obeying the commutation relations and f αβ js are arbitrary sets of numbers, obeying the evident restriction f αβ js = f βα js . It is easy to see that operatorŝ for all indices {j, k}. It is easy to see that in this case the "Hamiltonian" where η j and ω alpha are arbitrary constants, commutes with all H In the simplest nontrivial case of n = 2, one can always choose {f } such that f αβ jk = −f βα kj , and the equations (7) can be written in the form which immediately gives a overdetermined set of bilinear equations for {f }, It has been shown by Gaudin [7,8] that for n = 2 there is some so-called anisotropic solution to (10), where {u j } are arbitrary numbers. Now, following Gaudin, let us choose them as As β and consider the limit N 2 ≫ 1, where N 2 is an eigenvalue of the operatorN 2 . It is easy to see that in this limit b-operators can be treated as c-numbers; and send N 2 → ∞. In this limiting process, one easily finds And finally, if we will choose T (j) αβ as σ-representation of SL(2), we will obtain just the Dicke Hamiltonian! This derivation of the Dicke model proposed first by Gaudin [7] is not original of course. There are some recent papers in which the connection between it and the Gaudin models was exploited for construction the algebraic Bethe ansatz method for eigenvectors of the Dicke Hamiltonian and some generalisations which might be considered as the interaction term without the rotating wave approximation [9,10,11].
The above limiting procedure, however, does not work in the general case n ≥ 3. One also has in this case the overdetermined system of bilinear equations f αβ js f αλ sk + f αλ kj f βλ js + f αβ jk f βλ ks = 0, but, if n ≥ 3, there is a possibility of the choice of indices α = β = λ which excludes all "anisotropic" solutions of the type (11). There are only "isotropic" solutions of the type f αα js = f αβ js = 1 ν j −νs , and hence there are no parameters for N n → ∞. Hence it seems that the Gaudin approach does not lead to any integrable system of the type (4).
Surprisingly enough, one can construct integrable models of Gaudin type at n ≥ 3 by using only "isotropic" solution of (10). It will be done in the next section.

New integrable model with N sources and n − 1 modes
The receipt of constructing the integrable model of the type (4) is quite unusual. Let us choose the above H-operators as obeying the equality where x is some parameter andŜ jk have the following property: as Let us choose f αβ jk = A(ν j − ν k ) −1 , 1 ≤ j, k ≤ N , i.e. use the isotropic Gaudin solution at j, s, k ≤ N . Then the double sum vanishes and we are left witĥ There are no other restrictions to the numbers f αβ j,N +1 except symmetry. Hence we have i.e. f αβ j,N +1 = 0 unless α = n or β = n. The commutator H N j , H N k (16) is still too complicated. Let us make the next assumption: where {ν j } are arbitrary numbers. Then (16) can be recast in the form Let now choosê as Jordan-Schwinger representation of SL(n)). Hence the basis of the whole Hilbert space of the problem consists of the direct products of the basis vectors of the spaces in whichT j βλ , 1 ≤ j ≤ N act, and various basic vectors of above Jordan-Schwinger representation, Consider the action of the right-hand side of (17) on the subspace spanned by the vectors ϕ (L) nn with l n = L, L ≫ 1 such that They commute with H (N ) And finally, let us choose asT j αβ the matrix representation of SL(n), and add linear combination N +1 . It is easy to see that we obtain just the operator of generalized Dicke model (4) for n ≥ 3, with coupling constants Hence, under the above choice of the coupling constants, the most general model (4) becomes quantum integrable and the spectral problem might be simplified. However, we did not find the way for doing it except the simplest nontrivial case of N = 2 which is described in the next section.

Explicit solution for N = 2
In this case the Hamiltonian (4) under the conditions (20) can be written as Let us rewrite (21) as Eigenfunctions of (22) are of course common eigenfunctions of (n + 1) operators One gets from the relationsk 1 ψ = k 1 ψ,k 2 ψ = k 2 ψ the following set of algebraic equations for the coefficients {A, B, C}, They can be easily reduced to two equations for k 1 , k 2 : ω α N α + k 1 + k 2 .

Conclusions
In this paper, we found the most general quantum integrable model of interaction of (n − 1) modes of radiation with N dipole sources which comprises all known models of that type: Jaynes-Cummings model (N = 1, n = 2), its generalisation for arbitrary number of modes (N = 1), Dicke model (arbitrary N , n = 2). The commutative ring of operators which includes H (N ) n is found in explicit form. We did not use any "anisotropic" form of the Gaudin solution; the Hamiltonian of the model was obtained via some limiting procedure. We almost immediately got solution for N = 2, arbitrary n with the use of additional integrals of motion, but the most interesting case of solution for arbitrary N and n is unreachable at the present stage of investigation. We hope to come back to this problem in the future.
This paper has been presented for memorial volume of Vadim Kuznetsov. We both knew him personally -he was our guest in Dubna almost 17 years ago when he was a PhD student of Professor I.V. Komarov in Leningrad. We remember him as bright young man, very active in the field of quantum and classical integrable systems. Later he solved very complicated problem of integration of equations of motion for classical Toda chains with non-exponential ends [12] proposed by one of us (V.I.).