Systems with intensity-dependent conversion integrable by finite orthogonal polynomials

Abstract

We present exact solutions of a class of the nonlinear models which describe the parametric conversion of photons. Hamiltonians of these models are related to the classes of finite orthogonal polynomials. The spectra and exact expressions for eigenvectors of these Hamiltonians are obtained.

Transformation design and nonlinear Hamiltonians

“…We thus establish that this particular class of nonlinear Hamiltonians are exactly solvable. The connection between orthogonal polynomials and nonlinear optical process has been studied previously and general results have been obtained [9,10,11]. In particular, the mathematical results that we present in section 2 is a special case of the general theory presented in [9].…”

“…The connection between orthogonal polynomials and nonlinear optical process has been studied previously and general results have been obtained [9,10,11]. In particular, the mathematical results that we present in section 2 is a special case of the general theory presented in [9]. In [9] the general method is described for solving two boson systems in quantum optics via of orthogonal polynomials systems.…”

On the dimensions of oscillator algebras induced by orthogonal polynomials

“…The dynamics of (N + 1)-boson system is assumed to be governed by a Hamiltonian operator of the form [16,17]: H = h 0 (a * 0 a 0 , ..., a * N a N ) + g 0 (a * 0 a 0 , ..., a * N a N )a k 0 0 · · · a k N N (5.1) + a −k 0 0 · · · a −k N N g 0 (a * 0 a 0 , ..., a * N a N ), where (a 0 , ..., a N ) and (a * 0 , ..., a * N ) are bosonic annihilation and creation operators respectively with standard Heisenberg commutation relations [16]. The monomial a k 0 0 · · · a k N N , k 0 , ..., k N ∈ Z can be thought of as an operator which describes the subsequent creation and annihilation of the clusters of the bosonic modes.…”

“…For simplicity in (5.18) we may take U 1 = 1 and U 2 = c, some constant; then the necessary and sufficient condition for the reduction algebra to be isomorphic to the harmonic oscillator algebra becomes (5.19) |g 0 (x 0 , x 1 )| 2 = (α 0,0 x 0 + α 0,1 x 1 ) + c P k 0 (x 0 )P k 1 (x 1 ) , which implies that (5.20) g 0 (x, y) = exp(iθ) (α 0,0 x + α 0,1 y) + c P k 0 (x)P k 1 (y) 1 2 . Now let us look at a particular example given in [17], where H = h 0 (a * 0 a 0 , a * 1 a 1 ) + g 0 (a * 0 a 0 , a * 1 a 1 )a k 0 0 a * k 1 1 (5.21) +a * k 0 0 a k 1 1 g 0 (a * 0 a 0 , a * 1 a 1 ), with the matrix elements (5.22) α 0,0 = 1 k 0 , α 0,1 = 0, α 1,0 = k 1 , α 1,1 = k 0 .…”

Interference of composite bosons

“…Hamiltonians of the form of (10) have been studied before [10,11] and can be used in models of nonlinear couplers [12,13,14]. It can be verified thatM = kâ †â +nb †b commutes with (10).…”