Parametric amplification with a trilinear Hamiltonian

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Abstract
The trilinear Hamiltonian of Tucker and Walls, governing the parametric amplification, leads to three coupled nonlinear Heisenberg equations in terms of creation and annihilation operators. Using the Manley-Rowe relations the equations are reduced to a single nonlinear equation in terms of one of the number operators. Thereafter, it is solved by means of a perturbation method around the diagonal, allowing calculation of the mean number of photons and the variance of the amplified field. The short-time approximation, the parametric approximation, and the classical limit are reobtained by suitable limiting processes.