Photon number divergence in the quantum theory ofn-photon down conversion

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Abstract
The process of n-photon down conversion is considered both with a quantized pump and in the parametric approximation. In the parametric approximation it is formally found that the photon number diverges at a finite time for n>2. This divergence limits the times for which the parametric approximation is useful. The divergence does not occur if the pump is quantized. Furthermore, it is shown that the Hamiltonian with a quantized pump is essentially self-adjoint. This means that the questions that have been raised about the existence of the time-development operator in the parametric approximation do not arise when the pump is quantized.