Number and phase quantum fluctuations in the down-conversion with a quantum pump

The photon number and phase quantum fluctuations in the field produced by the down-conversion process with a quantum pump are studied. The fully quantum approach using the method of numerical diagonalization of the interaction Hamiltonian is applied to find the evolution of the system. The evolution of the photon number fluctuations, the joint number of photons probability distribution, the quadrature variances, the joint phase probability distribution, the marginal number and phase distributions for the signal mode, the number and phase uncertainty products and squeezing parameters are calculated and illustrated graphically. The results for the signal mode are compared to the corresponding results for the ideal squeezed vacuum to show the range of validity of the parametric approximation.