Effect of the pump state on the dynamics of the parametric amplifier

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Abstract
The parametric amplifier is usually studied under the assumption that the pump mode is in a large-amplitude coherent state. We consider what happens when this is no longer the case. We first derive some exact relations between the number fluctuations of the pump and signal, the pump and the idler, and also the signal and the idler. It is then determined how much phase noise can be tolerated in the initial pump state if two-mode squeezing is to be produced in the signal-idler field. The relation between the rotational symmetry in phase space of the initial pump state and that of the signal-idler subsystem is determined. We find that the sum phase distribution of the signal and idler has twice the rotational symmetry as that of the initial pump state. Finally, we generalize the parametric approximation to treat pump states which are not coherent states. We examine the effect of a phase-squeezed pump state and find that the pump squeezing can actually decrease the two-mode squeezing of the signal-idler field while enhancing a form of higher-order squeezing, sum squeezing, for these fields.