Limits to squeezing and phase information in the parametric amplifier

We investigate the time evolution of squeezing in the signal mode of the undamped parametric amplifier. Both the degenerate and nondegenerate cases are considered. These are the simplest models that give limits to squeezing caused by both the intrinsic pump noise and pump depletion. Conflicting analytic expressions have been reported. We resolve this by the use of exact numerical methods, employing both number state and stochastic techniques. Our central result is that in both cases we find that the minimum in the squeezed quadrature variance scales as N${\mathrm{¯}}^{\mathrm{-}1/2}$, where N¯ is the initial coherent pump photon number. This bound on the amount of squeezing obtainable, coupled with the loss in energy due to the conversion efficiency, serves to limit the phase information in the squeezed channel to no more than that of the coherent pump. We believe this to be a fundamental limit for using the squeezed radiation generated in coherently driven parametric amplifiers.