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Phase effects in coherently-stimulated down-conversion with a quantized pump field

We investigate the effect of the cumulative phase on the photon statistics of the three-mode state whose evolution is described by the trilinear Hamiltonian $\hat{H}_{I}=i\hbarκ\big(\hat{a}\hat{b}\hat{c}^{\dagger}-\hat{a}^{\dagger}\hat{b}^{\dagger}c\big)$, wherein the pump is taken to be quantized (and prepared in a coherent state) and the signal and idler modes are initially seeded with coherent states. We provide a brief review of the two-mode squeezed coherent states generated by non-degenerate coherently-stimulated parametric down-conversion, whereby the nonlinear crystal is driven by a strong classical field. The statistics of the resulting two mode state have been shown to depend greatly on the cumulative phase $Φ=θ_{s}+θ_{i}-2ϕ$ where $θ_{s\left(i\right)}$ are the signal(idler) coherent state phases and $2ϕ$ is the classical pump phase. Using perturbation theory, we analytically show for short times how the photon statistics and entanglement properties of the resultant state depends strictly on this phase combination. We also present numerical results of the relevant quantities to show the evolution of the three modes and provide a qualitative analysis of the steady state valid for long times.