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

TL;DR

This paper studies how the cumulative phase affects photon statistics and entanglement in a three-mode quantum state generated by a quantized pump in down-conversion, revealing phase-dependent behavior through analytical and numerical methods.

Contribution

It analytically and numerically demonstrates the phase dependence of photon statistics and entanglement in a three-mode quantum state with a quantized pump, extending understanding beyond classical pump assumptions.

Findings

Photon statistics depend on the cumulative phase $\

Entanglement properties are phase-sensitive for short times

Steady-state analysis shows long-term phase effects

Abstract

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 $\hat{H}_{I}=i\hbar\kappa\big(\hat{a}\hat{b}\hat{c}^{\dagger}-\hat{a}^{\dagger}\hat{b}^{\dagger}c\big)$, 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 $\Phi=\theta_{s}+\theta_{i}-2\phi$ where $\theta_{s\left(i\right)}$ are the signal(idler) coherent state phases and $2\phi$ 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.