Beyond the parametric approximation: Pump depletion, entanglement, and squeezing in macroscopic down-conversion

Abstract

We study the dynamics of the pump mode in the down-conversion Hamiltonian using the cumulant-expansion method, perturbation theory, and the full numerical simulation of systems with a pump mean photon number of up to $100000$. We particularly focus on the properties of the pump mode such as depletion, entanglement, and squeezing for an experimentally relevant initial state in which the pump mode is initialized in a coherent state. Through this analysis, we obtain the short-time behavior of various quantities and derive timescales at which the above-mentioned features, which cannot be understood through the parametric approximation, originate in the system. We also provide an entanglement witness involving moments of bosonic operators that can capture the entanglement of the pump mode. Finally, we study the photon-number statistics of the pump and the signal and idler modes to understand the general behavior of these modes for experimentally relevant timescales.