We explore the optical parametric amplifier, an optical device where a pump field creates a pair of lower-frequency fields: signal and idler. The pump field is usually treated classically, but this thesis focuses on scenarios where the pump must be treated quantum mechanically. One of these scenarios is the growing field of nonlinear interferometry, where the fundamental sensitivity of a probed relative phase can beat the classical bounds and reach the maximum limit allowed by quantum mechanics, the Heisenberg limit. Indeed, we show that a fully quantum nonlinear interferometer displays a Heisenberg scaling in terms of the mean number of input pump photons. This result goes beyond the well-accepted Heisenberg scaling with respect to the down-converted photons inside the interferometer, which predicts unphysical phase sensitivities starting at a particular input pump energy. Our theoretical findings are particularly useful when designing a nonlinear interferometer with bright pump fields or optimized optical parametric amplifiers for quantum metrology and quantum imaging applications. The quantum nature of the pump field may also play a central role concerning other physical phenomena, like Hawking radiation in the context of black holes. As suggested by several authors, both the optical parametric amplifier and Hawking radiation comprise the creation of fundamental particle pairs. Thus, if the optical parametric amplifier is fully treated quantum mechanically, we may get insight into an open problem in modern physics, namely the black hole information paradox. According to this paradox, the information stored in a black hole can be destroyed once the black hole has evaporated by emitting Hawking radiation, contradicting quantum mechanics. Despite the experimental efforts to build systems that reproduce event horizons and gravitational effects in the laboratory, the evaporation of black holes due to the emission of Hawking radiation remains a challenging task. In this thesis, we experimentally investigate the impact of an evolving pump field in an optical parametric amplifier by optimizing a parametric down-conversion process. We measure the pump and signal photon number properties, finding that the pump field gets chaotic and the signal coherent when the pump displays some sizeable depletion. We arrive at similar conclusions about the pump field from its measured Wigner function. Our experiment is the first step towards a successful experiment that could suggest that information in the black hole is not destroyed but encoded in the emitted Hawking radiation starting at some point in the black hole evolution. We finally discuss further experimental improvements to investigate the parallel between the optical parametric amplifier and Hawking radiation.