Quantum nonlinear dynamics of non-degenerate parametric amplification beyond the stiff-pump approximation

In circuit QED, non-degenerate parametric amplification is commonly realized using Josephson Parametric Converter (JPC) based nonlinear circuits. While such systems are often studied within a linear stiff-pump approximation, their nonlinear dynamics become important for strong input signals, nonlinearity strengths, and large gain operation. We present a theoretical analysis of non-degenerate parametric amplification going beyond this stiff-pump approximation, in particular accounting for the quantum dynamics of the pump mode. Within a regime of weak quantum fluctuations, strongly amplified input signals dominate the depletion of the pump, and the associated compression of amplifier gain is characterized analytically. By further allowing for fluctuations in the pump mode, we find that interactions between the idler-signal subspace and the common pump mode in the nonlinear regime can strongly modify the noise properties of the amplifier. Finally, we analyze nonlinear dynamics in the regime of strong quantum fluctuations. Employing a reduced nonlinear two-mode description enables full quantum simulations in this regime where pump depletion can be strongly influenced by amplified vacuum fluctuations, leading to additional gain compression and added noise.