Programmable optical parametric amplifier synthesizer for cubic phase states and amplified Schrodinger cat states

We introduce a programmable optical parametric amplifier (OPA) synthesizer that, under a heralded photon-number-resolving framework, generates high-fidelity cubic phase states and amplifies Schrodinger cat states. By systematically exploring both the catalytic configuration, where the idler input and output contain the same number of photons ($m=n$), and non-catalytic configurations ($m\neq n$), we discover two qualitatively different functionalities. First, with a coherent-state signal input, our protocol generates cubic phase states with fidelity exceeding 0.99 across a broad range of $(m,n)$ configurations. Second, using a Schr\"odinger cat state as the signal input, the same framework amplifies the cat state: an input cat with amplitude $\alpha_{\mathrm{in}}\le 1$ is transformed into an output squeezed cat with $\alpha_{\mathrm{out}}\ge 2$ while maintaining fidelity above 0.99. The catalytic configuration preserves the input parity and restores the idler state, whereas non-catalytic configurations enable parity-flipping amplification with higher success rates. Moreover, the amplified output can serve as a seed for subsequent amplification rounds, offering a self-seeding pathway to progressively larger cat states. Our protocol requires only moderate-gain OPA operation and low-order photon-number-resolving detection, providing a flexible and experimentally accessible platform for cubic phase state preparation and amplified squeezed cat state generation.