Engineering a cubic quantum nondemolition gate with mesoscopic optical parametric interactions

We propose a scheme to realize a cubic quantum nondemolition (QND) Hamiltonian with optical parametric interactions. We show that strongly squeezed fundamental and second-harmonic fields propagating in a χ (2) nonlinear medium effectively evolve under a cubic QND Hamiltonian. We highlight the versatility offered by such a Hamiltonian for engineering non-Gaussian quantum states, such as Schrödinger cat states and cubic phase states. We show that our scheme can be highly tolerant of overall detection inefficiency with an auxiliary high-gain phase-sensitive optical amplifier. Our proposal involves parametric interactions in a mesoscopic photon-number regime, significantly enhancing the effective nonlinear coupling from the native single-photon coupling rate while providing powerful means to fight photon propagation loss. Experimental numbers suggest that our scheme might be feasible in the near future, particularly with pulsed nonlinear nanophotonics.