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Stabilization of quantum manifolds is at the heart of error-protected quantuminformation storage and manipulation. Nonlinear driven-dissipative processesachieve such stabilization in a hardware efficient manner. Josephson circuitswith parametric pump drives implement these nonlinear interactions. In thisarticle, we propose a scheme to engineer a four-photon drive and dissipation ona harmonic oscillator by cascading experimentally demonstrated two-photonprocesses. This would stabilize a four-dimensional degenerate manifold in asuperconducting resonator. We analyze the performance of the scheme usingnumerical simulations of a realizable system with experimentally achievableparameters.

Generating higher order quantum dissipation from lower order parametric processes