  Nonlinear processes in the quantum regime are essential for many
applications, such as quantum-limited amplification, measurement and control of
quantum systems. In particular, the field of quantum error correction relies
heavily on high-order nonlinear interactions between various modes of a quantum
system. However, the required order of nonlinearity is often not directly
available or weak compared to dissipation present in the system. Here, we
experimentally demonstrate a route to obtain higher-order nonlinearity by
combining more easily available lower-order nonlinear processes, using a
generalization of the Raman transition. In particular, we show a transformation
of four photons of a high-Q superconducting resonator into two excitations of a
superconducting transmon mode and vice versa. The resulting six-quanta process
is obtained by cascading two fourth-order nonlinear processes through a virtual
state. We expect this type of process to become a key component of hardware
efficient quantum error correction using continuous-variable error correction
codes.
