  We study four- and higher-order wave mixing of continuous coherent waves on a
single superconducting artificial atom. Narrow side peaks of different orders
of nonlinearity resulting from elastic multi-photon scattering on the atom are
observed and investigated. We derive an analytical expression for the peak
amplitudes and show that the ratio of any two adjacent peaks is a function of
driving amplitudes and detuning. This is attributed to the photon distribution
in the coherent states and provides a measure for characterisation of photon
statistics in non-classical coherent waves. We also demonstrate an
Autler-Townes-like splitting of side peaks, the magnitude of which scales with
the scattering order.
