  Superconducting quantum systems (artificial atoms) have been recently
successfully used to demonstrate on-chip effects of quantum optics with single
atoms in the microwave range. In particular, a well-known effect of four-wave
mixing could reveal a series of features beyond classical physics, when a
non-linear medium is scaled down to a single quantum scatterer. Here we
demonstrate a phenomenon of the quantum wave mixing (QWM) on a single
superconducting artificial atom. In the QWM, the spectrum of elastically
scattered radiation is a direct map of the interacting superposed and coherent
photonic states. Moreover, the artificial atom visualises photon-state
statistics, distinguishing coherent, one- and two-photon superposed states with
the finite (quantized) number of peaks in the quantum regime. Our results may
give a new insight into nonlinear quantum effects in microwave optics with
artificial atoms.
