  A superconducting qubit in a waveguide behaves as a point-like nonlinear
element. If irradiated with nearly resonant microwave pulses, the qubit
undergoes quantum evolution and generates coherent fields at sideband
frequencies due to elastic scattering. This effect is called Quantum Wave
Mixing (QWM), and the number of emerged side components depends on the number
of interacting photons. By driving a superconducting qubit with short pulses
with alternating carrier frequencies, we control the maximal number of photons
simultaneously interacting with a two-level system by varying the number and
duration of applied pulses. Increasing the number of pulses results in
consecutive growth of the order of non-linearity, which manifests in additional
coherent side peaks appearing in the spectrum of scattered radiation while the
whole spectrum maintains its asymmetry.
