  Multiphoton up/down conversion in a transmon circuit, driven by a pair of
microwaves tuned near and far off the qubit resonance, has been observed. The
experimental realization of these high order non-linear processes is
accomplished in the three-photon regime, when the transmon is coupled to weak
bichromatic microwave fields with the same Rabi frequencies. A many-mode
Floquet formalism, with longitudinal coupling, is used to simulate the quantum
interferences in the absorption spectrum that manifest the multiphoton pumping
processes in the transmon qubit. An intuitive graph theoretic approach is used
to introduce effective Hamiltonians that elucidate main features of the Floquet
results. The analytical solutions also illustrate how controllability is
achievable for desired single- or multiphoton pumping processes in a wide
frequency range.
