  Photonic states of superconducting microwave cavities controlled by transmon
ancillas provide a platform for encoding and manipulating quantum information.
A key challenge in scaling up the platform is the requirement to communicate on
demand the information between the cavities. It has been recently demonstrated
that a tunable bilinear interaction between two cavities can be realized by
coupling them to a bichromatically-driven transmon ancilla, which allows
swapping and interfering the multi-photon states of the cavities [Gao et al.,
Phys. Rev. X 8, 021073(2018)]. Here, we explore both theoretically and
experimentally the regime of relatively strong drives on the ancilla needed to
achieve fast SWAP gates but which can also lead to undesired non-perturbative
effects that lower the SWAP fidelity. We develop a theoretical formalism based
on linear response theory that allows one to calculate the rate of
ancilla-induced interaction, decay and frequency shift of the cavities in terms
of a susceptibility matrix. We treat the drives non-perturbatively using
Floquet theory, and find that the interference of the two drives can strongly
alter the system dynamics even in the regime where the rotating wave
approximation applies. We identify two major sources of infidelity due to
ancilla decoherence. i) Ancilla dissipation and dephasing leads to incoherent
hopping among Floquet states which occurs even when the ancilla is at zero
temperature, resulting in a sudden change of the SWAP rate. ii) The cavities
inherit finite decay from the relatively lossy ancilla through the inverse
Purcell effect; the effect can be enhanced when the drive-induced AC Stark
shift pushes certain ancilla transition frequencies to the vicinity of the
cavity frequencies. The theoretical predictions agree quantitatively with the
experimental results, paving the way for using the theory to design future
experiments.
