  Parametric coupling between harmonic oscillators has enabled exquisite
measurement precision and control of linear resonators, being extensively
studied, for example, in cavity optomechanics. This level of control has been
made possible by using strong sideband drives, enhancing the coupling rate
while also linearizing the interaction. In this article, we demonstrate a new
paradigm of parametrically coupled microwave circuits replacing one linear
microwave cavity with a superconducting transmon qubit. Our system utilizes
photon-pressure coupling between the transmon qubit and a highly linear
microwave resonator, a microwave analog of the radiation-pressure interaction.
Applying a strong sideband drive results in an on-demand, non-linear
Jaynes-Cummings interaction with the linear resonator. We also observe a
single-photon coupling rate an order of magnitude larger than all decay rates,
placing the device in the single-photon strong coupling regime. This
demonstration of photon-pressure Jaynes-Cummings interactions paves the way for
developing novel photon-pressure quantum information processing hardware and
will enable exotic tests of quantum gravity in the future by interfacing this
new platform with mechanical resonators.
