  Photon-pressure coupling between two superconducting circuits is a promising
platform for investigating radiation-pressure coupling in novel parameter
regimes and for the development of radio-frequency (RF) quantum photonics and
quantum-limited RF sensing. So far, the intrinsic Josephson nonlinearity of
photon-pressure coupled circuits has not been considered a potential resource
for enhanced devices or novel experimental schemes. Here, we implement
photon-pressure coupling between a RF circuit and a microwave cavity containing
a superconducting quantum interference device (SQUID) which can be operated as
a Josephson parametric amplifier (JPA). We demonstrate a Kerr-based enhancement
of the photon-pressure single-photon coupling rate and an increase of the
cooperativity by one order of magnitude in the amplifier regime. In addition,
we characterize the upconverted and Kerr-amplified residual thermal
fluctuations of the RF circuit, and observe that the intracavity amplification
reduces the measurement imprecision. Finally, we demonstrate that RF mode
sideband-cooling is surprisingly not limited to the effective amplifier mode
temperature arising from quantum noise amplification, which we explain by
non-trivial bath dynamics due to a two-stage amplification process. Our results
demonstrate how Kerr nonlinearities and in particular Josephson parametric
amplification can be utilized as resource for enhanced photon-pressure systems
and Kerr cavity optomechanics.
