  Mechanical degrees of freedom are natural candidates for continuous-variable
quantum information processing and bosonic quantum simulations. These
applications, however, require the engineering of squeezing and nonlinearities
in the quantum regime. Here we demonstrate ground state squeezing of a
gigahertz-frequency mechanical resonator coupled to a superconducting qubit.
This is achieved by parametrically driving the qubit, which results in an
effective two-phonon drive. In addition, we show that the resonator mode
inherits a nonlinearity from the off-resonant coupling with the qubit, which
can be tuned by controlling the detuning. We thus realize a mechanical squeezed
Kerr oscillator, where we demonstrate the preparation of non-Gaussian quantum
states of motion with Wigner function negativities and high quantum Fisher
information. This shows that our results also have applications in quantum
metrology and sensing.
