  By applying a microwave drive to a specially designed Josephson circuit, we
have realized an elementary quantum optics model, the squeezed Kerr oscillator.
This model displays, as the squeezing amplitude is increased, a cross-over from
a single ground state regime to a doubly-degenerate ground state regime. In the
latter case, the ground state manifold is spanned by Schr\"odinger-cat states,
i.e. quantum superpositions of coherent states with opposite phases. For the
first time, having resolved up to the tenth excited state in a spectroscopic
experiment, we confirm that the proposed emergent static effective Hamiltonian
correctly describes the system, despite its driven character. We also find that
the lifetime of the coherent state components of the cat states increases in
steps as a function of the squeezing amplitude. We interpret the staircase
pattern as resulting from pairwise level kissing in the excited state spectrum.
Considering the Kerr-cat qubit encoded in this ground state manifold, we
achieve for the first time quantum nondemolition readout fidelities greater
than 99%, and enhancement of the phase-flip lifetime by more than two orders of
magnitude, while retaining universal quantum control. Our experiment
illustrates the crucial role of parametric drive Hamiltonian engineering for
hardware-efficient quantum computation.
