  Cat states are a valuable resource for quantum metrology applications,
promising to enable sensitivity down to the Heisenberg limit. Moreover,
Schr\"odinger cat states, based on a coherent superposition of coherent states,
show robustness against phase-flip errors making them a promising candidate for
bosonic quantum codes. A pathway to realize cat states is via utilizing single
Kerr-type anharmonicities as found in superconducting devices as well as in
Rydberg atoms. Such platforms nevertheless utilize only the second order
anharmonicity, which limits the time it takes for a cat state to be prepared.
Here we show how proper tuning of multiple higher order nonlinear interactions
leads to shorter cat state preparation time. We also discuss practical aspects
including an optimal control scheme which allows us to start the state
preparation from the vacuum state under standard single mode driving. Lastly,
we propose an ensemble of Rydberg atoms that exhibits higher order
nonlinearities as a platform to prepare cat states in the laboratory.
