  We theoretically propose a method for on-demand generation of traveling
Schr\"odinger cat states, namely, quantum superpositions of distinct coherent
states of traveling fields. This method is based on deterministic generation of
intracavity cat states using a Kerr-nonlinear parametric oscillator (KPO) via
quantum adiabatic evolution. We show that the cat states generated inside a KPO
can be released into an output mode by controlling the parametric pump
amplitude dynamically. We further show that the quality of the traveling cat
states can be improved by using a shortcut-to-adiabaticity technique.
