  The driven Kerr parametric oscillator, of interest to fundamental physics and
quantum technologies, exhibits an excited state quantum phase transition
(ESQPT) originating in an unstable classical periodic orbit. The main signature
of this type of ESQPT is a singularity in the level density in the vicinity of
the energy of the classical separatrix that divides the phase space into two
distinct regions. The quantum states with energies below the separatrix are
useful for quantum technologies, because they show a cat-like structure that
protects them against local decoherence processes. In this work, we show how
chaos arising from the interplay between the external drive and the
nonlinearities of the system destroys the ESQPT and eventually eliminates the
cat states. Our results demonstrate the importance of the analysis of
theoretical models for the design of new parametric oscillators with ever
larger nonlinearities.
