  The mathematical objects employed in physical theories do not always behave
well. Einstein's theory of space and time allows for spacetime singularities
and Van Hove singularities arise in condensed matter physics, while intensity,
phase and polarization singularities pervade wave physics. Within dissipative
systems governed by matrices, singularities occur at the exceptional points in
parameter space whereby some eigenvalues and eigenvectors coalesce
simultaneously. However, the nature of exceptional points arising in quantum
systems described within an open quantum systems approach has been much less
studied. Here we consider a quantum oscillator driven parametrically and
subject to loss. This squeezed system exhibits an exceptional point in the
dynamical equations describing its first and second moments, which acts as a
borderland between two phases with distinctive physical consequences. In
particular, we discuss how the populations, correlations, squeezed quadratures
and optical spectra crucially depend on being above or below the exceptional
point. We also remark upon the presence of a dissipative phase transition at a
critical point, which is associated with the closing of the Liouvillian gap.
Our results invite the experimental probing of quantum resonators under
two-photon driving, and perhaps a reappraisal of exceptional and critical
points within dissipative quantum systems more generally.
