  We propose a scheme for measuring topological properties in a
two-photon-driven Kerr-nonlinear resonator (KNR) subjected to a single-photon
modulation. The topological properties are revealed through the observation of
the Berry curvature and hence the first Chern number, as a nonadiabatic
response of the physical observable to the change rate of the control parameter
of the modulated drive. The parameter manifold, constructed from the system's
Hamiltonian that determines its dynamics constrained in the state space spanned
by the even and odd cat states as two basis states, is adjusted so that the
degeneracy crossing the manifold indicates a topological transition. The
scheme, with such continuous variable states in mesoscpic systems, provides a
new perspective for exploration of the geometry and the related topology with
complex systems.
