  A Kerr nonlinear oscillator (KNO) supports a pair of steady eigenstates,
coherent states with opposite phases, that are good for the encoding of
continuous variable qubit basis states. Arbitrary control of the KNO confined
within the steady state subspace allows extraction of the Berry curvature
through the linear response of the physical observable to the quench velocity
of the system, providing an effective method for the characterization of
topology in the KNO. As an alternative, the control adopting the "shortcut to
adiabaticity" to the KNO enables the exploration of the topology through
accelerated adiabatic eigenstate evolution to measure all three physical
observables. Topological transitions are revealed by the jump of the first
Chern number, obtained respectively from the integral of the Berry curvature
and of the new polar angle relation, over the whole parameter space. Our
strategy paves the way for measuring topological transitions in continuous
variable systems.
