  The creation of quantum coherences requires a system to be anharmonic. The
simplest such continuous 1D quantum system is the Kerr oscillator. It has a
number of interesting symmetries we derive. Its quantum dynamics is best
studied in phase space, using Wigner's distribution W and the associated Wigner
phase space current J. Expressions for the continuity equation governing its
time evolution are derived in terms of J and it is shown that J for Kerr
oscillators follows circles in phase space. Using J we also show that the
evolution's classical shear in phase space is quantum suppressed by an
effective `viscosity'. Quantifying this shear suppression provides measures to
contrast classical with quantum evolution and allows us to identify special
quantum states.
