  We experimentally studied evolution of quasi-eigenmodes as classical dynamics
undergoing a transition from being regular to chaotic in open quantum
billiards. In a deformation-variable microcavity we traced all high-Q cavity
modes in a wide range of frequency as the cavity deformation increased. By
employing an internal parameter we were able to obtain a mode-dynamics diagram
at a given deformation, showing avoided crossings between different mode
groups, and could directly observe the coupling strengths induced by ray chaos
among encountering modes. We also show that the observed mode-dynamics diagrams
reflect the underlying classical ray dynamics in the phase space.
