  Distributions of eigenmodes are widely concerned in both bounded and open
systems. In the realm of chaos, counting resonances can characterize the
underlying dynamics (regular vs. chaotic), and is often instrumental to
identify classical-to-quantum correspondence. Here, we study, both
theoretically and experimentally, the statistics of chaotic resonances in an
optical microcavity with a mixed phase space of both regular and chaotic
dynamics. Information on the number of chaotic modes is extracted by counting
regular modes, which couple to the former via dynamical tunneling. The
experimental data are in agreement with a known semiclassical prediction for
the dependence of the number of chaotic resonances on the number of open
channels, while they deviate significantly from a purely
random-matrix-theory-based treatment, in general. We ascribe this result to the
ballistic decay of the rays, which occurs within Ehrenfest time, and
importantly, within the timescale of transient chaos. The present approach may
provide a general tool for the statistical analysis of chaotic resonances in
open systems.
