  We determine with unprecedented accuracy the lowest 900 eigenvalues of two
quantum constant-width billiards from resonance spectra measured with flat,
superconducting microwave resonators. While the classical dynamics of the
constant-width billiards is unidirectional, a change of the direction of motion
is possible in the corresponding quantum system via dynamical tunneling. This
becomes manifest in a splitting of the vast majority of resonances into
doublets of nearly degenerate ones. The fluctuation properties of the two
respective spectra are demonstrated to coincide with those of a random-matrix
model for systems with violated time-reversal invariance and a mixed dynamics.
Furthermore, we investigate tunneling in terms of the splittings of the doublet
partners. On the basis of the random-matrix model we derive an analytical
expression for the splitting distribution which is generally applicable to
systems exhibiting dynamical tunneling between two regions with (predominantly)
chaotic dynamics.
