Geometry-dependent scattering through quantum billiards: Experiment and theory

We present experimental studies of geometry-specific quantum scattering in microwave billiards of a given shape. We perform full quantum-mechanical scattering calculations and find excellent agreement with experimental results. We also carry out semiclassical calculations where the conductance is given as a sum over all classical trajectories between the leads, each of the trajectories carrying a quantum-mechanical phase. We unambiguously demonstrate that the characteristic frequencies of the oscillations in the transmission and reflection amplitudes t and r are related to the length distribution of the classical trajectories between the leads, whereas the frequencies of the probabilities $T=|t{|}^{2}$ and $R=|r{|}^{2}$ can be understood in terms of the length difference distribution in the pairs of classical trajectories. We also discuss the effect of nonclassical ``ghost'' trajectories, i.e., trajectories that include classically forbidden reflection off the lead mouths.