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We report on experiments that were performed with microwave waveguide systemsand demonstrate that in the frequency range of a single transversal mode theymay serve as a model for closed and open quantum graphs. These consist of bondsthat are connected at vertices. On the bonds, they are governed by theone-dimensional Schr\"odinger equation with boundary conditions imposed at thevertices. The resulting transport properties through the vertices may beexpressed in terms of a vertex scattering matrix. Quantum graphs withincommensurate bond lengths attracted interest within the field of quantumchaos because, depending on the characteristics of the vertex scatteringmatrix, its wave dynamic may exhibit features of a typical quantum system withchaotic counterpart. In distinction to microwave networks, which serve as anexperimental model of quantum graphs with Neumann boundary conditions, thevertex scattering matrices associated with a waveguide system depend on thewavenumber and the wave functions can be determined experimentally. We analyzethe spectral properties of microwave waveguide systems with preserved andpartially violated time-reversal invariance, and the properties of theassociated wave functions. Furthermore, we study properties of the scatteringmatrix describing the measurement process within the frame work of randommatrix theory for quantum chaotic scattering systems.

Experimental study of closed and open microwave waveguide graphs with preserved and partially violated time-reversal invariance