7 research outputs found
Are Scattering Properties of Graphs Uniquely Connected to Their Shapes?
The famous question of Mark Kac "Can one hear the shape of a drum?"
addressing the unique connection between the shape of a planar region and the
spectrum of the corresponding Laplace operator can be legitimately extended to
scattering systems. In the modified version one asks whether the geometry of a
vibrating system can be determined by scattering experiments. We present the
first experimental approach to this problem in the case of microwave graphs
(networks) simulating quantum graphs. Our experimental results strongly
indicate a negative answer. To demonstrate this we consider scattering from a
pair of isospectral microwave networks consisting of vertices connected by
microwave coaxial cables and extended to scattering systems by connecting leads
to infinity to form isoscattering networks. We show that the amplitudes and
phases of the determinants of the scattering matrices of such networks are the
same within the experimental uncertainties. Furthermore, we demonstrate that
the scattering matrices of the networks are conjugated by the, so called,
transplantation relation.Comment: 3 figures; Physical Review Letters, 201
Experimental and numerical investigation of the reflection coefficient and the distributions of Wigner's reaction matrix for irregular graphs with absorption
We present the results of experimental and numerical study of the
distribution of the reflection coefficient P(R) and the distributions of the
imaginary P(v) and the real P(u) parts of the Wigner's reaction K matrix for
irregular fully connected hexagon networks (graphs) in the presence of strong
absorption. In the experiment we used microwave networks, which were built of
coaxial cables and attenuators connected by joints. In the numerical
calculations experimental networks were described by quantum fully connected
hexagon graphs. The presence of absorption introduced by attenuators was
modelled by optical potentials. The distribution of the reflection coefficient
P(R) and the distributions of the reaction K matrix were obtained from the
measurements and numerical calculations of the scattering matrix S of the
networks and graphs, respectively. We show that the experimental and numerical
results are in good agreement with the exact analytic ones obtained within the
framework of random matrix theory (RMT).Comment: 15 pages, 8 figure
Experimental simulation of quantum graphs by microwave networks
We present the results of experimental and theoretical study of irregular,
tetrahedral microwave networks consisting of coaxial cables (annular
waveguides) connected by T-joints. The spectra of the networks were measured in
the frequency range 0.0001-16 GHz in order to obtain their statistical
properties such as the integrated nearest neighbor spacing distribution and the
spectral rigidity. The comparison of our experimental and theoretical results
shows that microwave networks can simulate quantum graphs with time reversal
symmetry. In particular, we use the spectra of the microwave networks to study
the periodic orbits of the simulated quantum graphs. We also present
experimental study of directional microwave networks consisting of coaxial
cables and Faraday isolators for which the time reversal symmetry is broken. In
this case our experimental results indicate that spectral statistics of
directional microwave networks deviate from predictions of Gaussian orthogonal
ensembles (GOE) in random matrix theory approaching, especially for small
eigenfrequency spacing s, results for Gaussian unitary ensembles (GUE).
Experimental results are supported by the theoretical analysis of directional
graphs.Comment: 16 pages, 7 figures, to be published in Phys. Rev.