Design of 2D waveguide networks for the study of fundamental properties of Quantum Graphs

Transcription

1 Design of 2D waveguide networks for the study of fundamental properties of Quantum Graphs Introduction: what is a quantum graph? Areas of application of quantum graphs Motivation of our experiment Experimental realization of graphs with waveguides Simulation of graphs with CST MW Studio CST vs. quantum mechanical calculation Summary and outlook Supported by DFG within SFB 634 B. Dietz, T. Klaus, M. Miski-Oglu, T. Skipa, A. Richter, M. Wunderle

2 What is a Quantum Graph Variety of natural and social network structures and mathematical objects Human brain as a neural network Part of the Internet space source: psychiatry.cam.ac.uk Graphene and nanotube source: teachengineering.org CST-EUC 2015 Institute of Nuclear Physics SFB 634 T. Skipa 2

3 Areas of application of Quantum Graphs Material science: 1) Quantum wire circuits (nanotechnology and quantum computers) 2) Photonic crystals (wave propagation in periodic structures) 3) Thin waveguides (optical, acoustics, electromagnetic) Biology, human sciences, computing: 1) Biological networks (blood vessels, lungs, brain) 2) Social networks 3) Networks of data Quantum chemistry and physics: 1) π- electron orbitals in organic molecules 2) Qantum transport in multiply connected systems 3) Anderson localization 4) Quantum chaos: our motivation CST-EUC 2015 Institute of Nuclear Physics SFB 634 T. Skipa 3

4 Motivation: Quantum Graph as a chaotic system Vertex L ij Bond Tetrahedral Graph : network of bonds connecting vertices i,j = 1, 2, 3, 4 Free propagation of wave ψ ij (z) in a bond from vertex j to vertex i Scattering system is realized by attaching leads Incommensurable bond lengths lead to chaotic dynamics CST-EUC 2015 Institute of of Nuclear Physics SFB SFB 634 T. Skipa Tetyana 4 Skipa

5 Chaotic scattering on Quantum Graphs Lead 1 Lead 2 L ij Lead 3 To obtain an open scattering system leads are attached to vertices The S-matrix element S 21 describes scattering from lead 1 to lead 2 quantum-mechanical calculation Dimension of the S-matrix depends on the number of leads S-matrix depends on the bond lengths L ij and the individual scattering matrices at the vertices 1,2, 3, 4 CST-EUC 2015 Institute of Nuclear Physics SFB 634 T. Skipa 5

6 Experimental realization of Quantum Graphs ψij (z)=aijeikz + cji e ikz Bond of graph : - coaxial cable with propagating microwaves E ρ, z = f(ρ)e±ikz z π - rectangular waveguide, E x, z ~ sin( a x)e±ikz z Vertex of graph: - junction of coaxial cables - junction of waveguides CST-EUC 2015 Institute of Nuclear Physics SFB 634 T. Skipa 6

7 Realization of Quantum Graphs with coaxial cables Realization of tetrahedral graph with coaxial cables Drawback: - absorption losses in the walls and in the dielectric filling Idea: - realization of quantum graphs with superconducting waveguides (negligible ohmic losses within the walls of waveguides) CST-EUC 2015 Institute of Nuclear Physics SFB 634 T. Skipa 7

8 Realization of Quantum Graphs as 2D microwave WG networks Output Antenna Input Lead Mill the waveguides and junctions out of a brass plate Cover both top and base plate with lead to reduce the Ohmic losses Strong coupling to the environment via waveguide-to-coaxial adapters Weakly coupled field probe: thin wire antenna inserted into a hole in the top plate For frequencies below f max = ck 2π (28 GHz) the electric field vector is perpendicular to the top and bottom plate: Helmholtz equation is scalar and coincides with the Schrödinger eq. of quantum graph CST-EUC 2015 Institute of Nuclear Physics SFB 634 T. Skipa 8

9 Design and spectral properties of Quantum Graphs: CST Microwave Studio Graphs with undefined vertices 10-vertex Graph Looking for optimal design of the Graph Chaotic properties: variation of bond lengths (they must be all incommensurable) Eigenmode solver yields about 300 Eigenfrequencies in the experimental frequency range of GHz, tetrahedral mesh, ( tets) CST-EUC 2015 Institute of Nuclear Physics SFB 634 T. Skipa 9

10 Simulation of scattering properties: CST Microwave Studio Wire antenna: WG Ports: 1, 2, Frequency Domain Solver yields the transmission spectra S 21 2, S 32 2, S 42 2 Next: experimental realization of superconducting microwave graphs CST-EUC 2015 Institute of Nuclear Physics SFB 634 T. Skipa 10

11 Experimental realization of 2D Microwave Graphs 9 2 Input Lead Waveguides and junctions (of certified commercial dimensions) were milled out of a brass plate Open Graph: strong coupling to the environment via waveguide-to-coaxial adapters at input leads Closed Graph: close input leads with blind flanges and use weakly coupled field probe introduce thin wire antennas through drillings in the top plate CST-EUC 2015 Institute of Nuclear Physics SFB 634 T. Skipa 11

12 Superconducting Microwave Graphs: characteristics Top and base plate were covered with lead to achieve superconductivity Lead becomes superconducting at Tc= 7.2K Dimensions of the waveguide channels: height b = 4.318mm and width a = mm Cutoff frequency of the first propagating mode: 14 GHz, 𝑓𝑚𝑎𝑥 = 28 GHz 9 antennae are attached to the graph Single-mode propagation in a frequency range from 14 to 28 GHz Next: test of chaoticity determine resonance eigenfrequencies and study spectral properties of the closed graph CST-EUC 2015 Institute of Nuclear Physics SFB 634 T. Skipa 12

13 2.39 m 1.78 m Transmission Spectrum of Closed Graph 4 K 300 K Measured S-matrix: S 21 2 = P out,2 P in,1 Quality factor: Q 4K 10 5 Complete spectrum of 300 isolated resonances (eigenvalues) CST-EUC 2015 Institute of Nuclear Physics SFB 634 T. Skipa 13

14 Spectral properties of the Quantum Graphs: CST vs. quantum-mechanical calculations Quantum graph Experiment CST 300 resonance frequencies determined between 14 and 28 GHz and eigenfrequences calculated by CST Nearest-neighbor distribution of spacing P(s) between neighboring eigenfrequencies after rescaling to average spacing 1 Similar statistics for quantum (mathematical) graph, superconducting (experimental) graph and CST- simulated graph CST-EUC 2015 Institute of Nuclear Physics SFB 634 T. Skipa 14

15 Design and properties of complicated Graphs: CST Microwave Studio Step 1. Building up a completely flexible geometric model with 120 o bendings Step 2. Variation of bond lengths, looking for optimal chaotic design Step 3. Virtual scattering experiment, calculation of n-port S-matrix CST-EUC 2015 Institute of Nuclear Physics SFB 634 T. Skipa 15

16 Experimental realization of more complex Graphs Port 1 Port 2 Waveguide-to-coax adaptor Port 3 Next: Construction and measurements with more complex Quantum Graphs CST-EUC 2015 Institute of Nuclear Physics SFB 634 T. Skipa 16

17 Summary and Conclusions Benefits of using CST simulation for quantum graphs : Most appropriate possibility to design complicated microwave networks Finding optimal design (flexible parametric model) Testing of spectral and scattering properties Possibility to realize virtual scattering experiments prior construction17 (low cost) Result coincides with quantum-mechanical calculations (simple geometry of graphs) Any possible design can be realized CST-EUC 2015 Institute of Nuclear Physics SFB 634 T. Skipa 17

18 Thank you for your attention! CST-EUC 2015 Institute of Nuclear Physics SFB 634 T. Skipa 18