Microwave Experiments on Graphs Simulating Spin-1/2 System
In this work I study the statistical properties of the Gaussian symplectic ensemble (GSE) by means of microwave experiments on quantum graphs mimicking spin-1/2 systems. Additionally, the transport property of three terminal microwave graphs with orthogonal, unitary and symplectic symmetry is inv...
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|Zusammenfassung:||In this work I study the statistical properties of the Gaussian symplectic ensemble (GSE) by means of microwave experiments on quantum graphs mimicking spin-1/2 systems. Additionally, the transport property of three terminal microwave graphs with orthogonal, unitary and symplectic symmetry is investigated. In the first part of this thesis, following the spirit of the idea proposed by Joyner et al. we construct microwave quantum graphs to realize a antiunitary symmetry T that squares to minus one, T^2 = -1. This symmetry induces degenerate eigenvalues, which are called Kramers doublets. If the classical dynamics of the system is chaotic, statistical features of the spectrum can be well described by the corresponding statistics of random matrix Gaussian symplectic ensemble. Indeed, Kramers doublets are observed in reflection spectrum as expected from the scattering properties of a symplectic graph. The level spacing distribution of these doublets is compared with the corresponding random matrix predictions. Since the level spacing distribution accounts for the short range eigenvalue correlation, to study the spectral long range correlation the spectral two point correlation function and its Fourier transform, the spectral form factor are analyzed. In order to further examine the fluctuation of the eigenvalues smoothed quantities such as number variance and spectral rigidity are discussed. The graphs used in the experiment consist of two subgraphs coupled via one pair of connecting bonds. Theoretical study shows that the level spacing distribution for graphs with one pair of connecting bonds deviates by few percents from the random matrix prediction. This difference is too small to be resolved in the experiment. The one pair of bonds approximation is introduced to better understand the symplectic graph we used in the experiment. This model is extended to address more general cases of the symplectic graph. Finally, the parameter dependent dynamical transition of the statistical features of the spectrum from GSE via Gaussian unitary ensemble (GUE) to Gaussian orthogonal ensemble (GOE) is studied. In the second part of this thesis, the collaborative work with Dr. A. M. Martínez- Argüello from Mexico is briefly presented. A three terminal setup is proposed to study the universal transport properties of systems with orthogonal, unitary and symplectic symmetry. The probability distribution for a transport related quantity is predicted analytically and microwave graphs are constructed to test this prediction. The absorption within the system is modeled by effective Hamiltonian approach. The parameters of the absorption and coupling are extracted from the experimental autocorrelation function. This allowed a comparison between experiment and theory without any free parameters. Finally, a quantitative good agreement between experiment and theory was found for all three symmetry classes.|