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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Format: | Dissertation |
Sprog: | engelsk |
Udgivet: |
Philipps-Universität Marburg
2018
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Summary: | 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. |
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Fysisk beskrivelse: | 91 Seiten |
DOI: | 10.17192/z2018.0247 |