Understanding the connection of the atomic structure and the physical properties of materials remains one of the elementary questions of condensed-matter physics. One research line in this quest started with the discovery of quasicrystals by Shechtman et al. in 1982. It soon became clear that these materials with their 5-, 8-, 10- or 12-fold rotational symmetries, which are forbidden according to classical crystallography, can be described in terms of mathematical models for nonperiodic tilings of a plane proposed by Penrose and Ammann in the 1970s. Due to the missing translational symmetry of quasicrystals, till today only finite, relatively small systems or periodic approximants have been investigated by means of numerical calculations and theoretical results have mainly been obtained for one-dimensional systems. In this thesis we study d-dimensional quasiperiodic models, so-called labyrinth tilings, with separable Hamiltonians in the tight-binding approach. This method paves the way to study higher-dimensional, quantum mechanical solutions, which can be directly derived from the one-dimensional results. This allows the investigation of very large systems in two and three dimensions with up to 10^10 sites. In particular, we contemplate the class of metallic-mean sequences. Based on this model we focus on the electronic properties of quasicrystals with a special interest on the connection of the spectral and dynamical properties of the Hamiltonian. Hence, we investigate the characteristics of the eigenstates and wave functions and compare these with the wave-packet dynamics in the labyrinth tilings by numerical calculations and by a renormalization group approach in connection with perturbation theory. It turns out that many properties show a qualitatively similar behavior in different dimensions or are even independent of the dimension as e.g. the scaling behavior of the participation numbers and the mean square displacement of a wave packet. Further, we show that the structure of the labyrinth tilings and their transport properties are connected and obtain that certain moments of the spectral dimensions are related to the wave-packet dynamics. Besides this also the photonic properties are studied for one-dimensional quasiperiodic multilayer systems for oblique incidence of light, and we show that the characteristics of the transmission bands are related to the quasiperiodic structure. / Eine der elementaren Fragen der Physik kondensierter Materie beschäftigt sich mit dem Zusammenhang zwischen der atomaren Struktur und den physikalischen Eigenschaften von Materialien. Eine Forschungslinie in diesem Kontext begann mit der Entdeckung der Quasikristalle durch Shechtman et al. 1982. Es stellte sich bald heraus, dass diese Materialien mit ihren laut der klassischen Kristallographie verbotenen 5-, 8-, 10- oder 12-zähligen Rotationssymmetrien durch mathematische Modelle für die aperiodische Pflasterung der Ebene beschrieben werden können, die durch Penrose und Ammann in den 1970er Jahren vorgeschlagen wurden. Aufgrund der fehlenden Translationssymmetrie in Quasikristallen sind bis heute nur endliche, relativ kleine Systeme oder periodische Approximanten durch numerische Berechnungen untersucht worden und theoretische Ergebnisse wurden hauptsächlich für eindimensionale Systeme gewonnen. In dieser Arbeit werden d-dimensionale quasiperiodische Modelle, sogenannte Labyrinth-Pflasterungen, mit separablem Hamilton-Operator im Modell starker Bindung betrachtet. Diese Methode erlaubt es, quantenmechanische Lösungen in höheren Dimensionen direkt aus den eindimensionalen Ergebnissen abzuleiten und ermöglicht somit die Untersuchung von sehr großen Systemen in zwei und drei Dimensionen mit bis zu 10^10 Gitterpunkten. Insbesondere betrachten wir dabei quasiperiodische Folgen mit metallischem Schnitt. Basierend auf diesem Modell befassen wir uns im Speziellen mit den elektronischen Eigenschaften der Quasikristalle im Hinblick auf die Verbindung der spektralen und dynamischen Eigenschaften des Hamilton-Operators. Hierfür untersuchen wir die Eigenschaften der Eigenzustände und Wellenfunktionen und vergleichen diese mit der Dynamik von Wellenpaketen in den Labyrinth-Pflasterungen basierend auf numerischen Berechnungen und einem Renormierungsgruppen-Ansatz in Verbindung mit Störungstheorie. Dabei stellt sich heraus, dass viele Eigenschaften wie etwa das Skalenverhalten der Partizipationszahlen und der mittleren quadratischen Abweichung eines Wellenpakets für verschiedene Dimensionen ein qualitativ gleiches Verhalten zeigen oder sogar unabhängig von der Dimension sind. Zudem zeigen wir, dass die Struktur der Labyrinth-Pflasterungen und deren Transporteigenschaften sowie bestimmte Momente der spektralen Dimensionen und die Dynamik der Wellenpakete in Beziehung zueinander stehen. Darüber hinaus werden auch die photonischen Eigenschaften für eindimensionale quasiperiodische Mehrschichtsysteme für beliebige Einfallswinkel untersucht und der Verlauf der Transmissionsbänder mit der quasiperiodischen Struktur in Zusammenhang gebracht.

Geophysical methods are essential for exploration and monitoring of subsurface formations, e.g. in carbon dioxide sequestration or enhanced geothermal energy. One of the keys to their successful application is the knowledge of how the measured physical quantities are related to the desired reservoir parameters. The work presented in this thesis shows that the presence of carbon dioxide (CO2) in pore space gives rise to multiple processes all of which contribute to the electrical rock conductivity variation. Basically, three mechanisms take place: (1) CO2 partially replaces the pore water, which is equivalent to a decrease in water saturation. (2) CO2 chemically interacts with the pore water by dissolution and dissociation. These processes change both the chemical composition and the pH of the pore filling fluid. (3) The low-pH environment can give rise to mineral dissolution and/or precipitation processes and changes the properties of the grain-water interface. Investigations on the pore water phase show that the reactive nature of CO2 in all physical states significantly acts on the electrical conductivity of saline pore waters. The physico-chemical interaction appears in different manifestations depending mainly on the pore water composition (salinity, ion types) but also on both temperature and pressure. The complex behaviour includes a low- and a high-salinity regime originating from the conductivity increasing effect of CO2 dissociation, which is opposed by the conductivity decreasing effect of reduced ion activity caused by the enhanced mutual impediment of all solutes. These results are fundamental since the properties of the water phase significantly act on all conduction mechanisms in porous media. In order to predict the variation of pore water conductivity, both a semi-analytical formulation and an empirical relationship for correcting the pore water conductivity, which depends on salinity, pressure and temperature, are derived. The central part of the laboratory experiments covers the spectral complex conductivity of water-bearing sand during exposure to and flow-through by CO2 at pressures up to 30MPa and temperatures up to 80°C. It is shown that the impact of CO2 on the real part of conductivity of a clean quartz sand is dominated by the low- and high-salinity regime of the pore water. The obtained data further show that chemical interaction causes a reduction of interface conductivity, which could be related to the low pH in the acidic environment. This effect is described by a correction term, which is a constant value as a first approximation. When the impact of CO2 is taken into account, a correct reconstruction of fluid saturation from electrical measurements is possible. In addition, changes of the inner surface area, which are related to mineral dissolution or precipitation processes, can be quantified. Both the knowledge gained from the laboratory experiments and a new workflow for the description and incorporation of geological geometry models enable realistic finite element simulations. Those were conducted for three different electromagnetic methods applied in the geological scenario of a fictitious carbon dioxide sequestration site. The results show that electromagnetic methods can play an important role in monitoring CO2 sequestration. Compared to other geophysical methods, electromagnetic techniques are generally very sensitive to pore fluids. The proper configuration of sources and receivers for a suitable electromagnetic method that generates the appropriate current systems is essential. Its reactive nature causes CO2 to interact with a water-bearing porous rock in a much more complex manner than non-reactive gases. Without knowledge of the specific interactions between CO2 and rock, a determination of saturation and, consequently, a successful monitoring are possible only to a limited extend. The presented work provides fundamental laboratory investigations for the understanding of the electrical properties of rocks when the reactive gas CO2 enters the rock-water system. All laboratory results are put in the context of potential monitoring applications. The transfer from petrophysical investigations to the planning of an operational monitoring design by means of close-to-reality 3D FE simulations is accomplished.

Heterogeneous line broadening and spectral diffusion of the fluorescence emission spectra of perylene diimide molecules have been investigated by means of time dependent single molecule spectroscopy. The influence of temperature and environment has been studied and reveals strong correlation to spectral diffusion processes. We followed the freezing of the molecular mobility of quasi free molecules on the surface upon temperature lowering and by embedding into a poly(methyl methacrylate) (PMMA) polymer. Thereby changes of optical transition energies as a result of both intramolecular changes of conformation and external induced dynamics by the surrounding polymer matrix could be observed. Simulations of spectral fluctuations within a two-level system (TLS) model showed good agreement with the experimental findings.

This thesis is devoted to the study of operators on discrete structures. The operators are supposed to be self-adjoint and obey a certain translation invariance property. The discrete structures are given as Cayley graphs via finitely generated groups. Here, sofic groups and amenable groups are in the center of our considerations. Note that every finitely generated amenable group is sofic. We investigate the spectrum of a discrete self-adjoint operator by studying a sequence of finite dimensional analogues of these operators. In the setting of amenable groups we obtain these approximating operators by restricting the operator in question to finite subsets Qn , n ∈ N. These finite dimensional operators are self-adjoint and therefore admit a well-defined normalized eigenvalue counting function. The limit of the normalized eigenvalue counting functions when |Qn | → ∞ (if it exists) is called the integrated density of states (IDS). It is a distribution function of a probability measure encoding the distribution of the spectrum of the operator in question on the real axis. In this thesis, we prove the existence of the IDS in various geometric settings and for different types of operators. The models we consider include deterministic as well as random situations. Depending on the specific setting, we prove existence of the IDS as a weak limit of distribution functions or even as a uniform limit. Moreover, in certain situations we are able to express the IDS via a semi-explicit formula using the trace of the spectral projection of the original operator. This is sometimes referred to as the validity of the Pastur-Shubin trace formula. In the most general geometric setting we study, the operators are defined on Cayley graphs of sofic groups. Here we prove weak convergence of the eigenvalue counting functions and verify the validity of the Pastur-Shubin trace formula for random and non-random operators . These results apply to operators which not necessarily bounded or of finite hopping range. The methods are based on resolvent techniques. This theory is established without having an ergodic theorem for sofic groups at hand. Note that ergodic theory is the usual tool used in the proof of convergence results of this type. Specifying to operators on amenable groups we are able to prove stronger results. In the discrete case, we show that the IDS exists uniformly for a certain class of finite hopping range operators. This is obtained by using a Banach space-valued ergodic theorem. We show that this applies to eigenvalue counting functions, which implies their convergence with respect to the Banach space norm, in this case the supremum norm. Thus, the heart of this theory is the verification of the Banach space-valued ergodic theorem. Proceeding in two steps we first prove this result for so-called ST-amenable groups. Then, using results from the theory of ε-quasi tilings, we prove a version of the Banach space-valued ergodic theorem which is valid for all amenable groups. Focusing on random operators on amenable groups, we prove uniform existence of the IDS without the assumption that the operator needs to be of finite hopping range or bounded. Moreover, we verify the Pastur-Shubin trace formula. Here we present different techniques. First we show uniform convergence of the normalized eigenvalue counting functions adapting the technique of the Banach space-valued ergodic theorem from the deterministic setting. In a second approach we use weak convergence of the eigenvalue counting functions and additionally obtain control over the convergence at the jumps of the IDS. These ingredients are applied to verify uniform existence of the IDS. In both situations we employ results from the theory of large deviations, in order to deal with long-range interactions.

Im Rahmen der vorliegenden Dissertation wurden neuartige integrierte Einzelphotonenquellen (EPQ) und ihre Anwendung für die Quanteninformationsverarbeitung entwickelt und untersucht. Die Erzeugung von Einzelphotonen basiert auf einzelnen Defektzentren in nanometergroßen Diamantkristallen mit einzigartigen optischen Eigenschaften: Stabilität bei Zimmertemperatur ohne optisches Blinken. Diamantkristalle mit Größen bis unter 20nm wurden mit neuartigen „pick-and-place“ Techniken (z.B. mit einem Atomkraftmikroskop) in komplexe photonische Strukturen integriert. Zwei unterschiedliche Ansätze für die Realisierung der neuartigen EPQ wurden verfolgt. Beim ersten werden fluoreszierende Diamantkristalle in nano- und mikrometergroße Faser-basierte oder resonante Strukturen in einem „bottom-up“ Ansatz integriert, dadurch werden zusätzliche optische Komponenten überflüssig und das Gesamtsystem ultra-stabil und wartungsfrei. Der zweite Ansatz beruht auf einem Festkörperimmersionsmikroskop (FIM). Seine Festkörperimmersionslinse wirkt wie eine dielektrische Antenne für die Emission der Defektzentren. Es ermöglicht die höchsten bisher erreichten Photonenzählraten von Stickstoff-Fehlstellen von bis zu 2.4Mcts/s und Einsammeleffizienzen von bis zu 4.2%. Durch Anwendung des FIM bei cryogenen Temperaturen wurden neuartige Anwendungen und fundamentale Untersuchungen möglich, weil Photonenraten signifikant erhöht wurden. Die Bestimmung der spektralen Diffusionszeit eines einzelnen Defektzentrums (2.2µs) gab neue Erkenntnisse über die Ursachen von spektraler Diffusion. Spektrale Diffusion ist eine limitierende Eigenschaft für die Realisierung von Quanteninformationsanwendungen. Das Tisch-basierte FIM wurde außerdem als kompakte mobile EPQ mit Ausmaßen von nur 7x19x23cm^3 realisiert. Es wurde für ein Quantenkryptographie-Experiment implementiert, zum ersten Mal mit Siliziumdefektzentren. Des Weiteren wurde ein neues Konzept für die Erzeugung von infraroten EPQ entwickelt und realisiert. / The presented thesis covers the development and investigation of novel integrated single photon (SP) sources and their application for quantum information schemes. SP generation was based on single defect centers in diamond nanocrystals. Such defect centers offer unique optical properties as they are room temperature stable, non-blinking, and do not photo-bleach over time. The fluorescent nanocrystals are mechanically stable, their size down to 20nm enabled the development of novel nano-manipulation pick-and-place techniques, e.g., with an atomic force microscope, for integration into photonic structures. Two different approaches were pursued to realize novel SP sources. First, fluorescent diamond nanocrystals were integrated into nano- and micrometer scaled fiber devices and resonators, making them ultra-stable and maintenance free. Secondly, a solid immersion microscope (SIM) was developed. Its solid immersion lens acts as a dielectric antenna for the emission of defect centers, enabling the highest photon rates of up to 2.4Mcts/s and collection efficiencies of up to 4.2% from nitrogen vacancy defect centers achieved to date. Implementation of the SIM at cryogenic temperatures enabled novel applications and fundamental investigations due to increased photon rates. The determination of the spectral diffusion time of a single nitrogen vacancy defect center (2.2µs) gave new insights about the mechanisms causing spectral diffusion. Spectral diffusion is a limiting property for quantum information applications. The table-top SIM was integrated into a compact mobile SP system with dimension of only 7x19x23cm^3 while still maintaining record-high stable SP rates. This makes it interesting for various SP applications. First, a quantum key distribution scheme based on the BB84 protocol was implemented, for the first time also with silicon vacancy defect centers. Secondly, a conceptually novel scheme for the generation of infrared SPs was introduced and realized.

A new method, probability distribution of diffusivities (time scaled square displacements between succeeding video frames), was developed to analyze single molecule tracking (SMT) experiments. This method was then applied to SMT experiments on ultrathin liquid tetrakis(2-ethylhexoxy)silane (TEHOS) films on Si wafer with 100 nm thermally grown oxide, and on thin semectic liquid crystal films. Spatial maps of diffusivities from SMT experiments on 220 nm thick semectic liquid crystal films reveal structure related dynamics. The SMT experiments on ultrathin TEHOS films were complemented by fluorescence correlation spectroscopy (FCS). The observed strongly heterogeneous single molecule dynamics within those films can be explained by a three-layer model consisting of (i) dye molecules adsorbed to the substrate, (ii) slowly diffusing molecules in the laterally heterogeneous near-surface region of 1 - 2 molecular diameters, and (iii) freely diffusing dye molecules in the upper region of the film. FCS and SMT experiments reveal a strong influence of substrate heterogeneity on SM dynamics. Thereby chemisorption to substrate surface silanols plays an important role. Vertical mean first passage times (mfpt) in those films are below 1 µs. This appears as fast component in FCS autocorrelation curves, which further contain a contribution from lateral diffusion and from adsorption events. Therefore, the FCS curves are approximated by a tri-component function, which contains an exponential term related to the mfpt, the correlation function for translational diffusion and a stretched exponential term for the broad distribution of adsorption events. Lateral diffusion coefficients obtained by FCS on 10 nm thick TEHOS films, thereby, are effective diffusion coefficients from dye transients in the focal area. They strongly depend on the substrate heterogeneity. Variation of the frame times for the acquisition of SMT experiments in steps of 20 ms from 20 ms to 200 ms revealed a strong dependence of the corresponding probability distributions of diffusivities on time, in particular in the range between 20 ms and 100 ms. This points to average dwell times of the dye molecules in at least one type of the heterogeneous regions (e.g. on and above silanol clusters) in the range of few tens of milliseconds. Furthermore, time series of SM spectra from Nile Red in 25 nm thick poly-n-alkyl-methacrylate (PnAMA) films were studied. In analogy to translational diffusion, spectral diffusion (shifts in energetic positions of SM spectra) can be studied by probability distributions of spectral diffusivities, i.e. time scaled square energetic displacements. Simulations were run and analyzed to study contributions from noise and fitting uncertainty to spectral diffusion. Furthermore the effect of spectral jumps during acquisition of a SM spectrum was investigated. Probability distributions of spectral diffusivites of Nile Red probing vitreous PnAMA films reveal a two-level system. In contrast, such probability distributions obtained from Nile Red within a 25 nm thick poly-n-butylmethacrylate film around glass transition and in the melt state, display larger spectral jumps. Moreover, for longer alkyl side chains a solvent shift to higher energies is observed, which supports the idea of nanophase separation within those polymers.

A new method, probability distribution of diffusivities (time scaled square displacements between succeeding video frames), was developed to analyze single molecule tracking (SMT) experiments. This method was then applied to SMT experiments on ultrathin liquid tetrakis(2-ethylhexoxy)silane (TEHOS) films on Si wafer with 100 nm thermally grown oxide, and on thin semectic liquid crystal films. Spatial maps of diffusivities from SMT experiments on 220 nm thick semectic liquid crystal films reveal structure related dynamics. The SMT experiments on ultrathin TEHOS films were complemented by fluorescence correlation spectroscopy (FCS). The observed strongly heterogeneous single molecule dynamics within those films can be explained by a three-layer model consisting of (i) dye molecules adsorbed to the substrate, (ii) slowly diffusing molecules in the laterally heterogeneous near-surface region of 1 - 2 molecular diameters, and (iii) freely diffusing dye molecules in the upper region of the film. FCS and SMT experiments reveal a strong influence of substrate heterogeneity on SM dynamics. Thereby chemisorption to substrate surface silanols plays an important role. Vertical mean first passage times (mfpt) in those films are below 1 µs. This appears as fast component in FCS autocorrelation curves, which further contain a contribution from lateral diffusion and from adsorption events. Therefore, the FCS curves are approximated by a tri-component function, which contains an exponential term related to the mfpt, the correlation function for translational diffusion and a stretched exponential term for the broad distribution of adsorption events. Lateral diffusion coefficients obtained by FCS on 10 nm thick TEHOS films, thereby, are effective diffusion coefficients from dye transients in the focal area. They strongly depend on the substrate heterogeneity. Variation of the frame times for the acquisition of SMT experiments in steps of 20 ms from 20 ms to 200 ms revealed a strong dependence of the corresponding probability distributions of diffusivities on time, in particular in the range between 20 ms and 100 ms. This points to average dwell times of the dye molecules in at least one type of the heterogeneous regions (e.g. on and above silanol clusters) in the range of few tens of milliseconds. Furthermore, time series of SM spectra from Nile Red in 25 nm thick poly-n-alkyl-methacrylate (PnAMA) films were studied. In analogy to translational diffusion, spectral diffusion (shifts in energetic positions of SM spectra) can be studied by probability distributions of spectral diffusivities, i.e. time scaled square energetic displacements. Simulations were run and analyzed to study contributions from noise and fitting uncertainty to spectral diffusion. Furthermore the effect of spectral jumps during acquisition of a SM spectrum was investigated. Probability distributions of spectral diffusivites of Nile Red probing vitreous PnAMA films reveal a two-level system. In contrast, such probability distributions obtained from Nile Red within a 25 nm thick poly-n-butylmethacrylate film around glass transition and in the melt state, display larger spectral jumps. Moreover, for longer alkyl side chains a solvent shift to higher energies is observed, which supports the idea of nanophase separation within those polymers.